%------------------------------------------------------------------------------
% File     : ITP222_3 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_Definitions 00680_031305
% Version  : [Des22] axioms.
% English  :

% Refs     : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
%          : [Des22] Desharnais (2022), Email to Geoff Sutcliffe
% Source   : [Des22]
% Names    : 0063_VEBT_Definitions_00680_031305 [Des22]

% Status   : Theorem
% Rating   : 0.00 v8.1.0
% Syntax   : Number of formulae    : 12910 (4672 unt;2117 typ;   0 def)
%            Number of atoms       : 23714 (9032 equ)
%            Maximal formula atoms :   73 (   2 avg)
%            Number of connectives : 14234 (1989   ~; 503   |;1046   &)
%                                         (2838 <=>;7858  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   35 (   4 avg)
%            Maximal term depth    :   23 (   2 avg)
%            Number of FOOLs       : 1023 ( 514 fml; 509 var)
%            Number of X terms     :  598 (   0  []; 515 ite;  83 let)
%            Number of types       :  386 ( 385 usr)
%            Number of type conns  : 1904 (1276   >; 628   *;   0   +;   0  <<)
%            Number of predicates  :  127 ( 124 usr;   2 prp; 0-4 aty)
%            Number of functors    : 1620 (1620 usr; 468 con; 0-7 aty)
%            Number of variables   : 23778 (23202   !; 576   ?;23778   :)
% SPC      : TX0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            from the van Emde Boas Trees session in the Archive of Formal
%            proofs - 
%            www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
%            2022-02-17 17:57:01.541
%------------------------------------------------------------------------------
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tff(ty_n_t__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J,type,
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tff(ty_n_t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
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tff(ty_n_t__List__Olist_It__Code____Numeral__Ointeger_J,type,
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tff(ty_n_t__fun_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
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tff(ty_n_t__fun_It__List__Olist_It__Int__Oint_J_M_Eo_J,type,
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tff(ty_n_t__fun_I_Eo_Mt__Set__Oset_It__Real__Oreal_J_J,type,
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tff(ty_n_t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
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tff(ty_n_t__fun_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_Mt__Complex__Ocomplex_J,type,
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tff(ty_n_t__fun_It__Int__Oint_Mt__Complex__Ocomplex_J,type,
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tff(ty_n_t__fun_It__Complex__Ocomplex_Mt__Nat__Onat_J,type,
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tff(ty_n_t__fun_It__Complex__Ocomplex_Mt__Int__Oint_J,type,
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tff(ty_n_t__fun_I_Eo_Mt__VEBT____Definitions__OVEBT_J,type,
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tff(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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tff(ty_n_t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
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tff(ty_n_t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
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tff(ty_n_t__fun_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
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tff(ty_n_t__fun_It__Set__Oset_It__Int__Oint_J_M_Eo_J,type,
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tff(ty_n_t__fun_It__Code____Numeral__Ointeger_M_Eo_J,type,
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tff(ty_n_t__fun_I_Eo_Mt__Set__Oset_It__Nat__Onat_J_J,type,
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tff(ty_n_t__fun_I_Eo_Mt__Set__Oset_It__Int__Oint_J_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_Mt__fun_I_Eo_M_Eo_J_J,type,
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tff(ty_n_t__Set__Oset_It__Product____Type__Ounit_J,type,
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tff(ty_n_t__Set__Oset_It__Extended____Nat__Oenat_J,type,
    set_Extended_enat: $tType ).

tff(ty_n_t__fun_It__String__Ochar_Mt__Nat__Onat_J,type,
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tff(ty_n_t__fun_It__Real__Oreal_Mt__Real__Oreal_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_Mt__String__Ochar_J,type,
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tff(ty_n_t__fun_It__Extended____Nat__Oenat_M_Eo_J,type,
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tff(ty_n_t__fun_It__Real__Oreal_Mt__Rat__Orat_J,type,
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tff(ty_n_t__fun_It__Real__Oreal_Mt__Num__Onum_J,type,
    fun_real_num: $tType ).

tff(ty_n_t__fun_It__Real__Oreal_Mt__Nat__Onat_J,type,
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tff(ty_n_t__fun_It__Real__Oreal_Mt__Int__Oint_J,type,
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tff(ty_n_t__fun_It__Rat__Orat_Mt__Real__Oreal_J,type,
    fun_rat_real: $tType ).

tff(ty_n_t__fun_It__Num__Onum_Mt__Real__Oreal_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_Mt__Real__Oreal_J,type,
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tff(ty_n_t__fun_It__Int__Oint_Mt__Real__Oreal_J,type,
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tff(ty_n_t__List__Olist_It__Complex__Ocomplex_J,type,
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tff(ty_n_t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
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tff(ty_n_t__fun_It__Rat__Orat_Mt__Rat__Orat_J,type,
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tff(ty_n_t__fun_It__Rat__Orat_Mt__Num__Onum_J,type,
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tff(ty_n_t__fun_It__Rat__Orat_Mt__Nat__Onat_J,type,
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tff(ty_n_t__fun_It__Rat__Orat_Mt__Int__Oint_J,type,
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tff(ty_n_t__fun_It__Num__Onum_Mt__Rat__Orat_J,type,
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tff(ty_n_t__fun_It__Num__Onum_Mt__Num__Onum_J,type,
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tff(ty_n_t__fun_It__Num__Onum_Mt__Nat__Onat_J,type,
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tff(ty_n_t__fun_It__Num__Onum_Mt__Int__Oint_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_Mt__Rat__Orat_J,type,
    fun_nat_rat: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__Num__Onum_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_Mt__Nat__Onat_J,type,
    fun_nat_nat: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__Int__Oint_J,type,
    fun_nat_int: $tType ).

tff(ty_n_t__fun_It__List__Olist_I_Eo_J_M_Eo_J,type,
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tff(ty_n_t__fun_It__Int__Oint_Mt__Rat__Orat_J,type,
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tff(ty_n_t__fun_It__Int__Oint_Mt__Num__Onum_J,type,
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tff(ty_n_t__fun_It__Int__Oint_Mt__Nat__Onat_J,type,
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tff(ty_n_t__fun_It__Int__Oint_Mt__Int__Oint_J,type,
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tff(ty_n_t__Set__Oset_It__Complex__Ocomplex_J,type,
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tff(ty_n_t__Filter__Ofilter_It__Real__Oreal_J,type,
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tff(ty_n_t__fun_It__Complex__Ocomplex_M_Eo_J,type,
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tff(ty_n_t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
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tff(ty_n_t__fun_It__Set__Oset_I_Eo_J_M_Eo_J,type,
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tff(ty_n_t__fun_I_Eo_Mt__Set__Oset_I_Eo_J_J,type,
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tff(ty_n_t__Option__Ooption_It__Num__Onum_J,type,
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tff(ty_n_t__Filter__Ofilter_It__Nat__Onat_J,type,
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tff(ty_n_t__Set__Oset_It__String__Ochar_J,type,
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tff(ty_n_t__List__Olist_It__Real__Oreal_J,type,
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tff(ty_n_t__fun_I_Eo_Mt__String__Ochar_J,type,
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tff(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
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tff(ty_n_t__List__Olist_It__Num__Onum_J,type,
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tff(ty_n_t__List__Olist_It__Nat__Onat_J,type,
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tff(ty_n_t__List__Olist_It__Int__Oint_J,type,
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tff(ty_n_t__fun_It__Real__Oreal_M_Eo_J,type,
    fun_real_o: $tType ).

tff(ty_n_t__VEBT____Definitions__OVEBT,type,
    vEBT_VEBT: $tType ).

tff(ty_n_t__Set__Oset_It__Rat__Orat_J,type,
    set_rat: $tType ).

tff(ty_n_t__Set__Oset_It__Num__Onum_J,type,
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tff(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
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tff(ty_n_t__Set__Oset_It__Int__Oint_J,type,
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tff(ty_n_t__Code____Numeral__Ointeger,type,
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tff(ty_n_t__fun_It__Rat__Orat_M_Eo_J,type,
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tff(ty_n_t__fun_It__Num__Onum_M_Eo_J,type,
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tff(ty_n_t__fun_It__Nat__Onat_M_Eo_J,type,
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tff(ty_n_t__fun_It__Int__Oint_M_Eo_J,type,
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tff(ty_n_t__fun_I_Eo_Mt__Nat__Onat_J,type,
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tff(ty_n_t__fun_I_Eo_Mt__Int__Oint_J,type,
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tff(ty_n_t__Extended____Nat__Oenat,type,
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tff(ty_n_t__List__Olist_I_Eo_J,type,
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tff(ty_n_t__Complex__Ocomplex,type,
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tff(ty_n_t__Set__Oset_I_Eo_J,type,
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tff(ty_n_t__fun_I_Eo_M_Eo_J,type,
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tff(ty_n_t__String__Ochar,type,
    char: $tType ).

tff(ty_n_t__Real__Oreal,type,
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tff(ty_n_t__Rat__Orat,type,
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tff(ty_n_t__Num__Onum,type,
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tff(ty_n_t__Nat__Onat,type,
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tff(ty_n_t__Int__Oint,type,
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% Explicit typings (1732)
tff(sy_c_ATP_058Lamm__a____,type,
    aTP_Lamm_a: fun_VE3165209603783583043T_VEBT ).

tff(sy_c_ATP_058Lamm__aa____,type,
    aTP_Lamm_aa: nat > fun_VE3165209603783583043T_VEBT ).

tff(sy_c_ATP_058Lamm__ab____,type,
    aTP_Lamm_ab: set_real > fun_real_o ).

tff(sy_c_ATP_058Lamm__ac____,type,
    aTP_Lamm_ac: set_complex > fun_complex_o ).

tff(sy_c_ATP_058Lamm__ad____,type,
    aTP_Lamm_ad: set_list_nat > fun_list_nat_o ).

tff(sy_c_ATP_058Lamm__ae____,type,
    aTP_Lamm_ae: set_set_nat > fun_set_nat_o ).

tff(sy_c_ATP_058Lamm__af____,type,
    aTP_Lamm_af: set_nat > fun_nat_o ).

tff(sy_c_ATP_058Lamm__ag____,type,
    aTP_Lamm_ag: set_int > fun_int_o ).

tff(sy_c_ATP_058Lamm__ah____,type,
    aTP_Lamm_ah: fun_VE3165209603783583043T_VEBT ).

tff(sy_c_ATP_058Lamm__ai____,type,
    aTP_Lamm_ai: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aj____,type,
    aTP_Lamm_aj: extended_enat > fun_nat_o ).

tff(sy_c_ATP_058Lamm__ak____,type,
    aTP_Lamm_ak: nat > fun_na8797796813480665199d_enat ).

tff(sy_c_ATP_058Lamm__al____,type,
    aTP_Lamm_al: extended_enat > fun_na8797796813480665199d_enat ).

tff(sy_c_ATP_058Lamm__am____,type,
    aTP_Lamm_am: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__an____,type,
    aTP_Lamm_an: extended_enat > fun_nat_o ).

tff(sy_c_ATP_058Lamm__ao____,type,
    aTP_Lamm_ao: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__ap____,type,
    aTP_Lamm_ap: fun_real_real ).

tff(sy_c_ATP_058Lamm__aq____,type,
    aTP_Lamm_aq: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__ar____,type,
    aTP_Lamm_ar: fun_int_int ).

tff(sy_c_ATP_058Lamm__as____,type,
    aTP_Lamm_as: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__at____,type,
    aTP_Lamm_at: fun_real_real ).

tff(sy_c_ATP_058Lamm__au____,type,
    aTP_Lamm_au: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__av____,type,
    aTP_Lamm_av: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__aw____,type,
    aTP_Lamm_aw: fun_int_int ).

tff(sy_c_ATP_058Lamm__ax____,type,
    aTP_Lamm_ax: nat > fun_na8797796813480665199d_enat ).

tff(sy_c_ATP_058Lamm__ay____,type,
    aTP_Lamm_ay: extended_enat > fun_na8797796813480665199d_enat ).

tff(sy_c_ATP_058Lamm__az____,type,
    aTP_Lamm_az: set_real > fun_fu9050450510664235267real_o ).

tff(sy_c_ATP_058Lamm__ba____,type,
    aTP_Lamm_ba: set_complex > fun_fu1358370060594899587plex_o ).

tff(sy_c_ATP_058Lamm__bb____,type,
    aTP_Lamm_bb: set_list_nat > fun_fu1994797156054922755_nat_o ).

tff(sy_c_ATP_058Lamm__bc____,type,
    aTP_Lamm_bc: set_set_nat > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__bd____,type,
    aTP_Lamm_bd: set_nat > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__be____,type,
    aTP_Lamm_be: set_int > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__bf____,type,
    aTP_Lamm_bf: complex > fun_complex_o ).

tff(sy_c_ATP_058Lamm__bg____,type,
    aTP_Lamm_bg: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__bh____,type,
    aTP_Lamm_bh: fun_Co2081214437782444048eger_o ).

tff(sy_c_ATP_058Lamm__bi____,type,
    aTP_Lamm_bi: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__bj____,type,
    aTP_Lamm_bj: vEBT_VEBT > fun_VE3165209603783583043T_VEBT ).

tff(sy_c_ATP_058Lamm__bk____,type,
    aTP_Lamm_bk: vEBT_VEBT > fun_o_VEBT_VEBT ).

tff(sy_c_ATP_058Lamm__bl____,type,
    aTP_Lamm_bl: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__bm____,type,
    aTP_Lamm_bm: vEBT_VEBT > fun_nat_VEBT_VEBT ).

tff(sy_c_ATP_058Lamm__bn____,type,
    aTP_Lamm_bn: vEBT_VEBT > fun_int_VEBT_VEBT ).

tff(sy_c_ATP_058Lamm__bo____,type,
    aTP_Lamm_bo: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__bp____,type,
    aTP_Lamm_bp: nat > fun_na8797796813480665199d_enat ).

tff(sy_c_ATP_058Lamm__bq____,type,
    aTP_Lamm_bq: extended_enat > fun_na8797796813480665199d_enat ).

tff(sy_c_ATP_058Lamm__br____,type,
    aTP_Lamm_br: num > fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__bs____,type,
    aTP_Lamm_bs: num > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__bt____,type,
    aTP_Lamm_bt: num > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__bu____,type,
    aTP_Lamm_bu: set_real > fun_se2161501677553633980real_o ).

tff(sy_c_ATP_058Lamm__bv____,type,
    aTP_Lamm_bv: set_complex > fun_se9162176452187397432plex_o ).

tff(sy_c_ATP_058Lamm__bw____,type,
    aTP_Lamm_bw: set_list_nat > fun_se1865396762483378772_nat_o ).

tff(sy_c_ATP_058Lamm__bx____,type,
    aTP_Lamm_bx: set_set_nat > fun_se2389560805064864136_nat_o ).

tff(sy_c_ATP_058Lamm__by____,type,
    aTP_Lamm_by: set_nat > fun_se382663106265794036_nat_o ).

tff(sy_c_ATP_058Lamm__bz____,type,
    aTP_Lamm_bz: set_int > fun_se867122708194164540_int_o ).

tff(sy_c_ATP_058Lamm__ca____,type,
    aTP_Lamm_ca: real > fun_real_real ).

tff(sy_c_ATP_058Lamm__cb____,type,
    aTP_Lamm_cb: rat > fun_rat_rat ).

tff(sy_c_ATP_058Lamm__cc____,type,
    aTP_Lamm_cc: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__cd____,type,
    aTP_Lamm_cd: fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__ce____,type,
    aTP_Lamm_ce: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__cf____,type,
    aTP_Lamm_cf: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__cg____,type,
    aTP_Lamm_cg: fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__ch____,type,
    aTP_Lamm_ch: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__ci____,type,
    aTP_Lamm_ci: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__cj____,type,
    aTP_Lamm_cj: fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__ck____,type,
    aTP_Lamm_ck: fun_complex_o > fun_complex_o ).

tff(sy_c_ATP_058Lamm__cl____,type,
    aTP_Lamm_cl: fun_list_nat_o > fun_list_nat_o ).

tff(sy_c_ATP_058Lamm__cm____,type,
    aTP_Lamm_cm: fun_set_nat_o > fun_set_nat_o ).

tff(sy_c_ATP_058Lamm__cn____,type,
    aTP_Lamm_cn: fun_nat_o > fun_nat_o ).

tff(sy_c_ATP_058Lamm__co____,type,
    aTP_Lamm_co: fun_int_o > fun_int_o ).

tff(sy_c_ATP_058Lamm__cp____,type,
    aTP_Lamm_cp: set_real > fun_real_o ).

tff(sy_c_ATP_058Lamm__cq____,type,
    aTP_Lamm_cq: set_complex > fun_complex_o ).

tff(sy_c_ATP_058Lamm__cr____,type,
    aTP_Lamm_cr: set_list_nat > fun_list_nat_o ).

tff(sy_c_ATP_058Lamm__cs____,type,
    aTP_Lamm_cs: set_set_nat > fun_set_nat_o ).

tff(sy_c_ATP_058Lamm__ct____,type,
    aTP_Lamm_ct: set_nat > fun_nat_o ).

tff(sy_c_ATP_058Lamm__cu____,type,
    aTP_Lamm_cu: set_int > fun_int_o ).

tff(sy_c_ATP_058Lamm__cv____,type,
    aTP_Lamm_cv: fun_o_o > fun_fu138107234227827011_int_o ).

tff(sy_c_ATP_058Lamm__cw____,type,
    aTP_Lamm_cw: fun_o_int > fun_fu6453395667394816515nt_int ).

tff(sy_c_ATP_058Lamm__cx____,type,
    aTP_Lamm_cx: fun_int_o > fun_fu1481857817119020493_int_o ).

tff(sy_c_ATP_058Lamm__cy____,type,
    aTP_Lamm_cy: fun_int_int > fun_fu6092417073193843705nt_int ).

tff(sy_c_ATP_058Lamm__cz____,type,
    aTP_Lamm_cz: fun_Pr3238374608701176162_int_o > fun_fu4592178885959468770_int_o ).

tff(sy_c_ATP_058Lamm__da____,type,
    aTP_Lamm_da: fun_Pr982841511892225912nt_int > fun_fu717549361983437348nt_int ).

tff(sy_c_ATP_058Lamm__db____,type,
    aTP_Lamm_db: fun_o_8988054265482332260nt_int > fun_fu8427683764667544036nt_int ).

tff(sy_c_ATP_058Lamm__dc____,type,
    aTP_Lamm_dc: fun_in4211310926729683942nt_int > fun_fu918106826131887214nt_int ).

tff(sy_c_ATP_058Lamm__dd____,type,
    aTP_Lamm_dd: fun_Pr2478283820680522243nt_int > fun_fu3898734532474705027nt_int ).

tff(sy_c_ATP_058Lamm__de____,type,
    aTP_Lamm_de: fun_fu3275390434952296369_nat_o > fun_fu6384561554008368101_nat_o ).

tff(sy_c_ATP_058Lamm__df____,type,
    aTP_Lamm_df: fun_Pr2477239776880934807at_nat > fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__dg____,type,
    aTP_Lamm_dg: fun_Pr2028404973800837934_nat_o > fun_na5914473681949881652_nat_o ).

tff(sy_c_ATP_058Lamm__dh____,type,
    aTP_Lamm_dh: fun_Pr2478283820680522243nt_int > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__di____,type,
    aTP_Lamm_di: fun_Pr3238374608701176162_int_o > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__dj____,type,
    aTP_Lamm_dj: fun_Pr982841511892225912nt_int > fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__dk____,type,
    aTP_Lamm_dk: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__dl____,type,
    aTP_Lamm_dl: $o > fun_fu138107234227827011_int_o ).

tff(sy_c_ATP_058Lamm__dm____,type,
    aTP_Lamm_dm: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__dn____,type,
    aTP_Lamm_dn: fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__do____,type,
    aTP_Lamm_do: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__dp____,type,
    aTP_Lamm_dp: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__dq____,type,
    aTP_Lamm_dq: set_Pr448751882837621926eger_o > fun_Co480733329897759773un_o_o ).

tff(sy_c_ATP_058Lamm__dr____,type,
    aTP_Lamm_dr: set_Pr8218934625190621173um_num > fun_num_fun_num_o ).

tff(sy_c_ATP_058Lamm__ds____,type,
    aTP_Lamm_ds: set_Pr6200539531224447659at_num > fun_nat_fun_num_o ).

tff(sy_c_ATP_058Lamm__dt____,type,
    aTP_Lamm_dt: set_Pr1261947904930325089at_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__du____,type,
    aTP_Lamm_du: set_Pr958786334691620121nt_int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__dv____,type,
    aTP_Lamm_dv: fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__dw____,type,
    aTP_Lamm_dw: fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__dx____,type,
    aTP_Lamm_dx: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__dy____,type,
    aTP_Lamm_dy: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__dz____,type,
    aTP_Lamm_dz: fun_na7542918316964655096d_enat ).

tff(sy_c_ATP_058Lamm__ea____,type,
    aTP_Lamm_ea: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__eb____,type,
    aTP_Lamm_eb: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ec____,type,
    aTP_Lamm_ec: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ed____,type,
    aTP_Lamm_ed: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ee____,type,
    aTP_Lamm_ee: fun_nat_complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__ef____,type,
    aTP_Lamm_ef: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__eg____,type,
    aTP_Lamm_eg: fun_nat_real ).

tff(sy_c_ATP_058Lamm__eh____,type,
    aTP_Lamm_eh: fun_nat_complex ).

tff(sy_c_ATP_058Lamm__ei____,type,
    aTP_Lamm_ei: fun_nat_real ).

tff(sy_c_ATP_058Lamm__ej____,type,
    aTP_Lamm_ej: fun_nat_int ).

tff(sy_c_ATP_058Lamm__ek____,type,
    aTP_Lamm_ek: nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__el____,type,
    aTP_Lamm_el: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__em____,type,
    aTP_Lamm_em: nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__en____,type,
    aTP_Lamm_en: nat > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__eo____,type,
    aTP_Lamm_eo: fun_nat_real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__ep____,type,
    aTP_Lamm_ep: fun_nat_complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__eq____,type,
    aTP_Lamm_eq: complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__er____,type,
    aTP_Lamm_er: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__es____,type,
    aTP_Lamm_es: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__et____,type,
    aTP_Lamm_et: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__eu____,type,
    aTP_Lamm_eu: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ev____,type,
    aTP_Lamm_ev: fun_nat_complex > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ew____,type,
    aTP_Lamm_ew: fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__ex____,type,
    aTP_Lamm_ex: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ey____,type,
    aTP_Lamm_ey: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__ez____,type,
    aTP_Lamm_ez: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__fa____,type,
    aTP_Lamm_fa: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__fb____,type,
    aTP_Lamm_fb: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__fc____,type,
    aTP_Lamm_fc: fun_nat_complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__fd____,type,
    aTP_Lamm_fd: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__fe____,type,
    aTP_Lamm_fe: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__ff____,type,
    aTP_Lamm_ff: fun_nat_int > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__fg____,type,
    aTP_Lamm_fg: fun_nat_complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__fh____,type,
    aTP_Lamm_fh: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__fi____,type,
    aTP_Lamm_fi: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__fj____,type,
    aTP_Lamm_fj: fun_nat_complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__fk____,type,
    aTP_Lamm_fk: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__fl____,type,
    aTP_Lamm_fl: fun_nat_complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__fm____,type,
    aTP_Lamm_fm: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__fn____,type,
    aTP_Lamm_fn: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__fo____,type,
    aTP_Lamm_fo: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__fp____,type,
    aTP_Lamm_fp: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__fq____,type,
    aTP_Lamm_fq: fun_nat_complex > fun_co8378349014452122532t_real ).

tff(sy_c_ATP_058Lamm__fr____,type,
    aTP_Lamm_fr: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__fs____,type,
    aTP_Lamm_fs: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__ft____,type,
    aTP_Lamm_ft: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__fu____,type,
    aTP_Lamm_fu: fun_nat_complex > fun_na6632006986671646835omplex ).

tff(sy_c_ATP_058Lamm__fv____,type,
    aTP_Lamm_fv: fun_nat_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__fw____,type,
    aTP_Lamm_fw: real > fun_fu7631928937237476593t_real ).

tff(sy_c_ATP_058Lamm__fx____,type,
    aTP_Lamm_fx: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__fy____,type,
    aTP_Lamm_fy: fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__fz____,type,
    aTP_Lamm_fz: fun_nat_real > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__ga____,type,
    aTP_Lamm_ga: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__gb____,type,
    aTP_Lamm_gb: complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__gc____,type,
    aTP_Lamm_gc: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__gd____,type,
    aTP_Lamm_gd: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ge____,type,
    aTP_Lamm_ge: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__gf____,type,
    aTP_Lamm_gf: nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__gg____,type,
    aTP_Lamm_gg: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__gh____,type,
    aTP_Lamm_gh: nat > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__gi____,type,
    aTP_Lamm_gi: nat > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__gj____,type,
    aTP_Lamm_gj: nat > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__gk____,type,
    aTP_Lamm_gk: fun_nat_real ).

tff(sy_c_ATP_058Lamm__gl____,type,
    aTP_Lamm_gl: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__gm____,type,
    aTP_Lamm_gm: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__gn____,type,
    aTP_Lamm_gn: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__go____,type,
    aTP_Lamm_go: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__gp____,type,
    aTP_Lamm_gp: fun_nat_Code_integer > fun_nat_Code_integer ).

tff(sy_c_ATP_058Lamm__gq____,type,
    aTP_Lamm_gq: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__gr____,type,
    aTP_Lamm_gr: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__gs____,type,
    aTP_Lamm_gs: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__gt____,type,
    aTP_Lamm_gt: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__gu____,type,
    aTP_Lamm_gu: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__gv____,type,
    aTP_Lamm_gv: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__gw____,type,
    aTP_Lamm_gw: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__gx____,type,
    aTP_Lamm_gx: fun_real_real ).

tff(sy_c_ATP_058Lamm__gy____,type,
    aTP_Lamm_gy: fun_int_int ).

tff(sy_c_ATP_058Lamm__gz____,type,
    aTP_Lamm_gz: fun_Ex5852502975177633731d_enat ).

tff(sy_c_ATP_058Lamm__ha____,type,
    aTP_Lamm_ha: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__hb____,type,
    aTP_Lamm_hb: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__hc____,type,
    aTP_Lamm_hc: complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__hd____,type,
    aTP_Lamm_hd: complex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__he____,type,
    aTP_Lamm_he: rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__hf____,type,
    aTP_Lamm_hf: real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__hg____,type,
    aTP_Lamm_hg: int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__hh____,type,
    aTP_Lamm_hh: extended_enat > fun_na1154345429118027792d_enat ).

tff(sy_c_ATP_058Lamm__hi____,type,
    aTP_Lamm_hi: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__hj____,type,
    aTP_Lamm_hj: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__hk____,type,
    aTP_Lamm_hk: complex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__hl____,type,
    aTP_Lamm_hl: rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__hm____,type,
    aTP_Lamm_hm: real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__hn____,type,
    aTP_Lamm_hn: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__ho____,type,
    aTP_Lamm_ho: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__hp____,type,
    aTP_Lamm_hp: fun_real_o ).

tff(sy_c_ATP_058Lamm__hq____,type,
    aTP_Lamm_hq: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__hr____,type,
    aTP_Lamm_hr: fun_real_o ).

tff(sy_c_ATP_058Lamm__hs____,type,
    aTP_Lamm_hs: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__ht____,type,
    aTP_Lamm_ht: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__hu____,type,
    aTP_Lamm_hu: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__hv____,type,
    aTP_Lamm_hv: code_integer > fun_o_7148296245751160776un_o_o ).

tff(sy_c_ATP_058Lamm__hw____,type,
    aTP_Lamm_hw: num > fun_nu7188104440484974293_num_o ).

tff(sy_c_ATP_058Lamm__hx____,type,
    aTP_Lamm_hx: nat > fun_nu2628545618185965215_num_o ).

tff(sy_c_ATP_058Lamm__hy____,type,
    aTP_Lamm_hy: nat > fun_na7928631014926962227_nat_o ).

tff(sy_c_ATP_058Lamm__hz____,type,
    aTP_Lamm_hz: int > fun_in5570076352459579423_int_o ).

tff(sy_c_ATP_058Lamm__ia____,type,
    aTP_Lamm_ia: $o > fun_re6988797352886916895real_o ).

tff(sy_c_ATP_058Lamm__ib____,type,
    aTP_Lamm_ib: $o > fun_in5570076352459579423_int_o ).

tff(sy_c_ATP_058Lamm__ic____,type,
    aTP_Lamm_ic: real > fun_int_o ).

tff(sy_c_ATP_058Lamm__id____,type,
    aTP_Lamm_id: rat > fun_int_o ).

tff(sy_c_ATP_058Lamm__ie____,type,
    aTP_Lamm_ie: fun_na1668100129766719005at_nat > fun_Pr557098546508347797at_nat ).

tff(sy_c_ATP_058Lamm__if____,type,
    aTP_Lamm_if: fun_na5914473681949881652_nat_o > fun_Pr3455331679158536364_nat_o ).

tff(sy_c_ATP_058Lamm__ig____,type,
    aTP_Lamm_ig: fun_nat_complex ).

tff(sy_c_ATP_058Lamm__ih____,type,
    aTP_Lamm_ih: fun_nat_real ).

tff(sy_c_ATP_058Lamm__ii____,type,
    aTP_Lamm_ii: fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__ij____,type,
    aTP_Lamm_ij: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ik____,type,
    aTP_Lamm_ik: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__il____,type,
    aTP_Lamm_il: fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__im____,type,
    aTP_Lamm_im: complex > fun_real_o ).

tff(sy_c_ATP_058Lamm__in____,type,
    aTP_Lamm_in: fun_real_o > fun_real_o ).

tff(sy_c_ATP_058Lamm__io____,type,
    aTP_Lamm_io: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ip____,type,
    aTP_Lamm_ip: fun_nat_complex > fun_re2084383175013312036omplex ).

tff(sy_c_ATP_058Lamm__iq____,type,
    aTP_Lamm_iq: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ir____,type,
    aTP_Lamm_ir: fun_nat_real > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__is____,type,
    aTP_Lamm_is: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__it____,type,
    aTP_Lamm_it: fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__iu____,type,
    aTP_Lamm_iu: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__iv____,type,
    aTP_Lamm_iv: fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__iw____,type,
    aTP_Lamm_iw: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ix____,type,
    aTP_Lamm_ix: fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__iy____,type,
    aTP_Lamm_iy: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__iz____,type,
    aTP_Lamm_iz: fun_co8378349014452122532t_real ).

tff(sy_c_ATP_058Lamm__ja____,type,
    aTP_Lamm_ja: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__jb____,type,
    aTP_Lamm_jb: fun_co8378349014452122532t_real ).

tff(sy_c_ATP_058Lamm__jc____,type,
    aTP_Lamm_jc: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__jd____,type,
    aTP_Lamm_jd: fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__je____,type,
    aTP_Lamm_je: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__jf____,type,
    aTP_Lamm_jf: fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__jg____,type,
    aTP_Lamm_jg: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__jh____,type,
    aTP_Lamm_jh: fun_co8378349014452122532t_real ).

tff(sy_c_ATP_058Lamm__ji____,type,
    aTP_Lamm_ji: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__jj____,type,
    aTP_Lamm_jj: fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__jk____,type,
    aTP_Lamm_jk: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__jl____,type,
    aTP_Lamm_jl: fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__jm____,type,
    aTP_Lamm_jm: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__jn____,type,
    aTP_Lamm_jn: fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__jo____,type,
    aTP_Lamm_jo: fun_Pr3290071492984375169er_o_o > fun_Co480733329897759773un_o_o ).

tff(sy_c_ATP_058Lamm__jp____,type,
    aTP_Lamm_jp: fun_Pr6331313853612757382_num_o > fun_num_fun_num_o ).

tff(sy_c_ATP_058Lamm__jq____,type,
    aTP_Lamm_jq: fun_Pr5555124106300582480_num_o > fun_nat_fun_num_o ).

tff(sy_c_ATP_058Lamm__jr____,type,
    aTP_Lamm_jr: fun_Pr4796943543331071514_nat_o > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__js____,type,
    aTP_Lamm_js: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__jt____,type,
    aTP_Lamm_jt: real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ju____,type,
    aTP_Lamm_ju: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__jv____,type,
    aTP_Lamm_jv: complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__jw____,type,
    aTP_Lamm_jw: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__jx____,type,
    aTP_Lamm_jx: real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__jy____,type,
    aTP_Lamm_jy: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__jz____,type,
    aTP_Lamm_jz: complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__ka____,type,
    aTP_Lamm_ka: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__kb____,type,
    aTP_Lamm_kb: real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__kc____,type,
    aTP_Lamm_kc: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__kd____,type,
    aTP_Lamm_kd: complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__ke____,type,
    aTP_Lamm_ke: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__kf____,type,
    aTP_Lamm_kf: fun_complex_nat > fun_complex_complex ).

tff(sy_c_ATP_058Lamm__kg____,type,
    aTP_Lamm_kg: fun_int_nat > fun_int_int ).

tff(sy_c_ATP_058Lamm__kh____,type,
    aTP_Lamm_kh: fun_nat_nat > fun_nat_int ).

tff(sy_c_ATP_058Lamm__ki____,type,
    aTP_Lamm_ki: fun_nat_nat > fun_na8797796813480665199d_enat ).

tff(sy_c_ATP_058Lamm__kj____,type,
    aTP_Lamm_kj: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__kk____,type,
    aTP_Lamm_kk: fun_nat_nat > fun_nat_real ).

tff(sy_c_ATP_058Lamm__kl____,type,
    aTP_Lamm_kl: fun_complex_int > fun_complex_complex ).

tff(sy_c_ATP_058Lamm__km____,type,
    aTP_Lamm_km: fun_nat_int > fun_nat_real ).

tff(sy_c_ATP_058Lamm__kn____,type,
    aTP_Lamm_kn: fun_int_int > fun_int_real ).

tff(sy_c_ATP_058Lamm__ko____,type,
    aTP_Lamm_ko: fun_int_int > fun_int_rat ).

tff(sy_c_ATP_058Lamm__kp____,type,
    aTP_Lamm_kp: fun_int_int > fun_int_int ).

tff(sy_c_ATP_058Lamm__kq____,type,
    aTP_Lamm_kq: fun_complex_real > fun_complex_complex ).

tff(sy_c_ATP_058Lamm__kr____,type,
    aTP_Lamm_kr: fun_complex_real > fun_co7673397065115134084omplex ).

tff(sy_c_ATP_058Lamm__ks____,type,
    aTP_Lamm_ks: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__kt____,type,
    aTP_Lamm_kt: fun_int_int > fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__ku____,type,
    aTP_Lamm_ku: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__kv____,type,
    aTP_Lamm_kv: real > fun_fu5956788528830221615omplex ).

tff(sy_c_ATP_058Lamm__kw____,type,
    aTP_Lamm_kw: fun_complex_complex > fun_complex_real ).

tff(sy_c_ATP_058Lamm__kx____,type,
    aTP_Lamm_kx: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ky____,type,
    aTP_Lamm_ky: fun_co8378349014452122532t_real > fun_na5723442673449079972x_real ).

tff(sy_c_ATP_058Lamm__kz____,type,
    aTP_Lamm_kz: set_complex > fun_fu7426956082801719578t_real ).

tff(sy_c_ATP_058Lamm__la____,type,
    aTP_Lamm_la: fun_re4238438383274954274t_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__lb____,type,
    aTP_Lamm_lb: set_real > fun_fu5104398059618711576t_real ).

tff(sy_c_ATP_058Lamm__lc____,type,
    aTP_Lamm_lc: fun_int_fun_nat_real > fun_nat_fun_int_real ).

tff(sy_c_ATP_058Lamm__ld____,type,
    aTP_Lamm_ld: set_int > fun_fu7141299348810748056t_real ).

tff(sy_c_ATP_058Lamm__le____,type,
    aTP_Lamm_le: fun_re2084383175013312036omplex > fun_na1957955181367619364omplex ).

tff(sy_c_ATP_058Lamm__lf____,type,
    aTP_Lamm_lf: set_real > fun_fu847012557204443036omplex ).

tff(sy_c_ATP_058Lamm__lg____,type,
    aTP_Lamm_lg: fun_na1142814060743658952omplex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__lh____,type,
    aTP_Lamm_lh: set_nat > fun_fu7808106308057563584omplex ).

tff(sy_c_ATP_058Lamm__li____,type,
    aTP_Lamm_li: fun_in8056368165505483812omplex > fun_na5558608354090810916omplex ).

tff(sy_c_ATP_058Lamm__lj____,type,
    aTP_Lamm_lj: set_int > fun_fu6465309499045537820omplex ).

tff(sy_c_ATP_058Lamm__lk____,type,
    aTP_Lamm_lk: fun_nat_fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ll____,type,
    aTP_Lamm_ll: set_nat > fun_fu7979797992566127108at_nat ).

tff(sy_c_ATP_058Lamm__lm____,type,
    aTP_Lamm_lm: fun_co3258106519541737638omplex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__ln____,type,
    aTP_Lamm_ln: set_complex > fun_fu689363623238371998omplex ).

tff(sy_c_ATP_058Lamm__lo____,type,
    aTP_Lamm_lo: fun_nat_fun_nat_real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__lp____,type,
    aTP_Lamm_lp: set_nat > fun_fu2115079063411676732t_real ).

tff(sy_c_ATP_058Lamm__lq____,type,
    aTP_Lamm_lq: fun_int_fun_nat_int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__lr____,type,
    aTP_Lamm_lr: set_int > fun_fu688336635129761048at_int ).

tff(sy_c_ATP_058Lamm__ls____,type,
    aTP_Lamm_ls: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__lt____,type,
    aTP_Lamm_lt: fun_nat_complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__lu____,type,
    aTP_Lamm_lu: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__lv____,type,
    aTP_Lamm_lv: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__lw____,type,
    aTP_Lamm_lw: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__lx____,type,
    aTP_Lamm_lx: set_nat > fun_set_nat_o ).

tff(sy_c_ATP_058Lamm__ly____,type,
    aTP_Lamm_ly: set_int > fun_set_int_o ).

tff(sy_c_ATP_058Lamm__lz____,type,
    aTP_Lamm_lz: fun_rat_fun_rat_o ).

tff(sy_c_ATP_058Lamm__ma____,type,
    aTP_Lamm_ma: fun_num_fun_num_o ).

tff(sy_c_ATP_058Lamm__mb____,type,
    aTP_Lamm_mb: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__mc____,type,
    aTP_Lamm_mc: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__md____,type,
    aTP_Lamm_md: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__me____,type,
    aTP_Lamm_me: fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__mf____,type,
    aTP_Lamm_mf: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__mg____,type,
    aTP_Lamm_mg: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__mh____,type,
    aTP_Lamm_mh: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__mi____,type,
    aTP_Lamm_mi: nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__mj____,type,
    aTP_Lamm_mj: fun_co8378349014452122532t_real > fun_complex_real ).

tff(sy_c_ATP_058Lamm__mk____,type,
    aTP_Lamm_mk: fun_re4238438383274954274t_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__ml____,type,
    aTP_Lamm_ml: fun_int_fun_nat_real > fun_int_real ).

tff(sy_c_ATP_058Lamm__mm____,type,
    aTP_Lamm_mm: fun_re2084383175013312036omplex > fun_real_complex ).

tff(sy_c_ATP_058Lamm__mn____,type,
    aTP_Lamm_mn: fun_na1142814060743658952omplex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__mo____,type,
    aTP_Lamm_mo: fun_in8056368165505483812omplex > fun_int_complex ).

tff(sy_c_ATP_058Lamm__mp____,type,
    aTP_Lamm_mp: fun_nat_fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__mq____,type,
    aTP_Lamm_mq: fun_co3258106519541737638omplex > fun_complex_complex ).

tff(sy_c_ATP_058Lamm__mr____,type,
    aTP_Lamm_mr: fun_nat_fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ms____,type,
    aTP_Lamm_ms: fun_int_fun_nat_int > fun_int_int ).

tff(sy_c_ATP_058Lamm__mt____,type,
    aTP_Lamm_mt: nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__mu____,type,
    aTP_Lamm_mu: fun_nat_complex > fun_fu4528473944208963392omplex ).

tff(sy_c_ATP_058Lamm__mv____,type,
    aTP_Lamm_mv: fun_nat_complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__mw____,type,
    aTP_Lamm_mw: fun_nat_real > fun_fu163975356400620604t_real ).

tff(sy_c_ATP_058Lamm__mx____,type,
    aTP_Lamm_mx: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__my____,type,
    aTP_Lamm_my: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__mz____,type,
    aTP_Lamm_mz: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__na____,type,
    aTP_Lamm_na: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__nb____,type,
    aTP_Lamm_nb: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__nc____,type,
    aTP_Lamm_nc: fun_nat_complex > fun_fu6652553491520710942omplex ).

tff(sy_c_ATP_058Lamm__nd____,type,
    aTP_Lamm_nd: fun_nat_rat > fun_rat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ne____,type,
    aTP_Lamm_ne: fun_nat_rat > fun_fu6228432884689642900at_rat ).

tff(sy_c_ATP_058Lamm__nf____,type,
    aTP_Lamm_nf: fun_nat_rat > fun_fu4710970720072501148at_rat ).

tff(sy_c_ATP_058Lamm__ng____,type,
    aTP_Lamm_ng: fun_nat_int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__nh____,type,
    aTP_Lamm_nh: fun_nat_int > fun_fu1567874110255095356at_int ).

tff(sy_c_ATP_058Lamm__ni____,type,
    aTP_Lamm_ni: fun_nat_int > fun_fu5519318738670804632at_int ).

tff(sy_c_ATP_058Lamm__nj____,type,
    aTP_Lamm_nj: fun_nat_real > fun_fu8064956510386623256t_real ).

tff(sy_c_ATP_058Lamm__nk____,type,
    aTP_Lamm_nk: fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__nl____,type,
    aTP_Lamm_nl: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__nm____,type,
    aTP_Lamm_nm: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__nn____,type,
    aTP_Lamm_nn: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__no____,type,
    aTP_Lamm_no: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__np____,type,
    aTP_Lamm_np: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__nq____,type,
    aTP_Lamm_nq: extended_enat > fun_Ex8979681104671372885d_enat ).

tff(sy_c_ATP_058Lamm__nr____,type,
    aTP_Lamm_nr: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__ns____,type,
    aTP_Lamm_ns: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__nt____,type,
    aTP_Lamm_nt: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__nu____,type,
    aTP_Lamm_nu: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__nv____,type,
    aTP_Lamm_nv: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__nw____,type,
    aTP_Lamm_nw: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__nx____,type,
    aTP_Lamm_nx: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__ny____,type,
    aTP_Lamm_ny: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__nz____,type,
    aTP_Lamm_nz: complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__oa____,type,
    aTP_Lamm_oa: rat > fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ob____,type,
    aTP_Lamm_ob: int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__oc____,type,
    aTP_Lamm_oc: real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__od____,type,
    aTP_Lamm_od: nat > fun_fu3593947265352647275omplex ).

tff(sy_c_ATP_058Lamm__oe____,type,
    aTP_Lamm_oe: nat > fun_fu3593947265352647275omplex ).

tff(sy_c_ATP_058Lamm__of____,type,
    aTP_Lamm_of: nat > fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__og____,type,
    aTP_Lamm_og: nat > fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__oh____,type,
    aTP_Lamm_oh: nat > fun_fu960360787657196261at_int ).

tff(sy_c_ATP_058Lamm__oi____,type,
    aTP_Lamm_oi: nat > fun_fu960360787657196261at_int ).

tff(sy_c_ATP_058Lamm__oj____,type,
    aTP_Lamm_oj: nat > fun_fu5935951186182374353at_nat ).

tff(sy_c_ATP_058Lamm__ok____,type,
    aTP_Lamm_ok: nat > fun_fu5935951186182374353at_nat ).

tff(sy_c_ATP_058Lamm__ol____,type,
    aTP_Lamm_ol: nat > fun_fu1620311477532340197t_real ).

tff(sy_c_ATP_058Lamm__om____,type,
    aTP_Lamm_om: nat > fun_fu1620311477532340197t_real ).

tff(sy_c_ATP_058Lamm__on____,type,
    aTP_Lamm_on: nat > fun_co3732688428756473980omplex ).

tff(sy_c_ATP_058Lamm__oo____,type,
    aTP_Lamm_oo: nat > fun_co3732688428756473980omplex ).

tff(sy_c_ATP_058Lamm__op____,type,
    aTP_Lamm_op: nat > fun_ra8505079968215247908at_rat ).

tff(sy_c_ATP_058Lamm__oq____,type,
    aTP_Lamm_oq: nat > fun_ra8505079968215247908at_rat ).

tff(sy_c_ATP_058Lamm__or____,type,
    aTP_Lamm_or: nat > fun_re6135230694098607860t_real ).

tff(sy_c_ATP_058Lamm__os____,type,
    aTP_Lamm_os: nat > fun_re6135230694098607860t_real ).

tff(sy_c_ATP_058Lamm__ot____,type,
    aTP_Lamm_ot: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__ou____,type,
    aTP_Lamm_ou: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__ov____,type,
    aTP_Lamm_ov: nat > fun_in1521351555827773387at_int ).

tff(sy_c_ATP_058Lamm__ow____,type,
    aTP_Lamm_ow: nat > fun_co6147893885205795665omplex ).

tff(sy_c_ATP_058Lamm__ox____,type,
    aTP_Lamm_ox: nat > fun_ra3995598502166916815at_rat ).

tff(sy_c_ATP_058Lamm__oy____,type,
    aTP_Lamm_oy: nat > fun_Co8502721031311864414nteger ).

tff(sy_c_ATP_058Lamm__oz____,type,
    aTP_Lamm_oz: nat > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__pa____,type,
    aTP_Lamm_pa: fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__pb____,type,
    aTP_Lamm_pb: nat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__pc____,type,
    aTP_Lamm_pc: nat > fun_nat_Code_integer ).

tff(sy_c_ATP_058Lamm__pd____,type,
    aTP_Lamm_pd: nat > fun_nat_int ).

tff(sy_c_ATP_058Lamm__pe____,type,
    aTP_Lamm_pe: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__pf____,type,
    aTP_Lamm_pf: fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__pg____,type,
    aTP_Lamm_pg: nat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ph____,type,
    aTP_Lamm_ph: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__pi____,type,
    aTP_Lamm_pi: nat > fun_co3732688428756473980omplex ).

tff(sy_c_ATP_058Lamm__pj____,type,
    aTP_Lamm_pj: nat > fun_ra8505079968215247908at_rat ).

tff(sy_c_ATP_058Lamm__pk____,type,
    aTP_Lamm_pk: nat > fun_re6135230694098607860t_real ).

tff(sy_c_ATP_058Lamm__pl____,type,
    aTP_Lamm_pl: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__pm____,type,
    aTP_Lamm_pm: fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__pn____,type,
    aTP_Lamm_pn: nat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__po____,type,
    aTP_Lamm_po: nat > fun_nat_Code_integer ).

tff(sy_c_ATP_058Lamm__pp____,type,
    aTP_Lamm_pp: nat > fun_nat_int ).

tff(sy_c_ATP_058Lamm__pq____,type,
    aTP_Lamm_pq: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__pr____,type,
    aTP_Lamm_pr: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__ps____,type,
    aTP_Lamm_ps: fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__pt____,type,
    aTP_Lamm_pt: nat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__pu____,type,
    aTP_Lamm_pu: nat > fun_nat_int ).

tff(sy_c_ATP_058Lamm__pv____,type,
    aTP_Lamm_pv: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__pw____,type,
    aTP_Lamm_pw: fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__px____,type,
    aTP_Lamm_px: nat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__py____,type,
    aTP_Lamm_py: nat > fun_nat_int ).

tff(sy_c_ATP_058Lamm__pz____,type,
    aTP_Lamm_pz: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__qa____,type,
    aTP_Lamm_qa: fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__qb____,type,
    aTP_Lamm_qb: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__qc____,type,
    aTP_Lamm_qc: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__qd____,type,
    aTP_Lamm_qd: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__qe____,type,
    aTP_Lamm_qe: int > fun_in5570076352459579423_int_o ).

tff(sy_c_ATP_058Lamm__qf____,type,
    aTP_Lamm_qf: rat > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__qg____,type,
    aTP_Lamm_qg: int > fun_in5570076352459579423_int_o ).

tff(sy_c_ATP_058Lamm__qh____,type,
    aTP_Lamm_qh: rat > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__qi____,type,
    aTP_Lamm_qi: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__qj____,type,
    aTP_Lamm_qj: nat > fun_co1148497455491383662plex_o ).

tff(sy_c_ATP_058Lamm__qk____,type,
    aTP_Lamm_qk: nat > fun_complex_o ).

tff(sy_c_ATP_058Lamm__ql____,type,
    aTP_Lamm_ql: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__qm____,type,
    aTP_Lamm_qm: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__qn____,type,
    aTP_Lamm_qn: fun_nat_real ).

tff(sy_c_ATP_058Lamm__qo____,type,
    aTP_Lamm_qo: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__qp____,type,
    aTP_Lamm_qp: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__qq____,type,
    aTP_Lamm_qq: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__qr____,type,
    aTP_Lamm_qr: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__qs____,type,
    aTP_Lamm_qs: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__qt____,type,
    aTP_Lamm_qt: fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__qu____,type,
    aTP_Lamm_qu: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__qv____,type,
    aTP_Lamm_qv: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__qw____,type,
    aTP_Lamm_qw: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__qx____,type,
    aTP_Lamm_qx: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__qy____,type,
    aTP_Lamm_qy: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__qz____,type,
    aTP_Lamm_qz: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__ra____,type,
    aTP_Lamm_ra: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__rb____,type,
    aTP_Lamm_rb: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__rc____,type,
    aTP_Lamm_rc: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__rd____,type,
    aTP_Lamm_rd: complex > fun_na1554462459066888140plex_o ).

tff(sy_c_ATP_058Lamm__re____,type,
    aTP_Lamm_re: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__rf____,type,
    aTP_Lamm_rf: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__rg____,type,
    aTP_Lamm_rg: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__rh____,type,
    aTP_Lamm_rh: fun_nat_complex > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ri____,type,
    aTP_Lamm_ri: fun_nat_complex > fun_nat_real ).

tff(sy_c_ATP_058Lamm__rj____,type,
    aTP_Lamm_rj: fun_nat_complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__rk____,type,
    aTP_Lamm_rk: fun_Co1092983284983208416er_int ).

tff(sy_c_ATP_058Lamm__rl____,type,
    aTP_Lamm_rl: fun_Co8521658056209057742er_num ).

tff(sy_c_ATP_058Lamm__rm____,type,
    aTP_Lamm_rm: code_integer > fun_Co3544761417380347684eger_o ).

tff(sy_c_ATP_058Lamm__rn____,type,
    aTP_Lamm_rn: fun_Co3583066429914935172er_nat ).

tff(sy_c_ATP_058Lamm__ro____,type,
    aTP_Lamm_ro: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__rp____,type,
    aTP_Lamm_rp: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__rq____,type,
    aTP_Lamm_rq: fun_nat_o ).

tff(sy_c_ATP_058Lamm__rr____,type,
    aTP_Lamm_rr: fun_nat_o ).

tff(sy_c_ATP_058Lamm__rs____,type,
    aTP_Lamm_rs: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__rt____,type,
    aTP_Lamm_rt: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ru____,type,
    aTP_Lamm_ru: fun_nat_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__rv____,type,
    aTP_Lamm_rv: fun_nat_o > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__rw____,type,
    aTP_Lamm_rw: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__rx____,type,
    aTP_Lamm_rx: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__ry____,type,
    aTP_Lamm_ry: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__rz____,type,
    aTP_Lamm_rz: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__sa____,type,
    aTP_Lamm_sa: set_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__sb____,type,
    aTP_Lamm_sb: set_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__sc____,type,
    aTP_Lamm_sc: nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__sd____,type,
    aTP_Lamm_sd: nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__se____,type,
    aTP_Lamm_se: nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__sf____,type,
    aTP_Lamm_sf: nat > fun_na8563630686737515764_nat_o ).

tff(sy_c_ATP_058Lamm__sg____,type,
    aTP_Lamm_sg: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__sh____,type,
    aTP_Lamm_sh: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__si____,type,
    aTP_Lamm_si: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__sj____,type,
    aTP_Lamm_sj: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__sk____,type,
    aTP_Lamm_sk: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__sl____,type,
    aTP_Lamm_sl: nat > fun_na7928631014926962227_nat_o ).

tff(sy_c_ATP_058Lamm__sm____,type,
    aTP_Lamm_sm: fun_na5914473681949881652_nat_o ).

tff(sy_c_ATP_058Lamm__sn____,type,
    aTP_Lamm_sn: nat > fun_na7928631014926962227_nat_o ).

tff(sy_c_ATP_058Lamm__so____,type,
    aTP_Lamm_so: fun_na5914473681949881652_nat_o ).

tff(sy_c_ATP_058Lamm__sp____,type,
    aTP_Lamm_sp: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__sq____,type,
    aTP_Lamm_sq: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__sr____,type,
    aTP_Lamm_sr: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__ss____,type,
    aTP_Lamm_ss: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__st____,type,
    aTP_Lamm_st: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__su____,type,
    aTP_Lamm_su: fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__sv____,type,
    aTP_Lamm_sv: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__sw____,type,
    aTP_Lamm_sw: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__sx____,type,
    aTP_Lamm_sx: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__sy____,type,
    aTP_Lamm_sy: $o > fun_o_5488095450495718970o_char ).

tff(sy_c_ATP_058Lamm__sz____,type,
    aTP_Lamm_sz: $o > fun_o_6360995121869435791o_char ).

tff(sy_c_ATP_058Lamm__ta____,type,
    aTP_Lamm_ta: $o > fun_o_2683029531876921444o_char ).

tff(sy_c_ATP_058Lamm__tb____,type,
    aTP_Lamm_tb: $o > fun_o_3037882179896959673o_char ).

tff(sy_c_ATP_058Lamm__tc____,type,
    aTP_Lamm_tc: $o > fun_o_335877185394996366o_char ).

tff(sy_c_ATP_058Lamm__td____,type,
    aTP_Lamm_td: $o > fun_o_2016307092959984611o_char ).

tff(sy_c_ATP_058Lamm__te____,type,
    aTP_Lamm_te: $o > fun_Co681828205884031672o_char ).

tff(sy_c_ATP_058Lamm__tf____,type,
    aTP_Lamm_tf: fun_Co681828205884031672o_char ).

tff(sy_c_ATP_058Lamm__tg____,type,
    aTP_Lamm_tg: num > fun_num_int ).

tff(sy_c_ATP_058Lamm__th____,type,
    aTP_Lamm_th: fun_num_option_num ).

tff(sy_c_ATP_058Lamm__ti____,type,
    aTP_Lamm_ti: num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__tj____,type,
    aTP_Lamm_tj: fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__tk____,type,
    aTP_Lamm_tk: num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__tl____,type,
    aTP_Lamm_tl: fun_num_option_num ).

tff(sy_c_ATP_058Lamm__tm____,type,
    aTP_Lamm_tm: nat > fun_num_option_num ).

tff(sy_c_ATP_058Lamm__tn____,type,
    aTP_Lamm_tn: nat > fun_num_option_num ).

tff(sy_c_ATP_058Lamm__to____,type,
    aTP_Lamm_to: num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__tp____,type,
    aTP_Lamm_tp: fun_na7595719129879163854on_num ).

tff(sy_c_ATP_058Lamm__tq____,type,
    aTP_Lamm_tq: fun_real_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__tr____,type,
    aTP_Lamm_tr: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__ts____,type,
    aTP_Lamm_ts: fun_real_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__tt____,type,
    aTP_Lamm_tt: real > fun_real_real ).

tff(sy_c_ATP_058Lamm__tu____,type,
    aTP_Lamm_tu: fun_real_real > fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__tv____,type,
    aTP_Lamm_tv: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__tw____,type,
    aTP_Lamm_tw: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__tx____,type,
    aTP_Lamm_tx: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ty____,type,
    aTP_Lamm_ty: fun_nat_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__tz____,type,
    aTP_Lamm_tz: fun_na6724076748692969250l_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ua____,type,
    aTP_Lamm_ua: real > fun_fu5589980185390252312t_real ).

tff(sy_c_ATP_058Lamm__ub____,type,
    aTP_Lamm_ub: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__uc____,type,
    aTP_Lamm_uc: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__ud____,type,
    aTP_Lamm_ud: fun_na6724076748692969250l_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__ue____,type,
    aTP_Lamm_ue: nat > fun_fu3612569900928539882l_real ).

tff(sy_c_ATP_058Lamm__uf____,type,
    aTP_Lamm_uf: fun_na6724076748692969250l_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__ug____,type,
    aTP_Lamm_ug: fun_real_real ).

tff(sy_c_ATP_058Lamm__uh____,type,
    aTP_Lamm_uh: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ui____,type,
    aTP_Lamm_ui: fun_real_real ).

tff(sy_c_ATP_058Lamm__uj____,type,
    aTP_Lamm_uj: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__uk____,type,
    aTP_Lamm_uk: fun_nat_real ).

tff(sy_c_ATP_058Lamm__ul____,type,
    aTP_Lamm_ul: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__um____,type,
    aTP_Lamm_um: fun_nat_real ).

tff(sy_c_ATP_058Lamm__un____,type,
    aTP_Lamm_un: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__uo____,type,
    aTP_Lamm_uo: fun_nat_real ).

tff(sy_c_ATP_058Lamm__up____,type,
    aTP_Lamm_up: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__uq____,type,
    aTP_Lamm_uq: real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ur____,type,
    aTP_Lamm_ur: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__us____,type,
    aTP_Lamm_us: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ut____,type,
    aTP_Lamm_ut: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__uu____,type,
    aTP_Lamm_uu: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__uv____,type,
    aTP_Lamm_uv: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__uw____,type,
    aTP_Lamm_uw: fun_nat_real > fun_fu6119738925743952495t_real ).

tff(sy_c_ATP_058Lamm__ux____,type,
    aTP_Lamm_ux: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__uy____,type,
    aTP_Lamm_uy: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__uz____,type,
    aTP_Lamm_uz: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__va____,type,
    aTP_Lamm_va: real > fun_real_real ).

tff(sy_c_ATP_058Lamm__vb____,type,
    aTP_Lamm_vb: nat > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__vc____,type,
    aTP_Lamm_vc: fun_real_real ).

tff(sy_c_ATP_058Lamm__vd____,type,
    aTP_Lamm_vd: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__ve____,type,
    aTP_Lamm_ve: real > fun_real_real ).

tff(sy_c_ATP_058Lamm__vf____,type,
    aTP_Lamm_vf: fun_real_real > fun_real_o ).

tff(sy_c_ATP_058Lamm__vg____,type,
    aTP_Lamm_vg: fun_real_real > fun_fu3022329567491847689real_o ).

tff(sy_c_ATP_058Lamm__vh____,type,
    aTP_Lamm_vh: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__vi____,type,
    aTP_Lamm_vi: fun_nat_o > fun_nat_o ).

tff(sy_c_ATP_058Lamm__vj____,type,
    aTP_Lamm_vj: fun_nat_o > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__vk____,type,
    aTP_Lamm_vk: fun_real_o > fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__vl____,type,
    aTP_Lamm_vl: fun_real_o > fun_real_o ).

tff(sy_c_ATP_058Lamm__vm____,type,
    aTP_Lamm_vm: real > fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__vn____,type,
    aTP_Lamm_vn: fun_real_o > fun_real_o ).

tff(sy_c_ATP_058Lamm__vo____,type,
    aTP_Lamm_vo: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__vp____,type,
    aTP_Lamm_vp: nat > fun_li6914387844523535256_nat_o ).

tff(sy_c_ATP_058Lamm__vq____,type,
    aTP_Lamm_vq: set_int > fun_int_o ).

tff(sy_c_ATP_058Lamm__vr____,type,
    aTP_Lamm_vr: fun_real_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__vs____,type,
    aTP_Lamm_vs: fun_real_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__vt____,type,
    aTP_Lamm_vt: fun_real_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__vu____,type,
    aTP_Lamm_vu: real > fun_co1148497455491383662plex_o ).

tff(sy_c_ATP_058Lamm__vv____,type,
    aTP_Lamm_vv: fun_re3844104906058862192omplex ).

tff(sy_c_ATP_058Lamm__vw____,type,
    aTP_Lamm_vw: real > fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__vx____,type,
    aTP_Lamm_vx: fun_re5512370375968561900l_real ).

tff(sy_c_ATP_058Lamm__vy____,type,
    aTP_Lamm_vy: set_real > fun_re6988797352886916895real_o ).

tff(sy_c_ATP_058Lamm__vz____,type,
    aTP_Lamm_vz: set_complex > fun_co51012501629074457plex_o ).

tff(sy_c_ATP_058Lamm__wa____,type,
    aTP_Lamm_wa: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__wb____,type,
    aTP_Lamm_wb: fun_real_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__wc____,type,
    aTP_Lamm_wc: fun_nat_real > fun_fu3106432010075488275t_real ).

tff(sy_c_Archimedean__Field_Oceiling_001t__Rat__Orat,type,
    archim2889992004027027881ng_rat: fun_rat_int ).

tff(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
    archim7802044766580827645g_real: fun_real_int ).

tff(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Rat__Orat,type,
    archim3151403230148437115or_rat: fun_rat_int ).

tff(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Real__Oreal,type,
    archim6058952711729229775r_real: fun_real_int ).

tff(sy_c_Archimedean__Field_Ofrac_001t__Rat__Orat,type,
    archimedean_frac_rat: fun_rat_rat ).

tff(sy_c_Archimedean__Field_Ofrac_001t__Real__Oreal,type,
    archim2898591450579166408c_real: fun_real_real ).

tff(sy_c_Archimedean__Field_Oround_001t__Rat__Orat,type,
    archim7778729529865785530nd_rat: fun_rat_int ).

tff(sy_c_Archimedean__Field_Oround_001t__Real__Oreal,type,
    archim8280529875227126926d_real: fun_real_int ).

tff(sy_c_Binomial_Obinomial,type,
    binomial: fun_nat_fun_nat_nat ).

tff(sy_c_Binomial_Ogbinomial_001t__Complex__Ocomplex,type,
    gbinomial_complex: fun_co3258106519541737638omplex ).

tff(sy_c_Binomial_Ogbinomial_001t__Int__Oint,type,
    gbinomial_int: fun_int_fun_nat_int ).

tff(sy_c_Binomial_Ogbinomial_001t__Nat__Onat,type,
    gbinomial_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Binomial_Ogbinomial_001t__Rat__Orat,type,
    gbinomial_rat: rat > fun_nat_rat ).

tff(sy_c_Binomial_Ogbinomial_001t__Real__Oreal,type,
    gbinomial_real: fun_re4238438383274954274t_real ).

tff(sy_c_Bit__Operations_Oand__int__rel,type,
    bit_and_int_rel: fun_Pr9018117161326028078_int_o ).

tff(sy_c_Bit__Operations_Oand__not__num,type,
    bit_and_not_num: ( num * num ) > option_num ).

tff(sy_c_Bit__Operations_Oconcat__bit,type,
    bit_concat_bit: nat > fun_int_fun_int_int ).

tff(sy_c_Bit__Operations_Oor__not__num__neg,type,
    bit_or_not_num_neg: num > fun_num_num ).

tff(sy_c_Bit__Operations_Oor__not__num__neg__rel,type,
    bit_or3848514188828904588eg_rel: fun_Pr4799464687642144302_num_o ).

tff(sy_c_Bit__Operations_Oring__bit__operations__class_Onot_001t__Code____Numeral__Ointeger,type,
    bit_ri7632146776885996613nteger: code_integer > code_integer ).

tff(sy_c_Bit__Operations_Oring__bit__operations__class_Onot_001t__Int__Oint,type,
    bit_ri7919022796975470100ot_int: fun_int_int ).

tff(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Code____Numeral__Ointeger,type,
    bit_ri6519982836138164636nteger: ( nat * code_integer ) > code_integer ).

tff(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Int__Oint,type,
    bit_ri631733984087533419it_int: nat > fun_int_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Code____Numeral__Ointeger,type,
    bit_se3949692690581998587nteger: ( code_integer * code_integer ) > code_integer ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Int__Oint,type,
    bit_se725231765392027082nd_int: fun_int_fun_int_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Nat__Onat,type,
    bit_se727722235901077358nd_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Int__Oint,type,
    bit_se8568078237143864401it_int: nat > fun_int_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Nat__Onat,type,
    bit_se8570568707652914677it_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Code____Numeral__Ointeger,type,
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    sup_sup_set_set_int: ( set_set_int * set_set_int ) > set_set_int ).

tff(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    sup_sup_set_set_nat: ( set_set_nat * set_set_nat ) > set_set_nat ).

tff(sy_c_Limits_OBfun_001t__Nat__Onat_001t__Real__Oreal,type,
    bfun_nat_real: ( fun_nat_real * filter_nat ) > $o ).

tff(sy_c_Limits_Oat__infinity_001t__Real__Oreal,type,
    at_infinity_real: filter_real ).

tff(sy_c_List_Oappend_001t__Int__Oint,type,
    append_int: ( list_int * list_int ) > list_int ).

tff(sy_c_List_Oappend_001t__Nat__Onat,type,
    append_nat: ( list_nat * list_nat ) > list_nat ).

tff(sy_c_List_Odistinct_001t__Int__Oint,type,
    distinct_int: fun_list_int_o ).

tff(sy_c_List_Odistinct_001t__Nat__Onat,type,
    distinct_nat: fun_list_nat_o ).

tff(sy_c_List_Odrop_001t__Nat__Onat,type,
    drop_nat: ( nat * list_nat ) > list_nat ).

tff(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Nat__Onat,type,
    linord2614967742042102400et_nat: set_nat > list_nat ).

tff(sy_c_List_Olist_OCons_001t__Int__Oint,type,
    cons_int: ( int * list_int ) > list_int ).

tff(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
    cons_nat: ( nat * list_nat ) > list_nat ).

tff(sy_c_List_Olist_ONil_001t__Int__Oint,type,
    nil_int: list_int ).

tff(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
    nil_nat: list_nat ).

tff(sy_c_List_Olist_Ohd_001t__Nat__Onat,type,
    hd_nat: list_nat > nat ).

tff(sy_c_List_Olist_Omap_001_Eo_001_Eo,type,
    map_o_o: ( fun_o_o * list_o ) > list_o ).

tff(sy_c_List_Olist_Omap_001_Eo_001t__Int__Oint,type,
    map_o_int: ( fun_o_int * list_o ) > list_int ).

tff(sy_c_List_Olist_Omap_001_Eo_001t__Nat__Onat,type,
    map_o_nat: ( fun_o_nat * list_o ) > list_nat ).

tff(sy_c_List_Olist_Omap_001_Eo_001t__VEBT____Definitions__OVEBT,type,
    map_o_VEBT_VEBT: ( fun_o_VEBT_VEBT * list_o ) > list_VEBT_VEBT ).

tff(sy_c_List_Olist_Omap_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
    map_complex_complex: ( fun_complex_complex * list_complex ) > list_complex ).

tff(sy_c_List_Olist_Omap_001t__Int__Oint_001_Eo,type,
    map_int_o: ( fun_int_o * list_int ) > list_o ).

tff(sy_c_List_Olist_Omap_001t__Int__Oint_001t__Int__Oint,type,
    map_int_int: ( fun_int_int * list_int ) > list_int ).

tff(sy_c_List_Olist_Omap_001t__Int__Oint_001t__VEBT____Definitions__OVEBT,type,
    map_int_VEBT_VEBT: ( fun_int_VEBT_VEBT * list_int ) > list_VEBT_VEBT ).

tff(sy_c_List_Olist_Omap_001t__Nat__Onat_001_Eo,type,
    map_nat_o: ( fun_nat_o * list_nat ) > list_o ).

tff(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Int__Oint,type,
    map_nat_int: ( fun_nat_int * list_nat ) > list_int ).

tff(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
    map_nat_nat: ( fun_nat_nat * list_nat ) > list_nat ).

tff(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
    map_nat_VEBT_VEBT: ( fun_nat_VEBT_VEBT * list_nat ) > list_VEBT_VEBT ).

tff(sy_c_List_Olist_Omap_001t__Real__Oreal_001t__Real__Oreal,type,
    map_real_real: ( fun_real_real * list_real ) > list_real ).

tff(sy_c_List_Olist_Omap_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
    map_set_nat_set_nat: ( fun_set_nat_set_nat * list_set_nat ) > list_set_nat ).

tff(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001_Eo,type,
    map_VEBT_VEBT_o: ( fun_VEBT_VEBT_o * list_VEBT_VEBT ) > list_o ).

tff(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
    map_VEBT_VEBT_int: ( fun_VEBT_VEBT_int * list_VEBT_VEBT ) > list_int ).

tff(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    map_VEBT_VEBT_nat: ( fun_VEBT_VEBT_nat * list_VEBT_VEBT ) > list_nat ).

tff(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
    map_VE8901447254227204932T_VEBT: ( fun_VE3165209603783583043T_VEBT * list_VEBT_VEBT ) > list_VEBT_VEBT ).

tff(sy_c_List_Olist_Oset_001_Eo,type,
    set_o2: list_o > set_o ).

tff(sy_c_List_Olist_Oset_001t__Complex__Ocomplex,type,
    set_complex2: list_complex > set_complex ).

tff(sy_c_List_Olist_Oset_001t__Int__Oint,type,
    set_int2: list_int > set_int ).

tff(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
    set_nat2: list_nat > set_nat ).

tff(sy_c_List_Olist_Oset_001t__Real__Oreal,type,
    set_real2: list_real > set_real ).

tff(sy_c_List_Olist_Oset_001t__Set__Oset_It__Nat__Onat_J,type,
    set_set_nat2: list_set_nat > set_set_nat ).

tff(sy_c_List_Olist_Oset_001t__VEBT____Definitions__OVEBT,type,
    set_VEBT_VEBT2: list_VEBT_VEBT > set_VEBT_VEBT ).

tff(sy_c_List_Olist_Osize__list_001t__VEBT____Definitions__OVEBT,type,
    size_list_VEBT_VEBT: ( fun_VEBT_VEBT_nat * list_VEBT_VEBT ) > nat ).

tff(sy_c_List_Olist_Otl_001t__Nat__Onat,type,
    tl_nat: list_nat > list_nat ).

tff(sy_c_List_Onth_001_Eo,type,
    nth_o: list_o > fun_nat_o ).

tff(sy_c_List_Onth_001t__Code____Numeral__Ointeger,type,
    nth_Code_integer: list_Code_integer > fun_nat_Code_integer ).

tff(sy_c_List_Onth_001t__Complex__Ocomplex,type,
    nth_complex: list_complex > fun_nat_complex ).

tff(sy_c_List_Onth_001t__Int__Oint,type,
    nth_int: list_int > fun_nat_int ).

tff(sy_c_List_Onth_001t__Nat__Onat,type,
    nth_nat: list_nat > fun_nat_nat ).

tff(sy_c_List_Onth_001t__Num__Onum,type,
    nth_num: list_num > fun_nat_num ).

tff(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
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tff(sy_c_List_Onth_001t__Real__Oreal,type,
    nth_real: list_real > fun_nat_real ).

tff(sy_c_List_Onth_001t__Set__Oset_It__Nat__Onat_J,type,
    nth_set_nat: ( list_set_nat * nat ) > set_nat ).

tff(sy_c_List_Onth_001t__VEBT____Definitions__OVEBT,type,
    nth_VEBT_VEBT: list_VEBT_VEBT > fun_nat_VEBT_VEBT ).

tff(sy_c_List_Oproduct_001_Eo_001_Eo,type,
    product_o_o: ( list_o * list_o ) > list_P4002435161011370285od_o_o ).

tff(sy_c_List_Oproduct_001_Eo_001t__Int__Oint,type,
    product_o_int: ( list_o * list_int ) > list_P3795440434834930179_o_int ).

tff(sy_c_List_Oproduct_001_Eo_001t__Nat__Onat,type,
    product_o_nat: ( list_o * list_nat ) > list_P6285523579766656935_o_nat ).

tff(sy_c_List_Oproduct_001_Eo_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Oproduct_001t__Code____Numeral__Ointeger_001_Eo,type,
    produc3607205314601156340eger_o: ( list_Code_integer * list_o ) > list_P8526636022914148096eger_o ).

tff(sy_c_List_Oproduct_001t__Nat__Onat_001_Eo,type,
    product_nat_o: ( list_nat * list_o ) > list_P7333126701944960589_nat_o ).

tff(sy_c_List_Oproduct_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
    produc7156399406898700509T_VEBT: ( list_nat * list_VEBT_VEBT ) > list_P5647936690300460905T_VEBT ).

tff(sy_c_List_Oproduct_001t__Num__Onum_001t__Num__Onum,type,
    product_num_num: ( list_num * list_num ) > list_P3744719386663036955um_num ).

tff(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001_Eo,type,
    product_VEBT_VEBT_o: ( list_VEBT_VEBT * list_o ) > list_P3126845725202233233VEBT_o ).

tff(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
    produc7292646706713671643BT_int: ( list_VEBT_VEBT * list_int ) > list_P4547456442757143711BT_int ).

tff(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    produc7295137177222721919BT_nat: ( list_VEBT_VEBT * list_nat ) > list_P7037539587688870467BT_nat ).

tff(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Oremdups_001t__Nat__Onat,type,
    remdups_nat: list_nat > list_nat ).

tff(sy_c_List_Oreplicate_001_Eo,type,
    replicate_o: ( nat * $o ) > list_o ).

tff(sy_c_List_Oreplicate_001t__Complex__Ocomplex,type,
    replicate_complex: ( nat * complex ) > list_complex ).

tff(sy_c_List_Oreplicate_001t__Int__Oint,type,
    replicate_int: ( nat * int ) > list_int ).

tff(sy_c_List_Oreplicate_001t__Nat__Onat,type,
    replicate_nat: ( nat * nat ) > list_nat ).

tff(sy_c_List_Oreplicate_001t__Real__Oreal,type,
    replicate_real: ( nat * real ) > list_real ).

tff(sy_c_List_Oreplicate_001t__Set__Oset_It__Nat__Onat_J,type,
    replicate_set_nat: ( nat * set_nat ) > list_set_nat ).

tff(sy_c_List_Oreplicate_001t__VEBT____Definitions__OVEBT,type,
    replicate_VEBT_VEBT: ( nat * vEBT_VEBT ) > list_VEBT_VEBT ).

tff(sy_c_List_Osorted__wrt_001t__Int__Oint,type,
    sorted_wrt_int: fun_int_fun_int_o > fun_list_int_o ).

tff(sy_c_List_Osorted__wrt_001t__Nat__Onat,type,
    sorted_wrt_nat: fun_nat_fun_nat_o > fun_list_nat_o ).

tff(sy_c_List_Otake_001_Eo,type,
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tff(sy_c_List_Otake_001t__Complex__Ocomplex,type,
    take_complex: ( nat * list_complex ) > list_complex ).

tff(sy_c_List_Otake_001t__Int__Oint,type,
    take_int: ( nat * list_int ) > list_int ).

tff(sy_c_List_Otake_001t__Nat__Onat,type,
    take_nat: ( nat * list_nat ) > list_nat ).

tff(sy_c_List_Otake_001t__Real__Oreal,type,
    take_real: ( nat * list_real ) > list_real ).

tff(sy_c_List_Otake_001t__Set__Oset_It__Nat__Onat_J,type,
    take_set_nat: ( nat * list_set_nat ) > list_set_nat ).

tff(sy_c_List_Otake_001t__VEBT____Definitions__OVEBT,type,
    take_VEBT_VEBT: ( nat * list_VEBT_VEBT ) > list_VEBT_VEBT ).

tff(sy_c_List_Oupt,type,
    upt: ( nat * nat ) > list_nat ).

tff(sy_c_List_Oupto,type,
    upto: ( int * int ) > list_int ).

tff(sy_c_List_Oupto__aux,type,
    upto_aux: ( int * int * list_int ) > list_int ).

tff(sy_c_List_Oupto__rel,type,
    upto_rel: fun_Pr9018117161326028078_int_o ).

tff(sy_c_Nat_OSuc,type,
    suc: fun_nat_nat ).

tff(sy_c_Nat_Ocompow_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J,type,
    compow_fun_nat_nat: ( nat * fun_nat_nat ) > fun_nat_nat ).

tff(sy_c_Nat_Onat_Ocase__nat_001_Eo,type,
    case_nat_o: ( $o * fun_nat_o ) > fun_nat_o ).

tff(sy_c_Nat_Onat_Ocase__nat_001t__Nat__Onat,type,
    case_nat_nat: ( nat * fun_nat_nat ) > fun_nat_nat ).

tff(sy_c_Nat_Onat_Ocase__nat_001t__Option__Ooption_It__Num__Onum_J,type,
    case_nat_option_num: ( option_num * fun_nat_option_num * nat ) > option_num ).

tff(sy_c_Nat_Onat_Opred,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Code____Numeral__Ointeger,type,
    semiri4939895301339042750nteger: fun_nat_Code_integer ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Extended____Nat__Oenat,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
    semiri1314217659103216013at_int: fun_nat_int ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
    semiri1316708129612266289at_nat: fun_nat_nat ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Rat__Orat,type,
    semiri681578069525770553at_rat: fun_nat_rat ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
    semiri5074537144036343181t_real: fun_nat_real ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Complex__Ocomplex,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Int__Oint,type,
    semiri8420488043553186161ux_int: ( fun_int_int * nat ) > fun_int_int ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Nat__Onat,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Rat__Orat,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Real__Oreal,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_I_Eo_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Num__Onum_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Num__Onum_J,type,
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tff(sy_c_Nat__Bijection_Oprod__decode__aux,type,
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tff(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
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tff(sy_c_Nat__Bijection_Oset__decode,type,
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tff(sy_c_Nat__Bijection_Oset__encode,type,
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tff(sy_c_Nat__Bijection_Otriangle,type,
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tff(sy_c_NthRoot_Oroot,type,
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tff(sy_c_NthRoot_Osqrt,type,
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tff(sy_c_Num_OBitM,type,
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tff(sy_c_Num_Oinc,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Real__Oreal,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Num_Onum_OBit0,type,
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tff(sy_c_Num_Onum_OBit1,type,
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tff(sy_c_Num_Onum_OOne,type,
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tff(sy_c_Num_Onum_Osize__num,type,
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tff(sy_c_Num_Onum__of__nat,type,
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tff(sy_c_Num_Opow,type,
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tff(sy_c_Num_Opred__numeral,type,
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tff(sy_c_Option_Ooption_ONone_001t__Num__Onum,type,
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tff(sy_c_Option_Ooption_OSome_001t__Num__Onum,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Rat__Orat,type,
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    order_Greatest_nat: fun_nat_o > nat ).

tff(sy_c_Orderings_Oorder__class_Oantimono_001t__Nat__Onat_001t__Real__Oreal,type,
    order_9091379641038594480t_real: fun_nat_real > $o ).

tff(sy_c_Orderings_Oorder__class_Omono_001t__Nat__Onat_001t__Nat__Onat,type,
    order_mono_nat_nat: fun_nat_nat > $o ).

tff(sy_c_Orderings_Oorder__class_Omono_001t__Nat__Onat_001t__Real__Oreal,type,
    order_mono_nat_real: fun_nat_real > $o ).

tff(sy_c_Orderings_Oorder__class_Omono_001t__Real__Oreal_001t__Real__Oreal,type,
    order_mono_real_real: fun_real_real > $o ).

tff(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Nat__Onat_001t__Nat__Onat,type,
    order_5726023648592871131at_nat: fun_nat_nat > $o ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Extended____Nat__Oenat,type,
    top_to3028658606643905974d_enat: extended_enat ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_Eo_J,type,
    top_top_set_o: set_o ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Int__Oint_J,type,
    top_top_set_int: set_int ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
    top_top_set_nat: set_nat ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Ounit_J,type,
    top_to1996260823553986621t_unit: set_Product_unit ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
    top_top_set_real: set_real ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__String__Ochar_J,type,
    top_top_set_char: set_char ).

tff(sy_c_Power_Opower__class_Opower_001t__Code____Numeral__Ointeger,type,
    power_8256067586552552935nteger: code_integer > fun_nat_Code_integer ).

tff(sy_c_Power_Opower__class_Opower_001t__Complex__Ocomplex,type,
    power_power_complex: fun_co3258106519541737638omplex ).

tff(sy_c_Power_Opower__class_Opower_001t__Extended____Nat__Oenat,type,
    power_8040749407984259932d_enat: extended_enat > fun_na8797796813480665199d_enat ).

tff(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
    power_power_int: fun_int_fun_nat_int ).

tff(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
    power_power_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Power_Opower__class_Opower_001t__Rat__Orat,type,
    power_power_rat: rat > fun_nat_rat ).

tff(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
    power_power_real: fun_re4238438383274954274t_real ).

tff(sy_c_Product__Type_OPair_001_Eo_001_Eo,type,
    product_Pair_o_o: ( $o * $o ) > product_prod_o_o ).

tff(sy_c_Product__Type_OPair_001_Eo_001t__Int__Oint,type,
    product_Pair_o_int: ( $o * int ) > product_prod_o_int ).

tff(sy_c_Product__Type_OPair_001_Eo_001t__Nat__Onat,type,
    product_Pair_o_nat: ( $o * nat ) > product_prod_o_nat ).

tff(sy_c_Product__Type_OPair_001_Eo_001t__VEBT____Definitions__OVEBT,type,
    produc2982872950893828659T_VEBT: ( $o * vEBT_VEBT ) > produc2504756804600209347T_VEBT ).

tff(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001_Eo,type,
    produc6677183202524767010eger_o: fun_Co4602396739301704407eger_o ).

tff(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
    produc1086072967326762835nteger: ( code_integer * code_integer ) > produc8923325533196201883nteger ).

tff(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Int__Oint,type,
    product_Pair_int_int: fun_in1696471567231710611nt_int ).

tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
    product_Pair_nat_nat: fun_na4138050610385705875at_nat ).

tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Num__Onum,type,
    product_Pair_nat_num: fun_na157340750583166227at_num ).

tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    produc487386426758144856at_nat: ( nat * product_prod_nat_nat ) > produc7248412053542808358at_nat ).

tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
    produc1195630363706982562at_num: ( nat * product_prod_nat_num ) > produc2963631642982155120at_num ).

tff(sy_c_Product__Type_OPair_001t__Num__Onum_001t__Num__Onum,type,
    product_Pair_num_num: fun_nu4037332675543032467um_num ).

tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001_Eo,type,
    produc8721562602347293563VEBT_o: ( vEBT_VEBT * $o ) > produc334124729049499915VEBT_o ).

tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Extended____Nat__Oenat,type,
    produc581526299967858633d_enat: ( vEBT_VEBT * extended_enat ) > produc7272778201969148633d_enat ).

tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
    produc736041933913180425BT_int: ( vEBT_VEBT * int ) > produc4894624898956917775BT_int ).

tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    produc738532404422230701BT_nat: ( vEBT_VEBT * nat ) > produc9072475918466114483BT_nat ).

tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
    produc537772716801021591T_VEBT: ( vEBT_VEBT * vEBT_VEBT ) > produc8243902056947475879T_VEBT ).

tff(sy_c_Product__Type_OPair_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    produc2818902965508115497at_nat: ( fun_nat_fun_nat_nat * produc7248412053542808358at_nat ) > produc9217583074051984497at_nat ).

tff(sy_c_Product__Type_OPair_001t__fun_It__Nat__Onat_Mt__fun_It__Num__Onum_Mt__Num__Onum_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J_J,type,
    produc460779904311030791at_num: ( fun_nat_fun_num_num * produc2963631642982155120at_num ) > produc8114805097830739791at_num ).

tff(sy_c_Product__Type_Oapsnd_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
    produc6499014454317279255nteger: fun_fu4800041728187187127nteger ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001_Eo_001_Eo,type,
    produc7828578312038201481er_o_o: fun_Co480733329897759773un_o_o > fun_Pr3290071492984375169er_o_o ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001_Eo_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
    produc8281089916252573759eger_o: ( fun_Co4602396739301704407eger_o * produc6271795597528267376eger_o ) > produc6271795597528267376eger_o ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001_Eo_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    produc1043322548047392435omplex: ( fun_Co5015990272074005067omplex * produc6271795597528267376eger_o ) > set_complex ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001_Eo_001t__Set__Oset_It__Int__Oint_J,type,
    produc1253318751659547953et_int: ( fun_Co2587678820519358153et_int * produc6271795597528267376eger_o ) > set_int ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001_Eo_001t__Set__Oset_It__Nat__Onat_J,type,
    produc5431169771168744661et_nat: ( fun_Co7401227239307276909et_nat * produc6271795597528267376eger_o ) > set_nat ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001_Eo_001t__Set__Oset_It__Real__Oreal_J,type,
    produc242741666403216561t_real: ( fun_Co5368275265270938953t_real * produc6271795597528267376eger_o ) > set_real ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001_Eo_001t__String__Ochar,type,
    produc4188289175737317920o_char: ( fun_Co681828205884031672o_char * produc6271795597528267376eger_o ) > char ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Int__Oint,type,
    produc1553301316500091796er_int: ( fun_Co1092983284983208416er_int * produc8923325533196201883nteger ) > int ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
    produc1555791787009142072er_nat: ( fun_Co3583066429914935172er_nat * produc8923325533196201883nteger ) > nat ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Num__Onum,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
    produc9125791028180074456eger_o: ( fun_Co3544761417380347684eger_o * produc8923325533196201883nteger ) > produc6271795597528267376eger_o ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    produc6916734918728496179nteger: ( fun_Co2470855966376689427nteger * produc8923325533196201883nteger ) > produc8923325533196201883nteger ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Complex__Ocomplex_001t__Complex__Ocomplex_001_Eo,type,
    produc6771430404735790350plex_o: fun_co1148497455491383662plex_o > fun_Pr7716844367077540446plex_o ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001_Eo,type,
    produc4947309494688390418_int_o: fun_int_fun_int_o > fun_Pr3238374608701176162_int_o ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
    produc8211389475949308722nt_int: fun_int_fun_int_int > fun_Pr982841511892225912nt_int ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    produc4245557441103728435nt_int: fun_in1696471567231710611nt_int > fun_Pr2478283820680522243nt_int ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_Eo,type,
    produc6081775807080527818_nat_o: fun_nat_fun_nat_o > fun_Pr4796943543331071514_nat_o ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Complex__Ocomplex,type,
    produc1917071388513777916omplex: ( fun_na1142814060743658952omplex * product_prod_nat_nat ) > complex ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Extended____Nat__Oenat,type,
    produc2676513652042109336d_enat: ( fun_na7542918316964655096d_enat * product_prod_nat_nat ) > extended_enat ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Int__Oint,type,
    produc6840382203811409530at_int: ( fun_nat_fun_nat_int * product_prod_nat_nat ) > int ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
    produc6842872674320459806at_nat: fun_nat_fun_nat_nat > fun_Pr2602117396471556196at_nat ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    produc2626176000494625587at_nat: fun_na4138050610385705875at_nat > fun_Pr2118491707810114563at_nat ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Rat__Orat,type,
    produc6207742614233964070at_rat: ( fun_nat_fun_nat_rat * product_prod_nat_nat ) > rat ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Real__Oreal,type,
    produc1703576794950452218t_real: ( fun_nat_fun_nat_real * product_prod_nat_nat ) > real ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
    produc6395135316500328296_nat_o: fun_na5914473681949881652_nat_o > fun_Pr2028404973800837934_nat_o ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    produc4528391901023682385at_nat: fun_na1668100129766719005at_nat > fun_Pr2477239776880934807at_nat ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Num__Onum_001_Eo,type,
    produc4927758841916487424_num_o: fun_nat_fun_num_o > fun_Pr5555124106300582480_num_o ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Num__Onum_001t__Option__Ooption_It__Num__Onum_J,type,
    produc478579273971653890on_num: ( fun_na7595719129879163854on_num * product_prod_nat_num ) > option_num ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Num__Onum_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
    produc49306077274653107at_num: ( fun_na157340750583166227at_num * product_prod_nat_num ) > product_prod_nat_num ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Num__Onum_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    produc6231982587499038204omplex: ( fun_na3925014923085203400omplex * product_prod_nat_num ) > set_complex ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Num__Onum_001t__Set__Oset_It__Real__Oreal_J,type,
    produc1435849484188172666t_real: ( fun_na3666575854997362246t_real * product_prod_nat_num ) > set_real ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Num__Onum_001t__Num__Onum_001_Eo,type,
    produc5703948589228662326_num_o: fun_num_fun_num_o > fun_Pr6331313853612757382_num_o ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Num__Onum_001t__Num__Onum_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
    produc64540874165560627um_num: ( fun_nu4037332675543032467um_num * product_prod_num_num ) > product_prod_num_num ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Num__Onum_001t__Num__Onum_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    produc2866383454006189126omplex: ( fun_nu9041852840018610706omplex * product_prod_num_num ) > set_complex ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Num__Onum_001t__Num__Onum_001t__Set__Oset_It__Int__Oint_J,type,
    produc6406642877701697732et_int: ( fun_nu4940571614878387344et_int * product_prod_num_num ) > set_int ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Num__Onum_001t__Num__Onum_001t__Set__Oset_It__Nat__Onat_J,type,
    produc1361121860356118632et_nat: ( fun_nu530747996811530292et_nat * product_prod_num_num ) > set_nat ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Num__Onum_001t__Num__Onum_001t__Set__Oset_It__Real__Oreal_J,type,
    produc8296048397933160132t_real: ( fun_nu4698968662094489488t_real * product_prod_num_num ) > set_real ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Real__Oreal_001t__Real__Oreal_001_Eo,type,
    produc5414030515140494994real_o: fun_real_fun_real_o > fun_Pr4814023175418636514real_o ).

tff(sy_c_Product__Type_Oprod_Ofst_001t__Int__Oint_001t__Int__Oint,type,
    product_fst_int_int: fun_Pr982841511892225912nt_int ).

tff(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Nat__Onat,type,
    product_fst_nat_nat: fun_Pr2602117396471556196at_nat ).

tff(sy_c_Product__Type_Oprod_Osnd_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Product__Type_Oprod_Osnd_001t__Int__Oint_001t__Int__Oint,type,
    product_snd_int_int: fun_Pr982841511892225912nt_int ).

tff(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Nat__Onat,type,
    product_snd_nat_nat: fun_Pr2602117396471556196at_nat ).

tff(sy_c_Rat_OFrct,type,
    frct: product_prod_int_int > rat ).

tff(sy_c_Rat_Onormalize,type,
    normalize: fun_Pr2478283820680522243nt_int ).

tff(sy_c_Rat_Oof__int,type,
    of_int: fun_int_rat ).

tff(sy_c_Rat_Oquotient__of,type,
    quotient_of: rat > product_prod_int_int ).

tff(sy_c_Real__Vector__Spaces_OReals_001t__Complex__Ocomplex,type,
    real_V2521375963428798218omplex: set_complex ).

tff(sy_c_Real__Vector__Spaces_Obounded__linear_001t__Real__Oreal_001t__Real__Oreal,type,
    real_V5970128139526366754l_real: fun_real_real > $o ).

tff(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Complex__Ocomplex,type,
    real_V3694042436643373181omplex: complex > fun_complex_real ).

tff(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Real__Oreal,type,
    real_V975177566351809787t_real: real > fun_real_real ).

tff(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex,type,
    real_V1022390504157884413omplex: fun_complex_real ).

tff(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal,type,
    real_V7735802525324610683m_real: fun_real_real ).

tff(sy_c_Real__Vector__Spaces_Oof__real_001t__Complex__Ocomplex,type,
    real_V4546457046886955230omplex: fun_real_complex ).

tff(sy_c_Real__Vector__Spaces_Oof__real_001t__Real__Oreal,type,
    real_V1803761363581548252l_real: fun_real_real ).

tff(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Complex__Ocomplex,type,
    real_V2046097035970521341omplex: real > fun_complex_complex ).

tff(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Real__Oreal,type,
    real_V1485227260804924795R_real: real > fun_real_real ).

tff(sy_c_Rings_Odivide__class_Odivide_001t__Code____Numeral__Ointeger,type,
    divide6298287555418463151nteger: ( code_integer * code_integer ) > code_integer ).

tff(sy_c_Rings_Odivide__class_Odivide_001t__Complex__Ocomplex,type,
    divide1717551699836669952omplex: complex > fun_complex_complex ).

tff(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
    divide_divide_int: fun_int_fun_int_int ).

tff(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
    divide_divide_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Rings_Odivide__class_Odivide_001t__Rat__Orat,type,
    divide_divide_rat: rat > fun_rat_rat ).

tff(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
    divide_divide_real: real > fun_real_real ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Rings_Odvd__class_Odvd_001t__Complex__Ocomplex,type,
    dvd_dvd_complex: complex > fun_complex_o ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
    dvd_dvd_int: fun_int_fun_int_o ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
    dvd_dvd_nat: fun_nat_fun_nat_o ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Rat__Orat,type,
    dvd_dvd_rat: fun_rat_fun_rat_o ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Real__Oreal,type,
    dvd_dvd_real: fun_real_fun_real_o ).

tff(sy_c_Rings_Omodulo__class_Omodulo_001t__Code____Numeral__Ointeger,type,
    modulo364778990260209775nteger: ( code_integer * code_integer ) > code_integer ).

tff(sy_c_Rings_Omodulo__class_Omodulo_001t__Int__Oint,type,
    modulo_modulo_int: fun_int_fun_int_int ).

tff(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
    modulo_modulo_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Code____Numeral__Ointeger,type,
    zero_n356916108424825756nteger: $o > code_integer ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Complex__Ocomplex,type,
    zero_n1201886186963655149omplex: $o > complex ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Extended____Nat__Oenat,type,
    zero_n1046097342994218471d_enat: $o > extended_enat ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Int__Oint,type,
    zero_n2684676970156552555ol_int: fun_o_int ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Nat__Onat,type,
    zero_n2687167440665602831ol_nat: fun_o_nat ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Rat__Orat,type,
    zero_n2052037380579107095ol_rat: $o > rat ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Real__Oreal,type,
    zero_n3304061248610475627l_real: $o > real ).

tff(sy_c_Series_Osuminf_001t__Complex__Ocomplex,type,
    suminf_complex: fun_nat_complex > complex ).

tff(sy_c_Series_Osuminf_001t__Int__Oint,type,
    suminf_int: fun_nat_int > int ).

tff(sy_c_Series_Osuminf_001t__Nat__Onat,type,
    suminf_nat: fun_nat_nat > nat ).

tff(sy_c_Series_Osuminf_001t__Real__Oreal,type,
    suminf_real: fun_nat_real > real ).

tff(sy_c_Series_Osummable_001t__Complex__Ocomplex,type,
    summable_complex: fun_nat_complex > $o ).

tff(sy_c_Series_Osummable_001t__Int__Oint,type,
    summable_int: fun_nat_int > $o ).

tff(sy_c_Series_Osummable_001t__Nat__Onat,type,
    summable_nat: fun_nat_nat > $o ).

tff(sy_c_Series_Osummable_001t__Real__Oreal,type,
    summable_real: fun_nat_real > $o ).

tff(sy_c_Series_Osums_001t__Complex__Ocomplex,type,
    sums_complex: fun_nat_complex > fun_complex_o ).

tff(sy_c_Series_Osums_001t__Int__Oint,type,
    sums_int: fun_nat_int > fun_int_o ).

tff(sy_c_Series_Osums_001t__Nat__Onat,type,
    sums_nat: fun_nat_nat > fun_nat_o ).

tff(sy_c_Series_Osums_001t__Real__Oreal,type,
    sums_real: fun_nat_real > fun_real_o ).

tff(sy_c_Set_OCollect_001t__Code____Numeral__Ointeger,type,
    collect_Code_integer: fun_Code_integer_o > set_Code_integer ).

tff(sy_c_Set_OCollect_001t__Complex__Ocomplex,type,
    collect_complex: fun_complex_o > set_complex ).

tff(sy_c_Set_OCollect_001t__Int__Oint,type,
    collect_int: fun_int_o > set_int ).

tff(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
    collect_list_nat: fun_list_nat_o > set_list_nat ).

tff(sy_c_Set_OCollect_001t__Nat__Onat,type,
    collect_nat: fun_nat_o > set_nat ).

tff(sy_c_Set_OCollect_001t__Num__Onum,type,
    collect_num: fun_num_o > set_num ).

tff(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
    collec8663557070575231912omplex: fun_Pr7716844367077540446plex_o > set_Pr5085853215250843933omplex ).

tff(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    collec213857154873943460nt_int: fun_Pr3238374608701176162_int_o > set_Pr958786334691620121nt_int ).

tff(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
    collec3799799289383736868l_real: fun_Pr4814023175418636514real_o > set_Pr6218003697084177305l_real ).

tff(sy_c_Set_OCollect_001t__Rat__Orat,type,
    collect_rat: fun_rat_o > set_rat ).

tff(sy_c_Set_OCollect_001t__Real__Oreal,type,
    collect_real: fun_real_o > set_real ).

tff(sy_c_Set_OCollect_001t__Set__Oset_It__Int__Oint_J,type,
    collect_set_int: fun_set_int_o > set_set_int ).

tff(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
    collect_set_nat: fun_set_nat_o > set_set_nat ).

tff(sy_c_Set_OPow_001t__Nat__Onat,type,
    pow_nat: set_nat > set_set_nat ).

tff(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_I_Eo_J,type,
    image_o_set_o: ( fun_o_set_o * set_o ) > set_set_o ).

tff(sy_c_Set_Oimage_001t__Int__Oint_001t__Int__Oint,type,
    image_int_int: ( fun_int_int * set_int ) > set_int ).

tff(sy_c_Set_Oimage_001t__Int__Oint_001t__Set__Oset_It__Int__Oint_J,type,
    image_int_set_int: ( fun_int_set_int * set_int ) > set_set_int ).

tff(sy_c_Set_Oimage_001t__Nat__Onat_001t__Int__Oint,type,
    image_nat_int: ( fun_nat_int * set_nat ) > set_int ).

tff(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
    image_nat_nat: fun_nat_nat > fun_set_nat_set_nat ).

tff(sy_c_Set_Oimage_001t__Nat__Onat_001t__Real__Oreal,type,
    image_nat_real: ( fun_nat_real * set_nat ) > set_real ).

tff(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
    image_nat_set_nat: ( fun_nat_set_nat * set_nat ) > set_set_nat ).

tff(sy_c_Set_Oimage_001t__Nat__Onat_001t__String__Ochar,type,
    image_nat_char: ( fun_nat_char * set_nat ) > set_char ).

tff(sy_c_Set_Oimage_001t__Real__Oreal_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
    image_5971271580939081552omplex: ( fun_re3844104906058862192omplex * set_real ) > set_fi4554929511873752355omplex ).

tff(sy_c_Set_Oimage_001t__Real__Oreal_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
    image_2178119161166701260l_real: ( fun_re5512370375968561900l_real * set_real ) > set_fi7789364187291644575l_real ).

tff(sy_c_Set_Oimage_001t__Real__Oreal_001t__Real__Oreal,type,
    image_real_real: ( fun_real_real * set_real ) > set_real ).

tff(sy_c_Set_Oimage_001t__String__Ochar_001t__Nat__Onat,type,
    image_char_nat: ( fun_char_nat * set_char ) > set_nat ).

tff(sy_c_Set_Oinsert_001t__Int__Oint,type,
    insert_int: ( int * set_int ) > set_int ).

tff(sy_c_Set_Oinsert_001t__Nat__Onat,type,
    insert_nat: nat > fun_set_nat_set_nat ).

tff(sy_c_Set_Oinsert_001t__Real__Oreal,type,
    insert_real: ( real * set_real ) > set_real ).

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Complex__Ocomplex,type,
    set_fo1517530859248394432omplex: ( fun_na1591161823130732966omplex * nat * nat ) > fun_complex_complex ).

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Extended____Nat__Oenat,type,
    set_fo2538466533108834004d_enat: ( fun_na1154345429118027792d_enat * nat * nat * extended_enat ) > extended_enat ).

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Int__Oint,type,
    set_fo2581907887559384638at_int: ( fun_nat_fun_int_int * nat * nat ) > fun_int_int ).

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Nat__Onat,type,
    set_fo2584398358068434914at_nat: ( fun_nat_fun_nat_nat * nat ) > fun_nat_fun_nat_nat ).

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Num__Onum,type,
    set_fo8365102181078989356at_num: ( fun_nat_fun_num_num * nat ) > fun_nat_fun_num_num ).

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Rat__Orat,type,
    set_fo1949268297981939178at_rat: ( fun_nat_fun_rat_rat * nat * nat ) > fun_rat_rat ).

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Real__Oreal,type,
    set_fo3111899725591712190t_real: ( fun_na6724076748692969250l_real * nat ) > fun_na6724076748692969250l_real ).

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat__rel_001t__Nat__Onat,type,
    set_fo3699595496184130361el_nat: fun_Pr8463018485500633518_nat_o ).

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat__rel_001t__Num__Onum,type,
    set_fo256927282339908995el_num: fun_Pr7588637180273442094_num_o ).

tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Int__Oint,type,
    set_or1266510415728281911st_int: ( int * int ) > set_int ).

tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Nat__Onat,type,
    set_or1269000886237332187st_nat: ( nat * nat ) > set_nat ).

tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Real__Oreal,type,
    set_or1222579329274155063t_real: ( real * real ) > set_real ).

tff(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Int__Oint,type,
    set_or4662586982721622107an_int: ( int * int ) > set_int ).

tff(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
    set_or4665077453230672383an_nat: ( nat * nat ) > set_nat ).

tff(sy_c_Set__Interval_Oord__class_OatLeast_001t__Nat__Onat,type,
    set_ord_atLeast_nat: fun_nat_set_nat ).

tff(sy_c_Set__Interval_Oord__class_OatLeast_001t__Real__Oreal,type,
    set_ord_atLeast_real: real > set_real ).

tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Int__Oint,type,
    set_ord_atMost_int: int > set_int ).

tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
    set_ord_atMost_nat: fun_nat_set_nat ).

tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Num__Onum,type,
    set_ord_atMost_num: num > set_num ).

tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Rat__Orat,type,
    set_ord_atMost_rat: rat > set_rat ).

tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Real__Oreal,type,
    set_ord_atMost_real: real > set_real ).

tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Int__Oint_J,type,
    set_or58775011639299419et_int: set_int > set_set_int ).

tff(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Nat__Onat_J,type,
    set_or4236626031148496127et_nat: set_nat > set_set_nat ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Int__Oint,type,
    set_or6656581121297822940st_int: ( int * int ) > set_int ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Nat__Onat,type,
    set_or6659071591806873216st_nat: ( nat * nat ) > set_nat ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Int__Oint,type,
    set_or5832277885323065728an_int: ( int * int ) > set_int ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Nat__Onat,type,
    set_or5834768355832116004an_nat: ( nat * nat ) > set_nat ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Real__Oreal,type,
    set_or1633881224788618240n_real: ( real * real ) > set_real ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThan_001_Eo,type,
    set_or6416164934427428222Than_o: fun_o_set_o ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Int__Oint,type,
    set_or1207661135979820486an_int: fun_int_set_int ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Nat__Onat,type,
    set_or1210151606488870762an_nat: fun_nat_set_nat ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Real__Oreal,type,
    set_or5849166863359141190n_real: real > set_real ).

tff(sy_c_Set__Interval_Oord__class_OlessThan_001_Eo,type,
    set_ord_lessThan_o: fun_o_set_o ).

tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Int__Oint,type,
    set_ord_lessThan_int: fun_int_set_int ).

tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
    set_ord_lessThan_nat: fun_nat_set_nat ).

tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Real__Oreal,type,
    set_or5984915006950818249n_real: real > set_real ).

tff(sy_c_String_Oascii__of,type,
    ascii_of: char > char ).

tff(sy_c_String_Ochar_OChar,type,
    char2: ( $o * $o * $o * $o * $o * $o * $o ) > fun_o_char ).

tff(sy_c_String_Ochar__of__integer,type,
    char_of_integer: code_integer > char ).

tff(sy_c_String_Ocomm__semiring__1__class_Oof__char_001t__Nat__Onat,type,
    comm_s629917340098488124ar_nat: fun_char_nat ).

tff(sy_c_String_Ointeger__of__char,type,
    integer_of_char: char > code_integer ).

tff(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of_001t__Nat__Onat,type,
    unique3096191561947761185of_nat: fun_nat_char ).

tff(sy_c_Topological__Spaces_Ocontinuous_001t__Real__Oreal_001t__Real__Oreal,type,
    topolo4422821103128117721l_real: ( filter_real * fun_real_real ) > $o ).

tff(sy_c_Topological__Spaces_Ocontinuous__on_001t__Real__Oreal_001t__Real__Oreal,type,
    topolo5044208981011980120l_real: ( set_real * fun_real_real ) > $o ).

tff(sy_c_Topological__Spaces_Ogenerate__topology_001_Eo,type,
    topolo4667128019001906403logy_o: set_set_o > fun_set_o_o ).

tff(sy_c_Topological__Spaces_Ogenerate__topology_001t__Int__Oint,type,
    topolo1611008123915946401gy_int: set_set_int > fun_set_int_o ).

tff(sy_c_Topological__Spaces_Ogenerate__topology_001t__Nat__Onat,type,
    topolo1613498594424996677gy_nat: set_set_nat > fun_set_nat_o ).

tff(sy_c_Topological__Spaces_Omonoseq_001t__Code____Numeral__Ointeger,type,
    topolo2919662092509805066nteger: fun_nat_Code_integer > $o ).

tff(sy_c_Topological__Spaces_Omonoseq_001t__Int__Oint,type,
    topolo4899668324122417113eq_int: fun_nat_int > $o ).

tff(sy_c_Topological__Spaces_Omonoseq_001t__Nat__Onat,type,
    topolo4902158794631467389eq_nat: fun_nat_nat > $o ).

tff(sy_c_Topological__Spaces_Omonoseq_001t__Num__Onum,type,
    topolo1459490580787246023eq_num: fun_nat_num > $o ).

tff(sy_c_Topological__Spaces_Omonoseq_001t__Rat__Orat,type,
    topolo4267028734544971653eq_rat: fun_nat_rat > $o ).

tff(sy_c_Topological__Spaces_Omonoseq_001t__Real__Oreal,type,
    topolo6980174941875973593q_real: fun_nat_real > $o ).

tff(sy_c_Topological__Spaces_Omonoseq_001t__Set__Oset_It__Int__Oint_J,type,
    topolo3100542954746470799et_int: fun_nat_set_int > $o ).

tff(sy_c_Topological__Spaces_Oopen__class_Oopen_001_Eo,type,
    topolo9180104560040979295open_o: fun_set_o_o ).

tff(sy_c_Topological__Spaces_Oopen__class_Oopen_001t__Complex__Ocomplex,type,
    topolo4110288021797289639omplex: set_complex > $o ).

tff(sy_c_Topological__Spaces_Oopen__class_Oopen_001t__Int__Oint,type,
    topolo4325760605701065253en_int: fun_set_int_o ).

tff(sy_c_Topological__Spaces_Oopen__class_Oopen_001t__Nat__Onat,type,
    topolo4328251076210115529en_nat: fun_set_nat_o ).

tff(sy_c_Topological__Spaces_Oopen__class_Oopen_001t__Real__Oreal,type,
    topolo4860482606490270245n_real: set_real > $o ).

tff(sy_c_Topological__Spaces_Otopological__space__class_Oat__within_001t__Real__Oreal,type,
    topolo2177554685111907308n_real: ( real * set_real ) > filter_real ).

tff(sy_c_Topological__Spaces_Otopological__space__class_Onhds_001t__Real__Oreal,type,
    topolo2815343760600316023s_real: real > filter_real ).

tff(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy_001t__Complex__Ocomplex,type,
    topolo6517432010174082258omplex: fun_nat_complex > $o ).

tff(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy_001t__Real__Oreal,type,
    topolo4055970368930404560y_real: fun_nat_real > $o ).

tff(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Complex__Ocomplex,type,
    topolo896644834953643431omplex: filter6041513312241820739omplex ).

tff(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Real__Oreal,type,
    topolo1511823702728130853y_real: filter2146258269922977983l_real ).

tff(sy_c_Transcendental_Oarccos,type,
    arccos: fun_real_real ).

tff(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
    arcosh_real: fun_real_real ).

tff(sy_c_Transcendental_Oarcsin,type,
    arcsin: fun_real_real ).

tff(sy_c_Transcendental_Oarctan,type,
    arctan: fun_real_real ).

tff(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
    arsinh_real: fun_real_real ).

tff(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
    artanh_real: fun_real_real ).

tff(sy_c_Transcendental_Ocos_001t__Complex__Ocomplex,type,
    cos_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Ocos_001t__Real__Oreal,type,
    cos_real: fun_real_real ).

tff(sy_c_Transcendental_Ocos__coeff,type,
    cos_coeff: fun_nat_real ).

tff(sy_c_Transcendental_Ocosh_001t__Complex__Ocomplex,type,
    cosh_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Ocosh_001t__Real__Oreal,type,
    cosh_real: fun_real_real ).

tff(sy_c_Transcendental_Ocot_001t__Complex__Ocomplex,type,
    cot_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Ocot_001t__Real__Oreal,type,
    cot_real: fun_real_real ).

tff(sy_c_Transcendental_Odiffs_001t__Code____Numeral__Ointeger,type,
    diffs_Code_integer: fun_nat_Code_integer > fun_nat_Code_integer ).

tff(sy_c_Transcendental_Odiffs_001t__Complex__Ocomplex,type,
    diffs_complex: fun_nat_complex > fun_nat_complex ).

tff(sy_c_Transcendental_Odiffs_001t__Int__Oint,type,
    diffs_int: fun_nat_int > fun_nat_int ).

tff(sy_c_Transcendental_Odiffs_001t__Rat__Orat,type,
    diffs_rat: fun_nat_rat > fun_nat_rat ).

tff(sy_c_Transcendental_Odiffs_001t__Real__Oreal,type,
    diffs_real: fun_nat_real > fun_nat_real ).

tff(sy_c_Transcendental_Oexp_001t__Complex__Ocomplex,type,
    exp_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Oexp_001t__Real__Oreal,type,
    exp_real: fun_real_real ).

tff(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
    ln_ln_real: fun_real_real ).

tff(sy_c_Transcendental_Olog,type,
    log: real > fun_real_real ).

tff(sy_c_Transcendental_Opi,type,
    pi: real ).

tff(sy_c_Transcendental_Opowr_001t__Real__Oreal,type,
    powr_real: real > fun_real_real ).

tff(sy_c_Transcendental_Osin_001t__Complex__Ocomplex,type,
    sin_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Osin_001t__Real__Oreal,type,
    sin_real: fun_real_real ).

tff(sy_c_Transcendental_Osin__coeff,type,
    sin_coeff: fun_nat_real ).

tff(sy_c_Transcendental_Osinh_001t__Complex__Ocomplex,type,
    sinh_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Osinh_001t__Real__Oreal,type,
    sinh_real: fun_real_real ).

tff(sy_c_Transcendental_Otan_001t__Complex__Ocomplex,type,
    tan_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Otan_001t__Real__Oreal,type,
    tan_real: fun_real_real ).

tff(sy_c_Transcendental_Otanh_001t__Complex__Ocomplex,type,
    tanh_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Otanh_001t__Real__Oreal,type,
    tanh_real: fun_real_real ).

tff(sy_c_VEBT__Definitions_OVEBT_OLeaf,type,
    vEBT_Leaf: $o > fun_o_VEBT_VEBT ).

tff(sy_c_VEBT__Definitions_OVEBT_ONode,type,
    vEBT_Node: ( option4927543243414619207at_nat * nat * list_VEBT_VEBT ) > fun_VE3165209603783583043T_VEBT ).

tff(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
    vEBT_size_VEBT: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type,
    vEBT_V8194947554948674370ptions: vEBT_VEBT > fun_nat_o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oelim__dead,type,
    vEBT_VEBT_elim_dead: ( vEBT_VEBT * extended_enat ) > vEBT_VEBT ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oelim__dead__rel,type,
    vEBT_V312737461966249ad_rel: fun_Pr1263128411319240110enat_o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
    vEBT_VEBT_high: fun_nat_fun_nat_nat ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oin__children,type,
    vEBT_V5917875025757280293ildren: ( nat * list_VEBT_VEBT ) > fun_nat_o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
    vEBT_VEBT_low: fun_nat_fun_nat_nat ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima,type,
    vEBT_VEBT_membermima: vEBT_VEBT > fun_nat_o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
    vEBT_V4351362008482014158ma_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member,type,
    vEBT_V5719532721284313246member: vEBT_VEBT > fun_nat_o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
    vEBT_V5765760719290551771er_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H,type,
    vEBT_VEBT_valid: vEBT_VEBT > fun_nat_o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
    vEBT_VEBT_valid_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Definitions_Oinvar__vebt,type,
    vEBT_invar_vebt: vEBT_VEBT > fun_nat_o ).

tff(sy_c_VEBT__Definitions_Oset__vebt,type,
    vEBT_set_vebt: vEBT_VEBT > set_nat ).

tff(sy_c_VEBT__Definitions_Ovebt__buildup,type,
    vEBT_vebt_buildup: fun_nat_VEBT_VEBT ).

tff(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
    vEBT_v4011308405150292612up_rel: fun_nat_fun_nat_o ).

tff(sy_c_Wellfounded_Oaccp_001t__Nat__Onat,type,
    accp_nat: fun_nat_fun_nat_o > fun_nat_o ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    accp_P1096762738010456898nt_int: fun_Pr9018117161326028078_int_o > fun_Pr3238374608701176162_int_o ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    accp_P4275260045618599050at_nat: fun_Pr2028404973800837934_nat_o > fun_Pr4796943543331071514_nat_o ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
    accp_P3113834385874906142um_num: fun_Pr4799464687642144302_num_o > fun_Pr6331313853612757382_num_o ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Extended____Nat__Oenat_J,type,
    accp_P6183159247885693666d_enat: ( fun_Pr1263128411319240110enat_o * produc7272778201969148633d_enat ) > $o ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
    accp_P2887432264394892906BT_nat: fun_Pr8877776288780692068_nat_o > fun_Pr7546709192626087934_nat_o ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    accp_P4694664897847628410at_nat: ( fun_Pr8463018485500633518_nat_o * produc9217583074051984497at_nat ) > $o ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__fun_It__Nat__Onat_Mt__fun_It__Num__Onum_Mt__Num__Onum_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J_J_J,type,
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tff(sy_c_Wellfounded_Omeasure_001t__Int__Oint,type,
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tff(sy_c_Wellfounded_Omeasure_001t__Nat__Onat,type,
    measure_nat: fun_nat_nat > set_Pr1261947904930325089at_nat ).

tff(sy_c_Wellfounded_Omeasure_001t__Num__Onum,type,
    measure_num: fun_num_nat > set_Pr8218934625190621173um_num ).

tff(sy_c_aa_001_Eo_001_Eo,type,
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tff(sy_c_aa_001_Eo_001t__Int__Oint,type,
    aa_o_int: ( fun_o_int * $o ) > int ).

tff(sy_c_aa_001_Eo_001t__Nat__Onat,type,
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tff(sy_c_aa_001_Eo_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
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tff(sy_c_aa_001_Eo_001t__Set__Oset_It__Complex__Ocomplex_J,type,
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tff(sy_c_aa_001_Eo_001t__Set__Oset_It__Int__Oint_J,type,
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tff(sy_c_aa_001_Eo_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_aa_001_Eo_001t__Set__Oset_It__Real__Oreal_J,type,
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tff(sy_c_aa_001_Eo_001t__String__Ochar,type,
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tff(sy_c_aa_001_Eo_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__Num__Onum,type,
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tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
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tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
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tff(sy_c_aa_001t__List__Olist_I_Eo_J_001_Eo,type,
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tff(sy_c_aa_001t__Nat__Onat_001t__Num__Onum,type,
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    member_VEBT_VEBT: ( vEBT_VEBT * set_VEBT_VEBT ) > $o ).

tff(sy_v_dega____,type,
    dega: nat ).

tff(sy_v_m____,type,
    m: nat ).

tff(sy_v_na____,type,
    na: nat ).

tff(sy_v_summarya____,type,
    summarya: vEBT_VEBT ).

tff(sy_v_treeLista____,type,
    treeLista: list_VEBT_VEBT ).

% Relevant facts (10778)
tff(fact_0__C2_Ohyps_C_I3_J,axiom,
    m = na ).

% "2.hyps"(3)
tff(fact_1__C2_OIH_C_I2_J,axiom,
    vEBT_VEBT_elim_dead(summarya,extend5688581933313929465d_enat) = summarya ).

% "2.IH"(2)
tff(fact_2__C2_Ohyps_C_I5_J,axiom,
    ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(summarya),X_1) ).

% "2.hyps"(5)
tff(fact_3__C2_Ohyps_C_I1_J,axiom,
    aa_nat_o(vEBT_invar_vebt(summarya),m) ).

% "2.hyps"(1)
tff(fact_4_not__enat__eq,axiom,
    ! [X: extended_enat] :
      ( ! [Y: nat] : X != aa_nat_Extended_enat(extended_enat2,Y)
    <=> ( X = extend5688581933313929465d_enat ) ) ).

% not_enat_eq
tff(fact_5_not__infinity__eq,axiom,
    ! [X: extended_enat] :
      ( ( X != extend5688581933313929465d_enat )
    <=> ? [I: nat] : X = aa_nat_Extended_enat(extended_enat2,I) ) ).

% not_infinity_eq
tff(fact_6_enat_Osimps_I5_J,axiom,
    ! [F1: fun_nat_o,F2: $o] :
      ( aa_Extended_enat_o(aa_o_f2257658052187763793enat_o(aa_fun2142289715968585979enat_o(extended_case_enat_o,F1),(F2)),extend5688581933313929465d_enat)
    <=> (F2) ) ).

% enat.simps(5)
tff(fact_7_enat_Osimps_I5_J,axiom,
    ! [F1: fun_na8797796813480665199d_enat,F2: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(aa_fun2186929853097002313d_enat(extend3600170679010898289d_enat,F1),F2),extend5688581933313929465d_enat) = F2 ).

% enat.simps(5)
tff(fact_8__C2_OIH_C_I1_J,axiom,
    ! [X2: vEBT_VEBT] :
      ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(treeLista))
     => ( aa_nat_o(vEBT_invar_vebt(X2),na)
        & ( vEBT_VEBT_elim_dead(X2,extend5688581933313929465d_enat) = X2 ) ) ) ).

% "2.IH"(1)
tff(fact_9_enat_Odistinct_I2_J,axiom,
    ! [Nat: nat] : extend5688581933313929465d_enat != aa_nat_Extended_enat(extended_enat2,Nat) ).

% enat.distinct(2)
tff(fact_10_enat_Odistinct_I1_J,axiom,
    ! [Nat: nat] : aa_nat_Extended_enat(extended_enat2,Nat) != extend5688581933313929465d_enat ).

% enat.distinct(1)
tff(fact_11_enat_Oexhaust,axiom,
    ! [Y2: extended_enat] :
      ( ! [Nat2: nat] : Y2 != aa_nat_Extended_enat(extended_enat2,Nat2)
     => ( Y2 = extend5688581933313929465d_enat ) ) ).

% enat.exhaust
tff(fact_12_enat2__cases,axiom,
    ! [Y2: extended_enat,Ya: extended_enat] :
      ( ( ? [Nat2: nat] : Y2 = aa_nat_Extended_enat(extended_enat2,Nat2)
       => ! [Nata: nat] : Ya != aa_nat_Extended_enat(extended_enat2,Nata) )
     => ( ( ? [Nat2: nat] : Y2 = aa_nat_Extended_enat(extended_enat2,Nat2)
         => ( Ya != extend5688581933313929465d_enat ) )
       => ( ( ( Y2 = extend5688581933313929465d_enat )
           => ! [Nat2: nat] : Ya != aa_nat_Extended_enat(extended_enat2,Nat2) )
         => ~ ( ( Y2 = extend5688581933313929465d_enat )
             => ( Ya != extend5688581933313929465d_enat ) ) ) ) ) ).

% enat2_cases
tff(fact_13_enat3__cases,axiom,
    ! [Y2: extended_enat,Ya: extended_enat,Yb: extended_enat] :
      ( ( ? [Nat2: nat] : Y2 = aa_nat_Extended_enat(extended_enat2,Nat2)
       => ( ? [Nata: nat] : Ya = aa_nat_Extended_enat(extended_enat2,Nata)
         => ! [Natb: nat] : Yb != aa_nat_Extended_enat(extended_enat2,Natb) ) )
     => ( ( ? [Nat2: nat] : Y2 = aa_nat_Extended_enat(extended_enat2,Nat2)
         => ( ? [Nata: nat] : Ya = aa_nat_Extended_enat(extended_enat2,Nata)
           => ( Yb != extend5688581933313929465d_enat ) ) )
       => ( ( ? [Nat2: nat] : Y2 = aa_nat_Extended_enat(extended_enat2,Nat2)
           => ( ( Ya = extend5688581933313929465d_enat )
             => ! [Nata: nat] : Yb != aa_nat_Extended_enat(extended_enat2,Nata) ) )
         => ( ( ? [Nat2: nat] : Y2 = aa_nat_Extended_enat(extended_enat2,Nat2)
             => ( ( Ya = extend5688581933313929465d_enat )
               => ( Yb != extend5688581933313929465d_enat ) ) )
           => ( ( ( Y2 = extend5688581933313929465d_enat )
               => ( ? [Nat2: nat] : Ya = aa_nat_Extended_enat(extended_enat2,Nat2)
                 => ! [Nata: nat] : Yb != aa_nat_Extended_enat(extended_enat2,Nata) ) )
             => ( ( ( Y2 = extend5688581933313929465d_enat )
                 => ( ? [Nat2: nat] : Ya = aa_nat_Extended_enat(extended_enat2,Nat2)
                   => ( Yb != extend5688581933313929465d_enat ) ) )
               => ( ( ( Y2 = extend5688581933313929465d_enat )
                   => ( ( Ya = extend5688581933313929465d_enat )
                     => ! [Nat2: nat] : Yb != aa_nat_Extended_enat(extended_enat2,Nat2) ) )
                 => ~ ( ( Y2 = extend5688581933313929465d_enat )
                     => ( ( Ya = extend5688581933313929465d_enat )
                       => ( Yb != extend5688581933313929465d_enat ) ) ) ) ) ) ) ) ) ) ).

% enat3_cases
tff(fact_14__C2_Ohyps_C_I6_J,axiom,
    ! [X2: vEBT_VEBT] :
      ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(treeLista))
     => ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X2),X_1) ) ).

% "2.hyps"(6)
tff(fact_15_enat_Oinject,axiom,
    ! [Nat: nat,Nat3: nat] :
      ( ( aa_nat_Extended_enat(extended_enat2,Nat) = aa_nat_Extended_enat(extended_enat2,Nat3) )
    <=> ( Nat = Nat3 ) ) ).

% enat.inject
tff(fact_16_enat_Osimps_I4_J,axiom,
    ! [F1: fun_nat_o,F2: $o,Nat: nat] :
      ( aa_Extended_enat_o(aa_o_f2257658052187763793enat_o(aa_fun2142289715968585979enat_o(extended_case_enat_o,F1),(F2)),aa_nat_Extended_enat(extended_enat2,Nat))
    <=> aa_nat_o(F1,Nat) ) ).

% enat.simps(4)
tff(fact_17_enat_Osimps_I4_J,axiom,
    ! [F1: fun_na8797796813480665199d_enat,F2: extended_enat,Nat: nat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(aa_fun2186929853097002313d_enat(extend3600170679010898289d_enat,F1),F2),aa_nat_Extended_enat(extended_enat2,Nat)) = aa_nat_Extended_enat(F1,Nat) ).

% enat.simps(4)
tff(fact_18__C2_Ohyps_C_I4_J,axiom,
    dega = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,na),m) ).

% "2.hyps"(4)
tff(fact_19_enat__ex__split,axiom,
    ! [P: fun_Extended_enat_o] :
      ( ? [X_12: extended_enat] : aa_Extended_enat_o(P,X_12)
    <=> ( aa_Extended_enat_o(P,extend5688581933313929465d_enat)
        | ? [X3: nat] : aa_Extended_enat_o(P,aa_nat_Extended_enat(extended_enat2,X3)) ) ) ).

% enat_ex_split
tff(fact_20_the__enat_Osimps,axiom,
    ! [Nb: nat] : extended_the_enat(aa_nat_Extended_enat(extended_enat2,Nb)) = Nb ).

% the_enat.simps
tff(fact_21_valid__eq,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(T),D)
    <=> aa_nat_o(vEBT_invar_vebt(T),D) ) ).

% valid_eq
tff(fact_22_valid__eq1,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),D)
     => aa_nat_o(vEBT_VEBT_valid(T),D) ) ).

% valid_eq1
tff(fact_23_valid__eq2,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(T),D)
     => aa_nat_o(vEBT_invar_vebt(T),D) ) ).

% valid_eq2
tff(fact_24__C2_Ohyps_C_I2_J,axiom,
    size_s6755466524823107622T_VEBT(treeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m) ).

% "2.hyps"(2)
tff(fact_25__092_060open_062deg_Adiv_A2_A_061_An_092_060close_062,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = na ).

% \<open>deg div 2 = n\<close>
tff(fact_26_a,axiom,
    ! [I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m))
     => ( vEBT_VEBT_elim_dead(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeLista),I2),aa_nat_Extended_enat(extended_enat2,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeLista),I2) ) ) ).

% a
tff(fact_27__092_060open_0622_A_094_Am_A_061_A2_A_094_Adeg_Adiv_A2_A_094_A_Ideg_Adiv_A2_J_092_060close_062,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),dega)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% \<open>2 ^ m = 2 ^ deg div 2 ^ (deg div 2)\<close>
tff(fact_28_calculation,axiom,
    take_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),dega)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),map_VE8901447254227204932T_VEBT(aTP_Lamm_a,treeLista)) = treeLista ).

% calculation
tff(fact_29_elimnum,axiom,
    ! [Infoa: option4927543243414619207at_nat,Degb: nat,TreeListb: list_VEBT_VEBT,Summaryb: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Infoa,Degb,TreeListb),Summaryb)),Nb)
     => ( vEBT_VEBT_elim_dead(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Infoa,Degb,TreeListb),Summaryb),aa_nat_Extended_enat(extended_enat2,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Infoa,Degb,TreeListb),Summaryb) ) ) ).

% elimnum
tff(fact_30_infinity__ileE,axiom,
    ! [Ma: nat] : ~ aa_Extended_enat_o(ord_le2932123472753598470d_enat(extend5688581933313929465d_enat),aa_nat_Extended_enat(extended_enat2,Ma)) ).

% infinity_ileE
tff(fact_31_enat__ord__code_I5_J,axiom,
    ! [Nb: nat] : ~ aa_Extended_enat_o(ord_le2932123472753598470d_enat(extend5688581933313929465d_enat),aa_nat_Extended_enat(extended_enat2,Nb)) ).

% enat_ord_code(5)
tff(fact_32_inthall,axiom,
    ! [Xs: list_complex,P: fun_complex_o,Nb: nat] :
      ( ! [X4: complex] :
          ( member_complex(X4,set_complex2(Xs))
         => aa_complex_o(P,X4) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s3451745648224563538omplex(Xs))
       => aa_complex_o(P,aa_nat_complex(nth_complex(Xs),Nb)) ) ) ).

% inthall
tff(fact_33_inthall,axiom,
    ! [Xs: list_real,P: fun_real_o,Nb: nat] :
      ( ! [X4: real] :
          ( member_real(X4,set_real2(Xs))
         => aa_real_o(P,X4) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_real(Xs))
       => aa_real_o(P,aa_nat_real(nth_real(Xs),Nb)) ) ) ).

% inthall
tff(fact_34_inthall,axiom,
    ! [Xs: list_set_nat,P: fun_set_nat_o,Nb: nat] :
      ( ! [X4: set_nat] :
          ( member_set_nat(X4,set_set_nat2(Xs))
         => aa_set_nat_o(P,X4) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s3254054031482475050et_nat(Xs))
       => aa_set_nat_o(P,nth_set_nat(Xs,Nb)) ) ) ).

% inthall
tff(fact_35_inthall,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_o,Nb: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(Xs))
         => aa_VEBT_VEBT_o(P,X4) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s6755466524823107622T_VEBT(Xs))
       => aa_VEBT_VEBT_o(P,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),Nb)) ) ) ).

% inthall
tff(fact_36_inthall,axiom,
    ! [Xs: list_o,P: fun_o_o,Nb: nat] :
      ( ! [X4: $o] :
          ( member_o((X4),set_o2(Xs))
         => aa_o_o(P,(X4)) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_o(Xs))
       => aa_o_o(P,aa_nat_o(nth_o(Xs),Nb)) ) ) ).

% inthall
tff(fact_37_inthall,axiom,
    ! [Xs: list_nat,P: fun_nat_o,Nb: nat] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),set_nat2(Xs))
         => aa_nat_o(P,X4) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_nat(Xs))
       => aa_nat_o(P,aa_nat_nat(nth_nat(Xs),Nb)) ) ) ).

% inthall
tff(fact_38_inthall,axiom,
    ! [Xs: list_int,P: fun_int_o,Nb: nat] :
      ( ! [X4: int] :
          ( member_int(X4,set_int2(Xs))
         => aa_int_o(P,X4) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_int(Xs))
       => aa_int_o(P,aa_nat_int(nth_int(Xs),Nb)) ) ) ).

% inthall
tff(fact_39_VEBT_Oinject_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,Y11: option4927543243414619207at_nat,Y12: nat,Y13: list_VEBT_VEBT,Y14: vEBT_VEBT] :
      ( ( aa_VEB5832953974749077948T_VEBT(vEBT_Node(X11,X12,X13),X14) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Y11,Y12,Y13),Y14) )
    <=> ( ( X11 = Y11 )
        & ( X12 = Y12 )
        & ( X13 = Y13 )
        & ( X14 = Y14 ) ) ) ).

% VEBT.inject(1)
tff(fact_40_pow__sum,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B) ).

% pow_sum
tff(fact_41_enat__ord__code_I3_J,axiom,
    ! [Q: extended_enat] : aa_Extended_enat_o(ord_le2932123472753598470d_enat(Q),extend5688581933313929465d_enat) ).

% enat_ord_code(3)
tff(fact_42_enat__ord__simps_I5_J,axiom,
    ! [Q: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(extend5688581933313929465d_enat),Q)
    <=> ( Q = extend5688581933313929465d_enat ) ) ).

% enat_ord_simps(5)
tff(fact_43_b,axiom,
    map_VE8901447254227204932T_VEBT(aTP_Lamm_a,treeLista) = treeLista ).

% b
tff(fact_44_VEBT__internal_Oelim__dead_Osimps_I3_J,axiom,
    ! [Infoa: option4927543243414619207at_nat,Degb: nat,TreeListb: list_VEBT_VEBT,Summaryb: vEBT_VEBT,L: nat] : vEBT_VEBT_elim_dead(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Infoa,Degb,TreeListb),Summaryb),aa_nat_Extended_enat(extended_enat2,L)) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Infoa,Degb,take_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,L),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Degb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),map_VE8901447254227204932T_VEBT(aTP_Lamm_aa(Degb),TreeListb))),vEBT_VEBT_elim_dead(Summaryb,aa_nat_Extended_enat(extended_enat2,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,L),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Degb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))) ).

% VEBT_internal.elim_dead.simps(3)
tff(fact_45_VEBT__internal_Oelim__dead_Osimps_I2_J,axiom,
    ! [Infoa: option4927543243414619207at_nat,Degb: nat,TreeListb: list_VEBT_VEBT,Summaryb: vEBT_VEBT] : vEBT_VEBT_elim_dead(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Infoa,Degb,TreeListb),Summaryb),extend5688581933313929465d_enat) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Infoa,Degb,map_VE8901447254227204932T_VEBT(aTP_Lamm_aa(Degb),TreeListb)),vEBT_VEBT_elim_dead(Summaryb,extend5688581933313929465d_enat)) ).

% VEBT_internal.elim_dead.simps(2)
tff(fact_46_enat__ile,axiom,
    ! [Nb: extended_enat,Ma: nat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(Nb),aa_nat_Extended_enat(extended_enat2,Ma))
     => ? [K: nat] : Nb = aa_nat_Extended_enat(extended_enat2,K) ) ).

% enat_ile
tff(fact_47_enat__ord__simps_I3_J,axiom,
    ! [Q: extended_enat] : aa_Extended_enat_o(ord_le2932123472753598470d_enat(Q),extend5688581933313929465d_enat) ).

% enat_ord_simps(3)
tff(fact_48_case__enat__def,axiom,
    extended_case_enat_o = extended_rec_enat_o ).

% case_enat_def
tff(fact_49_case__enat__def,axiom,
    extend3600170679010898289d_enat = extend1611788537373416385d_enat ).

% case_enat_def
tff(fact_50_add__self__div__2,axiom,
    ! [Ma: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Ma)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Ma ).

% add_self_div_2
tff(fact_51_mem__Collect__eq,axiom,
    ! [A: real,P: fun_real_o] :
      ( member_real(A,collect_real(P))
    <=> aa_real_o(P,A) ) ).

% mem_Collect_eq
tff(fact_52_mem__Collect__eq,axiom,
    ! [A: complex,P: fun_complex_o] :
      ( member_complex(A,collect_complex(P))
    <=> aa_complex_o(P,A) ) ).

% mem_Collect_eq
tff(fact_53_mem__Collect__eq,axiom,
    ! [A: list_nat,P: fun_list_nat_o] :
      ( member_list_nat(A,collect_list_nat(P))
    <=> aa_list_nat_o(P,A) ) ).

% mem_Collect_eq
tff(fact_54_mem__Collect__eq,axiom,
    ! [A: set_nat,P: fun_set_nat_o] :
      ( member_set_nat(A,collect_set_nat(P))
    <=> aa_set_nat_o(P,A) ) ).

% mem_Collect_eq
tff(fact_55_mem__Collect__eq,axiom,
    ! [A: nat,P: fun_nat_o] :
      ( aa_set_nat_o(member_nat(A),collect_nat(P))
    <=> aa_nat_o(P,A) ) ).

% mem_Collect_eq
tff(fact_56_mem__Collect__eq,axiom,
    ! [A: int,P: fun_int_o] :
      ( member_int(A,collect_int(P))
    <=> aa_int_o(P,A) ) ).

% mem_Collect_eq
tff(fact_57_Collect__mem__eq,axiom,
    ! [A2: set_real] : collect_real(aTP_Lamm_ab(A2)) = A2 ).

% Collect_mem_eq
tff(fact_58_Collect__mem__eq,axiom,
    ! [A2: set_complex] : collect_complex(aTP_Lamm_ac(A2)) = A2 ).

% Collect_mem_eq
tff(fact_59_Collect__mem__eq,axiom,
    ! [A2: set_list_nat] : collect_list_nat(aTP_Lamm_ad(A2)) = A2 ).

% Collect_mem_eq
tff(fact_60_Collect__mem__eq,axiom,
    ! [A2: set_set_nat] : collect_set_nat(aTP_Lamm_ae(A2)) = A2 ).

% Collect_mem_eq
tff(fact_61_Collect__mem__eq,axiom,
    ! [A2: set_nat] : collect_nat(aTP_Lamm_af(A2)) = A2 ).

% Collect_mem_eq
tff(fact_62_Collect__mem__eq,axiom,
    ! [A2: set_int] : collect_int(aTP_Lamm_ag(A2)) = A2 ).

% Collect_mem_eq
tff(fact_63_Collect__cong,axiom,
    ! [P: fun_complex_o,Q2: fun_complex_o] :
      ( ! [X4: complex] :
          ( aa_complex_o(P,X4)
        <=> aa_complex_o(Q2,X4) )
     => ( collect_complex(P) = collect_complex(Q2) ) ) ).

% Collect_cong
tff(fact_64_Collect__cong,axiom,
    ! [P: fun_list_nat_o,Q2: fun_list_nat_o] :
      ( ! [X4: list_nat] :
          ( aa_list_nat_o(P,X4)
        <=> aa_list_nat_o(Q2,X4) )
     => ( collect_list_nat(P) = collect_list_nat(Q2) ) ) ).

% Collect_cong
tff(fact_65_Collect__cong,axiom,
    ! [P: fun_set_nat_o,Q2: fun_set_nat_o] :
      ( ! [X4: set_nat] :
          ( aa_set_nat_o(P,X4)
        <=> aa_set_nat_o(Q2,X4) )
     => ( collect_set_nat(P) = collect_set_nat(Q2) ) ) ).

% Collect_cong
tff(fact_66_Collect__cong,axiom,
    ! [P: fun_nat_o,Q2: fun_nat_o] :
      ( ! [X4: nat] :
          ( aa_nat_o(P,X4)
        <=> aa_nat_o(Q2,X4) )
     => ( collect_nat(P) = collect_nat(Q2) ) ) ).

% Collect_cong
tff(fact_67_Collect__cong,axiom,
    ! [P: fun_int_o,Q2: fun_int_o] :
      ( ! [X4: int] :
          ( aa_int_o(P,X4)
        <=> aa_int_o(Q2,X4) )
     => ( collect_int(P) = collect_int(Q2) ) ) ).

% Collect_cong
tff(fact_68_nth__map,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,F: fun_VEBT_VEBT_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s6755466524823107622T_VEBT(Xs))
     => ( aa_nat_nat(nth_nat(map_VEBT_VEBT_nat(F,Xs)),Nb) = aa_VEBT_VEBT_nat(F,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),Nb)) ) ) ).

% nth_map
tff(fact_69_nth__map,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,F: fun_VEBT_VEBT_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s6755466524823107622T_VEBT(Xs))
     => ( aa_nat_int(nth_int(map_VEBT_VEBT_int(F,Xs)),Nb) = aa_VEBT_VEBT_int(F,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),Nb)) ) ) ).

% nth_map
tff(fact_70_nth__map,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,F: fun_VE3165209603783583043T_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s6755466524823107622T_VEBT(Xs))
     => ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(map_VE8901447254227204932T_VEBT(F,Xs)),Nb) = aa_VEB5832953974749077948T_VEBT(F,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),Nb)) ) ) ).

% nth_map
tff(fact_71_nth__map,axiom,
    ! [Nb: nat,Xs: list_o,F: fun_o_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_o(Xs))
     => ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(map_o_VEBT_VEBT(F,Xs)),Nb) = aa_o_VEBT_VEBT(F,aa_nat_o(nth_o(Xs),Nb)) ) ) ).

% nth_map
tff(fact_72_nth__map,axiom,
    ! [Nb: nat,Xs: list_o,F: fun_o_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_o(Xs))
     => ( aa_nat_nat(nth_nat(map_o_nat(F,Xs)),Nb) = aa_o_nat(F,aa_nat_o(nth_o(Xs),Nb)) ) ) ).

% nth_map
tff(fact_73_nth__map,axiom,
    ! [Nb: nat,Xs: list_o,F: fun_o_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_o(Xs))
     => ( aa_nat_int(nth_int(map_o_int(F,Xs)),Nb) = aa_o_int(F,aa_nat_o(nth_o(Xs),Nb)) ) ) ).

% nth_map
tff(fact_74_nth__map,axiom,
    ! [Nb: nat,Xs: list_nat,F: fun_nat_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_nat(Xs))
     => ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(map_nat_VEBT_VEBT(F,Xs)),Nb) = aa_nat_VEBT_VEBT(F,aa_nat_nat(nth_nat(Xs),Nb)) ) ) ).

% nth_map
tff(fact_75_nth__map,axiom,
    ! [Nb: nat,Xs: list_nat,F: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_nat(Xs))
     => ( aa_nat_int(nth_int(map_nat_int(F,Xs)),Nb) = aa_nat_int(F,aa_nat_nat(nth_nat(Xs),Nb)) ) ) ).

% nth_map
tff(fact_76_nth__map,axiom,
    ! [Nb: nat,Xs: list_nat,F: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_nat(Xs))
     => ( aa_nat_nat(nth_nat(map_nat_nat(F,Xs)),Nb) = aa_nat_nat(F,aa_nat_nat(nth_nat(Xs),Nb)) ) ) ).

% nth_map
tff(fact_77_nth__map,axiom,
    ! [Nb: nat,Xs: list_int,F: fun_int_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_int(Xs))
     => ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(map_int_VEBT_VEBT(F,Xs)),Nb) = aa_int_VEBT_VEBT(F,aa_nat_int(nth_int(Xs),Nb)) ) ) ).

% nth_map
tff(fact_78_nth__take,axiom,
    ! [I2: nat,Nb: nat,Xs: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Nb)
     => ( aa_nat_int(nth_int(take_int(Nb,Xs)),I2) = aa_nat_int(nth_int(Xs),I2) ) ) ).

% nth_take
tff(fact_79_nth__take,axiom,
    ! [I2: nat,Nb: nat,Xs: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Nb)
     => ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(take_VEBT_VEBT(Nb,Xs)),I2) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I2) ) ) ).

% nth_take
tff(fact_80_nth__take,axiom,
    ! [I2: nat,Nb: nat,Xs: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Nb)
     => ( aa_nat_nat(nth_nat(take_nat(Nb,Xs)),I2) = aa_nat_nat(nth_nat(Xs),I2) ) ) ).

% nth_take
tff(fact_81_div__exp__eq,axiom,
    ! [A: nat,Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb))) ).

% div_exp_eq
tff(fact_82_div__exp__eq,axiom,
    ! [A: int,Ma: nat,Nb: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Ma))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb))) ).

% div_exp_eq
tff(fact_83_field__less__half__sum,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(X),Y2)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ).

% field_less_half_sum
tff(fact_84_field__less__half__sum,axiom,
    ! [X: rat,Y2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(X),Y2)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ) ).

% field_less_half_sum
tff(fact_85_high__bound__aux,axiom,
    ! [Ma: nat,Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) ) ).

% high_bound_aux
tff(fact_86_length__map,axiom,
    ! [F: fun_VE3165209603783583043T_VEBT,Xs: list_VEBT_VEBT] : size_s6755466524823107622T_VEBT(map_VE8901447254227204932T_VEBT(F,Xs)) = size_s6755466524823107622T_VEBT(Xs) ).

% length_map
tff(fact_87_length__map,axiom,
    ! [F: fun_o_VEBT_VEBT,Xs: list_o] : size_s6755466524823107622T_VEBT(map_o_VEBT_VEBT(F,Xs)) = size_size_list_o(Xs) ).

% length_map
tff(fact_88_length__map,axiom,
    ! [F: fun_nat_VEBT_VEBT,Xs: list_nat] : size_s6755466524823107622T_VEBT(map_nat_VEBT_VEBT(F,Xs)) = size_size_list_nat(Xs) ).

% length_map
tff(fact_89_length__map,axiom,
    ! [F: fun_int_VEBT_VEBT,Xs: list_int] : size_s6755466524823107622T_VEBT(map_int_VEBT_VEBT(F,Xs)) = size_size_list_int(Xs) ).

% length_map
tff(fact_90_length__map,axiom,
    ! [F: fun_VEBT_VEBT_o,Xs: list_VEBT_VEBT] : size_size_list_o(map_VEBT_VEBT_o(F,Xs)) = size_s6755466524823107622T_VEBT(Xs) ).

% length_map
tff(fact_91_length__map,axiom,
    ! [F: fun_o_o,Xs: list_o] : size_size_list_o(map_o_o(F,Xs)) = size_size_list_o(Xs) ).

% length_map
tff(fact_92_length__map,axiom,
    ! [F: fun_nat_o,Xs: list_nat] : size_size_list_o(map_nat_o(F,Xs)) = size_size_list_nat(Xs) ).

% length_map
tff(fact_93_length__map,axiom,
    ! [F: fun_int_o,Xs: list_int] : size_size_list_o(map_int_o(F,Xs)) = size_size_list_int(Xs) ).

% length_map
tff(fact_94_length__map,axiom,
    ! [F: fun_VEBT_VEBT_nat,Xs: list_VEBT_VEBT] : size_size_list_nat(map_VEBT_VEBT_nat(F,Xs)) = size_s6755466524823107622T_VEBT(Xs) ).

% length_map
tff(fact_95_length__map,axiom,
    ! [F: fun_o_nat,Xs: list_o] : size_size_list_nat(map_o_nat(F,Xs)) = size_size_list_o(Xs) ).

% length_map
tff(fact_96_map__eq__conv,axiom,
    ! [F: fun_VE3165209603783583043T_VEBT,Xs: list_VEBT_VEBT,G: fun_VE3165209603783583043T_VEBT] :
      ( ( map_VE8901447254227204932T_VEBT(F,Xs) = map_VE8901447254227204932T_VEBT(G,Xs) )
    <=> ! [X3: vEBT_VEBT] :
          ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(Xs))
         => ( aa_VEB5832953974749077948T_VEBT(F,X3) = aa_VEB5832953974749077948T_VEBT(G,X3) ) ) ) ).

% map_eq_conv
tff(fact_97_map__eq__conv,axiom,
    ! [F: fun_nat_nat,Xs: list_nat,G: fun_nat_nat] :
      ( ( map_nat_nat(F,Xs) = map_nat_nat(G,Xs) )
    <=> ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),set_nat2(Xs))
         => ( aa_nat_nat(F,X3) = aa_nat_nat(G,X3) ) ) ) ).

% map_eq_conv
tff(fact_98_nat__add__left__cancel__less,axiom,
    ! [K2: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb) ) ).

% nat_add_left_cancel_less
tff(fact_99_less__exp,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% less_exp
tff(fact_100_semiring__norm_I83_J,axiom,
    ! [Nb: num] : one != aa_num_num(bit0,Nb) ).

% semiring_norm(83)
tff(fact_101_semiring__norm_I85_J,axiom,
    ! [Ma: num] : aa_num_num(bit0,Ma) != one ).

% semiring_norm(85)
tff(fact_102_semiring__norm_I6_J,axiom,
    ! [Ma: num,Nb: num] : aa_num_num(plus_plus_num(aa_num_num(bit0,Ma)),aa_num_num(bit0,Nb)) = aa_num_num(bit0,aa_num_num(plus_plus_num(Ma),Nb)) ).

% semiring_norm(6)
tff(fact_103_semiring__norm_I78_J,axiom,
    ! [Ma: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit0,Ma)),aa_num_num(bit0,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Ma),Nb) ) ).

% semiring_norm(78)
tff(fact_104_semiring__norm_I87_J,axiom,
    ! [Ma: num,Nb: num] :
      ( ( aa_num_num(bit0,Ma) = aa_num_num(bit0,Nb) )
    <=> ( Ma = Nb ) ) ).

% semiring_norm(87)
tff(fact_105_semiring__norm_I75_J,axiom,
    ! [Ma: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Ma),one) ).

% semiring_norm(75)
tff(fact_106_plus__enat__simps_I2_J,axiom,
    ! [Q: extended_enat] : plus_p3455044024723400733d_enat(extend5688581933313929465d_enat,Q) = extend5688581933313929465d_enat ).

% plus_enat_simps(2)
tff(fact_107_plus__enat__simps_I3_J,axiom,
    ! [Q: extended_enat] : plus_p3455044024723400733d_enat(Q,extend5688581933313929465d_enat) = extend5688581933313929465d_enat ).

% plus_enat_simps(3)
tff(fact_108_enat__ord__simps_I4_J,axiom,
    ! [Q: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(Q),extend5688581933313929465d_enat)
    <=> ( Q != extend5688581933313929465d_enat ) ) ).

% enat_ord_simps(4)
tff(fact_109_enat__ord__simps_I6_J,axiom,
    ! [Q: extended_enat] : ~ aa_Extended_enat_o(ord_le72135733267957522d_enat(extend5688581933313929465d_enat),Q) ).

% enat_ord_simps(6)
tff(fact_110_high__def,axiom,
    ! [X: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% high_def
tff(fact_111_map__ident,axiom,
    ! [X2: list_VEBT_VEBT] : map_VE8901447254227204932T_VEBT(aTP_Lamm_ah,X2) = X2 ).

% map_ident
tff(fact_112_map__ident,axiom,
    ! [X2: list_nat] : map_nat_nat(aTP_Lamm_ai,X2) = X2 ).

% map_ident
tff(fact_113_semiring__norm_I2_J,axiom,
    aa_num_num(plus_plus_num(one),one) = aa_num_num(bit0,one) ).

% semiring_norm(2)
tff(fact_114_semiring__norm_I76_J,axiom,
    ! [Nb: num] : aa_num_o(aa_num_fun_num_o(ord_less_num,one),aa_num_num(bit0,Nb)) ).

% semiring_norm(76)
tff(fact_115_enat__ord__number_I2_J,axiom,
    ! [Ma: num,Nb: num] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(numera1916890842035813515d_enat(Ma)),numera1916890842035813515d_enat(Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Ma)),aa_num_nat(numeral_numeral_nat,Nb)) ) ).

% enat_ord_number(2)
tff(fact_116_enat__ord__simps_I2_J,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(aa_nat_Extended_enat(extended_enat2,Ma)),aa_nat_Extended_enat(extended_enat2,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb) ) ).

% enat_ord_simps(2)
tff(fact_117_plus__enat__simps_I1_J,axiom,
    ! [Ma: nat,Nb: nat] : plus_p3455044024723400733d_enat(aa_nat_Extended_enat(extended_enat2,Ma),aa_nat_Extended_enat(extended_enat2,Nb)) = aa_nat_Extended_enat(extended_enat2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb)) ).

% plus_enat_simps(1)
tff(fact_118_numeral__less__enat__iff,axiom,
    ! [Ma: num,Nb: nat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(numera1916890842035813515d_enat(Ma)),aa_nat_Extended_enat(extended_enat2,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Ma)),Nb) ) ).

% numeral_less_enat_iff
tff(fact_119_enat__add__left__cancel__less,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(plus_p3455044024723400733d_enat(A,B)),plus_p3455044024723400733d_enat(A,C))
    <=> ( ( A != extend5688581933313929465d_enat )
        & aa_Extended_enat_o(ord_le72135733267957522d_enat(B),C) ) ) ).

% enat_add_left_cancel_less
tff(fact_120_plus__eq__infty__iff__enat,axiom,
    ! [Ma: extended_enat,Nb: extended_enat] :
      ( ( plus_p3455044024723400733d_enat(Ma,Nb) = extend5688581933313929465d_enat )
    <=> ( ( Ma = extend5688581933313929465d_enat )
        | ( Nb = extend5688581933313929465d_enat ) ) ) ).

% plus_eq_infty_iff_enat
tff(fact_121_enat__add__left__cancel,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( ( plus_p3455044024723400733d_enat(A,B) = plus_p3455044024723400733d_enat(A,C) )
    <=> ( ( A = extend5688581933313929465d_enat )
        | ( B = C ) ) ) ).

% enat_add_left_cancel
tff(fact_122_numeral__ne__infinity,axiom,
    ! [K2: num] : numera1916890842035813515d_enat(K2) != extend5688581933313929465d_enat ).

% numeral_ne_infinity
tff(fact_123_enat__iless,axiom,
    ! [Nb: extended_enat,Ma: nat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(Nb),aa_nat_Extended_enat(extended_enat2,Ma))
     => ? [K: nat] : Nb = aa_nat_Extended_enat(extended_enat2,K) ) ).

% enat_iless
tff(fact_124_less__enat__def,axiom,
    ! [Ma: extended_enat,Nb: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(Ma),Nb)
    <=> aa_Extended_enat_o(aa_o_f2257658052187763793enat_o(aa_fun2142289715968585979enat_o(extended_case_enat_o,aTP_Lamm_aj(Nb)),$false),Ma) ) ).

% less_enat_def
tff(fact_125_numeral__eq__enat,axiom,
    ! [K2: num] : numera1916890842035813515d_enat(K2) = aa_nat_Extended_enat(extended_enat2,aa_num_nat(numeral_numeral_nat,K2)) ).

% numeral_eq_enat
tff(fact_126_less__enatE,axiom,
    ! [Nb: extended_enat,Ma: nat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(Nb),aa_nat_Extended_enat(extended_enat2,Ma))
     => ~ ! [K: nat] :
            ( ( Nb = aa_nat_Extended_enat(extended_enat2,K) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),Ma) ) ) ).

% less_enatE
tff(fact_127_infinity__ilessE,axiom,
    ! [Ma: nat] : ~ aa_Extended_enat_o(ord_le72135733267957522d_enat(extend5688581933313929465d_enat),aa_nat_Extended_enat(extended_enat2,Ma)) ).

% infinity_ilessE
tff(fact_128_less__infinityE,axiom,
    ! [Nb: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(Nb),extend5688581933313929465d_enat)
     => ~ ! [K: nat] : Nb != aa_nat_Extended_enat(extended_enat2,K) ) ).

% less_infinityE
tff(fact_129_enat__ord__code_I4_J,axiom,
    ! [Ma: nat] : aa_Extended_enat_o(ord_le72135733267957522d_enat(aa_nat_Extended_enat(extended_enat2,Ma)),extend5688581933313929465d_enat) ).

% enat_ord_code(4)
tff(fact_130_enat__add__left__cancel__le,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(plus_p3455044024723400733d_enat(A,B)),plus_p3455044024723400733d_enat(A,C))
    <=> ( ( A = extend5688581933313929465d_enat )
        | aa_Extended_enat_o(ord_le2932123472753598470d_enat(B),C) ) ) ).

% enat_add_left_cancel_le
tff(fact_131_nat__neq__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( Ma != Nb )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Ma) ) ) ).

% nat_neq_iff
tff(fact_132_less__not__refl,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Nb) ).

% less_not_refl
tff(fact_133_less__not__refl2,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Ma)
     => ( Ma != Nb ) ) ).

% less_not_refl2
tff(fact_134_less__not__refl3,axiom,
    ! [S: nat,T: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,S),T)
     => ( S != T ) ) ).

% less_not_refl3
tff(fact_135_less__irrefl__nat,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Nb) ).

% less_irrefl_nat
tff(fact_136_nat__less__induct,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( ! [N: nat] :
          ( ! [M: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
             => aa_nat_o(P,M) )
         => aa_nat_o(P,N) )
     => aa_nat_o(P,Nb) ) ).

% nat_less_induct
tff(fact_137_infinite__descent,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( ! [N: nat] :
          ( ~ aa_nat_o(P,N)
         => ? [M: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
              & ~ aa_nat_o(P,M) ) )
     => aa_nat_o(P,Nb) ) ).

% infinite_descent
tff(fact_138_linorder__neqE__nat,axiom,
    ! [X: nat,Y2: nat] :
      ( ( X != Y2 )
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y2),X) ) ) ).

% linorder_neqE_nat
tff(fact_139_size__neq__size__imp__neq,axiom,
    ! [X: list_VEBT_VEBT,Y2: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(X) != size_s6755466524823107622T_VEBT(Y2) )
     => ( X != Y2 ) ) ).

% size_neq_size_imp_neq
tff(fact_140_size__neq__size__imp__neq,axiom,
    ! [X: list_o,Y2: list_o] :
      ( ( size_size_list_o(X) != size_size_list_o(Y2) )
     => ( X != Y2 ) ) ).

% size_neq_size_imp_neq
tff(fact_141_size__neq__size__imp__neq,axiom,
    ! [X: list_nat,Y2: list_nat] :
      ( ( size_size_list_nat(X) != size_size_list_nat(Y2) )
     => ( X != Y2 ) ) ).

% size_neq_size_imp_neq
tff(fact_142_size__neq__size__imp__neq,axiom,
    ! [X: list_int,Y2: list_int] :
      ( ( size_size_list_int(X) != size_size_list_int(Y2) )
     => ( X != Y2 ) ) ).

% size_neq_size_imp_neq
tff(fact_143_size__neq__size__imp__neq,axiom,
    ! [X: num,Y2: num] :
      ( ( aa_num_nat(size_size_num,X) != aa_num_nat(size_size_num,Y2) )
     => ( X != Y2 ) ) ).

% size_neq_size_imp_neq
tff(fact_144_Ex__list__of__length,axiom,
    ! [Nb: nat] :
    ? [Xs2: list_VEBT_VEBT] : size_s6755466524823107622T_VEBT(Xs2) = Nb ).

% Ex_list_of_length
tff(fact_145_Ex__list__of__length,axiom,
    ! [Nb: nat] :
    ? [Xs2: list_o] : size_size_list_o(Xs2) = Nb ).

% Ex_list_of_length
tff(fact_146_Ex__list__of__length,axiom,
    ! [Nb: nat] :
    ? [Xs2: list_nat] : size_size_list_nat(Xs2) = Nb ).

% Ex_list_of_length
tff(fact_147_Ex__list__of__length,axiom,
    ! [Nb: nat] :
    ? [Xs2: list_int] : size_size_list_int(Xs2) = Nb ).

% Ex_list_of_length
tff(fact_148_neq__if__length__neq,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) != size_s6755466524823107622T_VEBT(Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_149_neq__if__length__neq,axiom,
    ! [Xs: list_o,Ys: list_o] :
      ( ( size_size_list_o(Xs) != size_size_list_o(Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_150_neq__if__length__neq,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( size_size_list_nat(Xs) != size_size_list_nat(Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_151_neq__if__length__neq,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( size_size_list_int(Xs) != size_size_list_int(Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_152_take__equalityI,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ! [I3: nat] : take_VEBT_VEBT(I3,Xs) = take_VEBT_VEBT(I3,Ys)
     => ( Xs = Ys ) ) ).

% take_equalityI
tff(fact_153_take__equalityI,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ! [I3: nat] : take_nat(I3,Xs) = take_nat(I3,Ys)
     => ( Xs = Ys ) ) ).

% take_equalityI
tff(fact_154_plus__enat__def,axiom,
    ! [Ma: extended_enat,Nb: extended_enat] : plus_p3455044024723400733d_enat(Ma,Nb) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(aa_fun2186929853097002313d_enat(extend3600170679010898289d_enat,aTP_Lamm_al(Nb)),extend5688581933313929465d_enat),Ma) ).

% plus_enat_def
tff(fact_155_list_Omap__ident,axiom,
    ! [T: list_VEBT_VEBT] : map_VE8901447254227204932T_VEBT(aTP_Lamm_ah,T) = T ).

% list.map_ident
tff(fact_156_list_Omap__ident,axiom,
    ! [T: list_nat] : map_nat_nat(aTP_Lamm_ai,T) = T ).

% list.map_ident
tff(fact_157_power__divide,axiom,
    ! [A: complex,B: complex,Nb: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(divide1717551699836669952omplex(A),B)),Nb) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),Nb)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,B),Nb)) ).

% power_divide
tff(fact_158_power__divide,axiom,
    ! [A: real,B: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(divide_divide_real(A),B)),Nb) = aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)) ).

% power_divide
tff(fact_159_power__divide,axiom,
    ! [A: rat,B: rat,Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(divide_divide_rat(A),B)),Nb) = aa_rat_rat(divide_divide_rat(aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb)) ).

% power_divide
tff(fact_160_add__lessD1,axiom,
    ! [I2: nat,J: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),J)),K2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),K2) ) ).

% add_lessD1
tff(fact_161_add__less__mono,axiom,
    ! [I2: nat,J: nat,K2: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ) ).

% add_less_mono
tff(fact_162_not__add__less1,axiom,
    ! [I2: nat,J: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),J)),I2) ).

% not_add_less1
tff(fact_163_not__add__less2,axiom,
    ! [J: nat,I2: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I2)),I2) ).

% not_add_less2
tff(fact_164_add__less__mono1,axiom,
    ! [I2: nat,J: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K2)) ) ).

% add_less_mono1
tff(fact_165_trans__less__add1,axiom,
    ! [I2: nat,J: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),Ma)) ) ).

% trans_less_add1
tff(fact_166_trans__less__add2,axiom,
    ! [I2: nat,J: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),J)) ) ).

% trans_less_add2
tff(fact_167_less__add__eq__less,axiom,
    ! [K2: nat,L: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),L)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),L) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),Nb) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb) ) ) ).

% less_add_eq_less
tff(fact_168_length__induct,axiom,
    ! [P: fun_list_VEBT_VEBT_o,Xs: list_VEBT_VEBT] :
      ( ! [Xs2: list_VEBT_VEBT] :
          ( ! [Ys2: list_VEBT_VEBT] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_s6755466524823107622T_VEBT(Ys2)),size_s6755466524823107622T_VEBT(Xs2))
             => aa_list_VEBT_VEBT_o(P,Ys2) )
         => aa_list_VEBT_VEBT_o(P,Xs2) )
     => aa_list_VEBT_VEBT_o(P,Xs) ) ).

% length_induct
tff(fact_169_length__induct,axiom,
    ! [P: fun_list_o_o,Xs: list_o] :
      ( ! [Xs2: list_o] :
          ( ! [Ys2: list_o] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_o(Ys2)),size_size_list_o(Xs2))
             => aa_list_o_o(P,Ys2) )
         => aa_list_o_o(P,Xs2) )
     => aa_list_o_o(P,Xs) ) ).

% length_induct
tff(fact_170_length__induct,axiom,
    ! [P: fun_list_nat_o,Xs: list_nat] :
      ( ! [Xs2: list_nat] :
          ( ! [Ys2: list_nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_nat(Ys2)),size_size_list_nat(Xs2))
             => aa_list_nat_o(P,Ys2) )
         => aa_list_nat_o(P,Xs2) )
     => aa_list_nat_o(P,Xs) ) ).

% length_induct
tff(fact_171_length__induct,axiom,
    ! [P: fun_list_int_o,Xs: list_int] :
      ( ! [Xs2: list_int] :
          ( ! [Ys2: list_int] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_int(Ys2)),size_size_list_int(Xs2))
             => aa_list_int_o(P,Ys2) )
         => aa_list_int_o(P,Xs2) )
     => aa_list_int_o(P,Xs) ) ).

% length_induct
tff(fact_172_list_Omap__cong,axiom,
    ! [X: list_VEBT_VEBT,Ya: list_VEBT_VEBT,F: fun_VE3165209603783583043T_VEBT,G: fun_VE3165209603783583043T_VEBT] :
      ( ( X = Ya )
     => ( ! [Z: vEBT_VEBT] :
            ( member_VEBT_VEBT(Z,set_VEBT_VEBT2(Ya))
           => ( aa_VEB5832953974749077948T_VEBT(F,Z) = aa_VEB5832953974749077948T_VEBT(G,Z) ) )
       => ( map_VE8901447254227204932T_VEBT(F,X) = map_VE8901447254227204932T_VEBT(G,Ya) ) ) ) ).

% list.map_cong
tff(fact_173_list_Omap__cong,axiom,
    ! [X: list_nat,Ya: list_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( ( X = Ya )
     => ( ! [Z: nat] :
            ( aa_set_nat_o(member_nat(Z),set_nat2(Ya))
           => ( aa_nat_nat(F,Z) = aa_nat_nat(G,Z) ) )
       => ( map_nat_nat(F,X) = map_nat_nat(G,Ya) ) ) ) ).

% list.map_cong
tff(fact_174_list_Omap__cong0,axiom,
    ! [X: list_VEBT_VEBT,F: fun_VE3165209603783583043T_VEBT,G: fun_VE3165209603783583043T_VEBT] :
      ( ! [Z: vEBT_VEBT] :
          ( member_VEBT_VEBT(Z,set_VEBT_VEBT2(X))
         => ( aa_VEB5832953974749077948T_VEBT(F,Z) = aa_VEB5832953974749077948T_VEBT(G,Z) ) )
     => ( map_VE8901447254227204932T_VEBT(F,X) = map_VE8901447254227204932T_VEBT(G,X) ) ) ).

% list.map_cong0
tff(fact_175_list_Omap__cong0,axiom,
    ! [X: list_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( ! [Z: nat] :
          ( aa_set_nat_o(member_nat(Z),set_nat2(X))
         => ( aa_nat_nat(F,Z) = aa_nat_nat(G,Z) ) )
     => ( map_nat_nat(F,X) = map_nat_nat(G,X) ) ) ).

% list.map_cong0
tff(fact_176_list_Oinj__map__strong,axiom,
    ! [X: list_VEBT_VEBT,Xa: list_VEBT_VEBT,F: fun_VE3165209603783583043T_VEBT,Fa: fun_VE3165209603783583043T_VEBT] :
      ( ! [Z: vEBT_VEBT,Za: vEBT_VEBT] :
          ( member_VEBT_VEBT(Z,set_VEBT_VEBT2(X))
         => ( member_VEBT_VEBT(Za,set_VEBT_VEBT2(Xa))
           => ( ( aa_VEB5832953974749077948T_VEBT(F,Z) = aa_VEB5832953974749077948T_VEBT(Fa,Za) )
             => ( Z = Za ) ) ) )
     => ( ( map_VE8901447254227204932T_VEBT(F,X) = map_VE8901447254227204932T_VEBT(Fa,Xa) )
       => ( X = Xa ) ) ) ).

% list.inj_map_strong
tff(fact_177_list_Oinj__map__strong,axiom,
    ! [X: list_nat,Xa: list_nat,F: fun_nat_nat,Fa: fun_nat_nat] :
      ( ! [Z: nat,Za: nat] :
          ( aa_set_nat_o(member_nat(Z),set_nat2(X))
         => ( aa_set_nat_o(member_nat(Za),set_nat2(Xa))
           => ( ( aa_nat_nat(F,Z) = aa_nat_nat(Fa,Za) )
             => ( Z = Za ) ) ) )
     => ( ( map_nat_nat(F,X) = map_nat_nat(Fa,Xa) )
       => ( X = Xa ) ) ) ).

% list.inj_map_strong
tff(fact_178_map__ext,axiom,
    ! [Xs: list_VEBT_VEBT,F: fun_VE3165209603783583043T_VEBT,G: fun_VE3165209603783583043T_VEBT] :
      ( ! [X4: vEBT_VEBT] :
          ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(Xs))
         => ( aa_VEB5832953974749077948T_VEBT(F,X4) = aa_VEB5832953974749077948T_VEBT(G,X4) ) )
     => ( map_VE8901447254227204932T_VEBT(F,Xs) = map_VE8901447254227204932T_VEBT(G,Xs) ) ) ).

% map_ext
tff(fact_179_map__ext,axiom,
    ! [Xs: list_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),set_nat2(Xs))
         => ( aa_nat_nat(F,X4) = aa_nat_nat(G,X4) ) )
     => ( map_nat_nat(F,Xs) = map_nat_nat(G,Xs) ) ) ).

% map_ext
tff(fact_180_map__idI,axiom,
    ! [Xs: list_complex,F: fun_complex_complex] :
      ( ! [X4: complex] :
          ( member_complex(X4,set_complex2(Xs))
         => ( aa_complex_complex(F,X4) = X4 ) )
     => ( map_complex_complex(F,Xs) = Xs ) ) ).

% map_idI
tff(fact_181_map__idI,axiom,
    ! [Xs: list_real,F: fun_real_real] :
      ( ! [X4: real] :
          ( member_real(X4,set_real2(Xs))
         => ( aa_real_real(F,X4) = X4 ) )
     => ( map_real_real(F,Xs) = Xs ) ) ).

% map_idI
tff(fact_182_map__idI,axiom,
    ! [Xs: list_set_nat,F: fun_set_nat_set_nat] :
      ( ! [X4: set_nat] :
          ( member_set_nat(X4,set_set_nat2(Xs))
         => ( aa_set_nat_set_nat(F,X4) = X4 ) )
     => ( map_set_nat_set_nat(F,Xs) = Xs ) ) ).

% map_idI
tff(fact_183_map__idI,axiom,
    ! [Xs: list_VEBT_VEBT,F: fun_VE3165209603783583043T_VEBT] :
      ( ! [X4: vEBT_VEBT] :
          ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(Xs))
         => ( aa_VEB5832953974749077948T_VEBT(F,X4) = X4 ) )
     => ( map_VE8901447254227204932T_VEBT(F,Xs) = Xs ) ) ).

% map_idI
tff(fact_184_map__idI,axiom,
    ! [Xs: list_nat,F: fun_nat_nat] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),set_nat2(Xs))
         => ( aa_nat_nat(F,X4) = X4 ) )
     => ( map_nat_nat(F,Xs) = Xs ) ) ).

% map_idI
tff(fact_185_map__idI,axiom,
    ! [Xs: list_int,F: fun_int_int] :
      ( ! [X4: int] :
          ( member_int(X4,set_int2(Xs))
         => ( aa_int_int(F,X4) = X4 ) )
     => ( map_int_int(F,Xs) = Xs ) ) ).

% map_idI
tff(fact_186_map__cong,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,F: fun_VE3165209603783583043T_VEBT,G: fun_VE3165209603783583043T_VEBT] :
      ( ( Xs = Ys )
     => ( ! [X4: vEBT_VEBT] :
            ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(Ys))
           => ( aa_VEB5832953974749077948T_VEBT(F,X4) = aa_VEB5832953974749077948T_VEBT(G,X4) ) )
       => ( map_VE8901447254227204932T_VEBT(F,Xs) = map_VE8901447254227204932T_VEBT(G,Ys) ) ) ) ).

% map_cong
tff(fact_187_map__cong,axiom,
    ! [Xs: list_nat,Ys: list_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( ( Xs = Ys )
     => ( ! [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),set_nat2(Ys))
           => ( aa_nat_nat(F,X4) = aa_nat_nat(G,X4) ) )
       => ( map_nat_nat(F,Xs) = map_nat_nat(G,Ys) ) ) ) ).

% map_cong
tff(fact_188_ex__map__conv,axiom,
    ! [Ys: list_VEBT_VEBT,F: fun_VE3165209603783583043T_VEBT] :
      ( ? [Xs3: list_VEBT_VEBT] : Ys = map_VE8901447254227204932T_VEBT(F,Xs3)
    <=> ! [X3: vEBT_VEBT] :
          ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(Ys))
         => ? [Xa2: vEBT_VEBT] : X3 = aa_VEB5832953974749077948T_VEBT(F,Xa2) ) ) ).

% ex_map_conv
tff(fact_189_ex__map__conv,axiom,
    ! [Ys: list_nat,F: fun_nat_nat] :
      ( ? [Xs3: list_nat] : Ys = map_nat_nat(F,Xs3)
    <=> ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),set_nat2(Ys))
         => ? [Xa2: nat] : X3 = aa_nat_nat(F,Xa2) ) ) ).

% ex_map_conv
tff(fact_190_map__eq__imp__length__eq,axiom,
    ! [F: fun_VE3165209603783583043T_VEBT,Xs: list_VEBT_VEBT,G: fun_VE3165209603783583043T_VEBT,Ys: list_VEBT_VEBT] :
      ( ( map_VE8901447254227204932T_VEBT(F,Xs) = map_VE8901447254227204932T_VEBT(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_191_map__eq__imp__length__eq,axiom,
    ! [F: fun_VE3165209603783583043T_VEBT,Xs: list_VEBT_VEBT,G: fun_o_VEBT_VEBT,Ys: list_o] :
      ( ( map_VE8901447254227204932T_VEBT(F,Xs) = map_o_VEBT_VEBT(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_o(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_192_map__eq__imp__length__eq,axiom,
    ! [F: fun_VEBT_VEBT_nat,Xs: list_VEBT_VEBT,G: fun_nat_nat,Ys: list_nat] :
      ( ( map_VEBT_VEBT_nat(F,Xs) = map_nat_nat(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_nat(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_193_map__eq__imp__length__eq,axiom,
    ! [F: fun_VE3165209603783583043T_VEBT,Xs: list_VEBT_VEBT,G: fun_nat_VEBT_VEBT,Ys: list_nat] :
      ( ( map_VE8901447254227204932T_VEBT(F,Xs) = map_nat_VEBT_VEBT(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_nat(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_194_map__eq__imp__length__eq,axiom,
    ! [F: fun_VE3165209603783583043T_VEBT,Xs: list_VEBT_VEBT,G: fun_int_VEBT_VEBT,Ys: list_int] :
      ( ( map_VE8901447254227204932T_VEBT(F,Xs) = map_int_VEBT_VEBT(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_int(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_195_map__eq__imp__length__eq,axiom,
    ! [F: fun_o_VEBT_VEBT,Xs: list_o,G: fun_VE3165209603783583043T_VEBT,Ys: list_VEBT_VEBT] :
      ( ( map_o_VEBT_VEBT(F,Xs) = map_VE8901447254227204932T_VEBT(G,Ys) )
     => ( size_size_list_o(Xs) = size_s6755466524823107622T_VEBT(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_196_map__eq__imp__length__eq,axiom,
    ! [F: fun_o_nat,Xs: list_o,G: fun_nat_nat,Ys: list_nat] :
      ( ( map_o_nat(F,Xs) = map_nat_nat(G,Ys) )
     => ( size_size_list_o(Xs) = size_size_list_nat(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_197_map__eq__imp__length__eq,axiom,
    ! [F: fun_nat_VEBT_VEBT,Xs: list_nat,G: fun_VE3165209603783583043T_VEBT,Ys: list_VEBT_VEBT] :
      ( ( map_nat_VEBT_VEBT(F,Xs) = map_VE8901447254227204932T_VEBT(G,Ys) )
     => ( size_size_list_nat(Xs) = size_s6755466524823107622T_VEBT(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_198_map__eq__imp__length__eq,axiom,
    ! [F: fun_nat_nat,Xs: list_nat,G: fun_VEBT_VEBT_nat,Ys: list_VEBT_VEBT] :
      ( ( map_nat_nat(F,Xs) = map_VEBT_VEBT_nat(G,Ys) )
     => ( size_size_list_nat(Xs) = size_s6755466524823107622T_VEBT(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_199_map__eq__imp__length__eq,axiom,
    ! [F: fun_nat_nat,Xs: list_nat,G: fun_o_nat,Ys: list_o] :
      ( ( map_nat_nat(F,Xs) = map_o_nat(G,Ys) )
     => ( size_size_list_nat(Xs) = size_size_list_o(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_200_in__set__takeD,axiom,
    ! [X: complex,Nb: nat,Xs: list_complex] :
      ( member_complex(X,set_complex2(take_complex(Nb,Xs)))
     => member_complex(X,set_complex2(Xs)) ) ).

% in_set_takeD
tff(fact_201_in__set__takeD,axiom,
    ! [X: real,Nb: nat,Xs: list_real] :
      ( member_real(X,set_real2(take_real(Nb,Xs)))
     => member_real(X,set_real2(Xs)) ) ).

% in_set_takeD
tff(fact_202_in__set__takeD,axiom,
    ! [X: set_nat,Nb: nat,Xs: list_set_nat] :
      ( member_set_nat(X,set_set_nat2(take_set_nat(Nb,Xs)))
     => member_set_nat(X,set_set_nat2(Xs)) ) ).

% in_set_takeD
tff(fact_203_in__set__takeD,axiom,
    ! [X: vEBT_VEBT,Nb: nat,Xs: list_VEBT_VEBT] :
      ( member_VEBT_VEBT(X,set_VEBT_VEBT2(take_VEBT_VEBT(Nb,Xs)))
     => member_VEBT_VEBT(X,set_VEBT_VEBT2(Xs)) ) ).

% in_set_takeD
tff(fact_204_in__set__takeD,axiom,
    ! [X: nat,Nb: nat,Xs: list_nat] :
      ( aa_set_nat_o(member_nat(X),set_nat2(take_nat(Nb,Xs)))
     => aa_set_nat_o(member_nat(X),set_nat2(Xs)) ) ).

% in_set_takeD
tff(fact_205_in__set__takeD,axiom,
    ! [X: int,Nb: nat,Xs: list_int] :
      ( member_int(X,set_int2(take_int(Nb,Xs)))
     => member_int(X,set_int2(Xs)) ) ).

% in_set_takeD
tff(fact_206_take__map,axiom,
    ! [Nb: nat,F: fun_nat_VEBT_VEBT,Xs: list_nat] : take_VEBT_VEBT(Nb,map_nat_VEBT_VEBT(F,Xs)) = map_nat_VEBT_VEBT(F,take_nat(Nb,Xs)) ).

% take_map
tff(fact_207_take__map,axiom,
    ! [Nb: nat,F: fun_VEBT_VEBT_nat,Xs: list_VEBT_VEBT] : take_nat(Nb,map_VEBT_VEBT_nat(F,Xs)) = map_VEBT_VEBT_nat(F,take_VEBT_VEBT(Nb,Xs)) ).

% take_map
tff(fact_208_take__map,axiom,
    ! [Nb: nat,F: fun_VE3165209603783583043T_VEBT,Xs: list_VEBT_VEBT] : take_VEBT_VEBT(Nb,map_VE8901447254227204932T_VEBT(F,Xs)) = map_VE8901447254227204932T_VEBT(F,take_VEBT_VEBT(Nb,Xs)) ).

% take_map
tff(fact_209_take__map,axiom,
    ! [Nb: nat,F: fun_nat_nat,Xs: list_nat] : take_nat(Nb,map_nat_nat(F,Xs)) = map_nat_nat(F,take_nat(Nb,Xs)) ).

% take_map
tff(fact_210_nth__equalityI,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) )
     => ( ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s6755466524823107622T_VEBT(Xs))
           => ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I3) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Ys),I3) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_211_nth__equalityI,axiom,
    ! [Xs: list_o,Ys: list_o] :
      ( ( size_size_list_o(Xs) = size_size_list_o(Ys) )
     => ( ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_o(Xs))
           => ( aa_nat_o(nth_o(Xs),I3)
            <=> aa_nat_o(nth_o(Ys),I3) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_212_nth__equalityI,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( size_size_list_nat(Xs) = size_size_list_nat(Ys) )
     => ( ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_nat(Xs))
           => ( aa_nat_nat(nth_nat(Xs),I3) = aa_nat_nat(nth_nat(Ys),I3) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_213_nth__equalityI,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( size_size_list_int(Xs) = size_size_list_int(Ys) )
     => ( ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_int(Xs))
           => ( aa_nat_int(nth_int(Xs),I3) = aa_nat_int(nth_int(Ys),I3) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_214_Skolem__list__nth,axiom,
    ! [K2: nat,P: fun_na6887940067898170868VEBT_o] :
      ( ! [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),K2)
         => ? [X_12: vEBT_VEBT] : aa_VEBT_VEBT_o(aa_nat1925708948643195573VEBT_o(P,I),X_12) )
    <=> ? [Xs3: list_VEBT_VEBT] :
          ( ( size_s6755466524823107622T_VEBT(Xs3) = K2 )
          & ! [I: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),K2)
             => aa_VEBT_VEBT_o(aa_nat1925708948643195573VEBT_o(P,I),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs3),I)) ) ) ) ).

% Skolem_list_nth
tff(fact_215_Skolem__list__nth,axiom,
    ! [K2: nat,P: fun_nat_fun_o_o] :
      ( ! [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),K2)
         => ? [X_12: $o] : aa_o_o(aa_nat_fun_o_o(P,I),(X_12)) )
    <=> ? [Xs3: list_o] :
          ( ( size_size_list_o(Xs3) = K2 )
          & ! [I: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),K2)
             => aa_o_o(aa_nat_fun_o_o(P,I),aa_nat_o(nth_o(Xs3),I)) ) ) ) ).

% Skolem_list_nth
tff(fact_216_Skolem__list__nth,axiom,
    ! [K2: nat,P: fun_nat_fun_nat_o] :
      ( ! [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),K2)
         => ? [X_12: nat] : aa_nat_o(aa_nat_fun_nat_o(P,I),X_12) )
    <=> ? [Xs3: list_nat] :
          ( ( size_size_list_nat(Xs3) = K2 )
          & ! [I: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),K2)
             => aa_nat_o(aa_nat_fun_nat_o(P,I),aa_nat_nat(nth_nat(Xs3),I)) ) ) ) ).

% Skolem_list_nth
tff(fact_217_Skolem__list__nth,axiom,
    ! [K2: nat,P: fun_nat_fun_int_o] :
      ( ! [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),K2)
         => ? [X_12: int] : aa_int_o(aa_nat_fun_int_o(P,I),X_12) )
    <=> ? [Xs3: list_int] :
          ( ( size_size_list_int(Xs3) = K2 )
          & ! [I: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),K2)
             => aa_int_o(aa_nat_fun_int_o(P,I),aa_nat_int(nth_int(Xs3),I)) ) ) ) ).

% Skolem_list_nth
tff(fact_218_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( Xs = Ys )
    <=> ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) )
        & ! [I: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
           => ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Ys),I) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_219_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_o,Ys: list_o] :
      ( ( Xs = Ys )
    <=> ( ( size_size_list_o(Xs) = size_size_list_o(Ys) )
        & ! [I: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
           => ( aa_nat_o(nth_o(Xs),I)
            <=> aa_nat_o(nth_o(Ys),I) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_220_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( Xs = Ys )
    <=> ( ( size_size_list_nat(Xs) = size_size_list_nat(Ys) )
        & ! [I: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(Xs))
           => ( aa_nat_nat(nth_nat(Xs),I) = aa_nat_nat(nth_nat(Ys),I) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_221_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( Xs = Ys )
    <=> ( ( size_size_list_int(Xs) = size_size_list_int(Ys) )
        & ! [I: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
           => ( aa_nat_int(nth_int(Xs),I) = aa_nat_int(nth_int(Ys),I) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_222_numeral__Bit0__div__2,axiom,
    ! [Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_num_nat(numeral_numeral_nat,Nb) ).

% numeral_Bit0_div_2
tff(fact_223_numeral__Bit0__div__2,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_num_int(numeral_numeral_int,Nb) ).

% numeral_Bit0_div_2
tff(fact_224_all__set__conv__all__nth,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( ! [X3: vEBT_VEBT] :
          ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(Xs))
         => aa_VEBT_VEBT_o(P,X3) )
    <=> ! [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
         => aa_VEBT_VEBT_o(P,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)) ) ) ).

% all_set_conv_all_nth
tff(fact_225_all__set__conv__all__nth,axiom,
    ! [Xs: list_o,P: fun_o_o] :
      ( ! [X3: $o] :
          ( member_o((X3),set_o2(Xs))
         => aa_o_o(P,(X3)) )
    <=> ! [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
         => aa_o_o(P,aa_nat_o(nth_o(Xs),I)) ) ) ).

% all_set_conv_all_nth
tff(fact_226_all__set__conv__all__nth,axiom,
    ! [Xs: list_nat,P: fun_nat_o] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),set_nat2(Xs))
         => aa_nat_o(P,X3) )
    <=> ! [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(Xs))
         => aa_nat_o(P,aa_nat_nat(nth_nat(Xs),I)) ) ) ).

% all_set_conv_all_nth
tff(fact_227_all__set__conv__all__nth,axiom,
    ! [Xs: list_int,P: fun_int_o] :
      ( ! [X3: int] :
          ( member_int(X3,set_int2(Xs))
         => aa_int_o(P,X3) )
    <=> ! [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
         => aa_int_o(P,aa_nat_int(nth_int(Xs),I)) ) ) ).

% all_set_conv_all_nth
tff(fact_228_all__nth__imp__all__set,axiom,
    ! [Xs: list_complex,P: fun_complex_o,X: complex] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s3451745648224563538omplex(Xs))
         => aa_complex_o(P,aa_nat_complex(nth_complex(Xs),I3)) )
     => ( member_complex(X,set_complex2(Xs))
       => aa_complex_o(P,X) ) ) ).

% all_nth_imp_all_set
tff(fact_229_all__nth__imp__all__set,axiom,
    ! [Xs: list_real,P: fun_real_o,X: real] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_real(Xs))
         => aa_real_o(P,aa_nat_real(nth_real(Xs),I3)) )
     => ( member_real(X,set_real2(Xs))
       => aa_real_o(P,X) ) ) ).

% all_nth_imp_all_set
tff(fact_230_all__nth__imp__all__set,axiom,
    ! [Xs: list_set_nat,P: fun_set_nat_o,X: set_nat] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s3254054031482475050et_nat(Xs))
         => aa_set_nat_o(P,nth_set_nat(Xs,I3)) )
     => ( member_set_nat(X,set_set_nat2(Xs))
       => aa_set_nat_o(P,X) ) ) ).

% all_nth_imp_all_set
tff(fact_231_all__nth__imp__all__set,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_o,X: vEBT_VEBT] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s6755466524823107622T_VEBT(Xs))
         => aa_VEBT_VEBT_o(P,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I3)) )
     => ( member_VEBT_VEBT(X,set_VEBT_VEBT2(Xs))
       => aa_VEBT_VEBT_o(P,X) ) ) ).

% all_nth_imp_all_set
tff(fact_232_all__nth__imp__all__set,axiom,
    ! [Xs: list_o,P: fun_o_o,X: $o] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_o(Xs))
         => aa_o_o(P,aa_nat_o(nth_o(Xs),I3)) )
     => ( member_o((X),set_o2(Xs))
       => aa_o_o(P,(X)) ) ) ).

% all_nth_imp_all_set
tff(fact_233_all__nth__imp__all__set,axiom,
    ! [Xs: list_nat,P: fun_nat_o,X: nat] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_nat(Xs))
         => aa_nat_o(P,aa_nat_nat(nth_nat(Xs),I3)) )
     => ( aa_set_nat_o(member_nat(X),set_nat2(Xs))
       => aa_nat_o(P,X) ) ) ).

% all_nth_imp_all_set
tff(fact_234_all__nth__imp__all__set,axiom,
    ! [Xs: list_int,P: fun_int_o,X: int] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_int(Xs))
         => aa_int_o(P,aa_nat_int(nth_int(Xs),I3)) )
     => ( member_int(X,set_int2(Xs))
       => aa_int_o(P,X) ) ) ).

% all_nth_imp_all_set
tff(fact_235_in__set__conv__nth,axiom,
    ! [X: complex,Xs: list_complex] :
      ( member_complex(X,set_complex2(Xs))
    <=> ? [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s3451745648224563538omplex(Xs))
          & ( aa_nat_complex(nth_complex(Xs),I) = X ) ) ) ).

% in_set_conv_nth
tff(fact_236_in__set__conv__nth,axiom,
    ! [X: real,Xs: list_real] :
      ( member_real(X,set_real2(Xs))
    <=> ? [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(Xs))
          & ( aa_nat_real(nth_real(Xs),I) = X ) ) ) ).

% in_set_conv_nth
tff(fact_237_in__set__conv__nth,axiom,
    ! [X: set_nat,Xs: list_set_nat] :
      ( member_set_nat(X,set_set_nat2(Xs))
    <=> ? [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s3254054031482475050et_nat(Xs))
          & ( nth_set_nat(Xs,I) = X ) ) ) ).

% in_set_conv_nth
tff(fact_238_in__set__conv__nth,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( member_VEBT_VEBT(X,set_VEBT_VEBT2(Xs))
    <=> ? [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
          & ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I) = X ) ) ) ).

% in_set_conv_nth
tff(fact_239_in__set__conv__nth,axiom,
    ! [X: $o,Xs: list_o] :
      ( member_o((X),set_o2(Xs))
    <=> ? [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
          & ( aa_nat_o(nth_o(Xs),I)
          <=> (X) ) ) ) ).

% in_set_conv_nth
tff(fact_240_in__set__conv__nth,axiom,
    ! [X: nat,Xs: list_nat] :
      ( aa_set_nat_o(member_nat(X),set_nat2(Xs))
    <=> ? [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(Xs))
          & ( aa_nat_nat(nth_nat(Xs),I) = X ) ) ) ).

% in_set_conv_nth
tff(fact_241_in__set__conv__nth,axiom,
    ! [X: int,Xs: list_int] :
      ( member_int(X,set_int2(Xs))
    <=> ? [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
          & ( aa_nat_int(nth_int(Xs),I) = X ) ) ) ).

% in_set_conv_nth
tff(fact_242_list__ball__nth,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s6755466524823107622T_VEBT(Xs))
     => ( ! [X4: vEBT_VEBT] :
            ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(Xs))
           => aa_VEBT_VEBT_o(P,X4) )
       => aa_VEBT_VEBT_o(P,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),Nb)) ) ) ).

% list_ball_nth
tff(fact_243_list__ball__nth,axiom,
    ! [Nb: nat,Xs: list_o,P: fun_o_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_o(Xs))
     => ( ! [X4: $o] :
            ( member_o((X4),set_o2(Xs))
           => aa_o_o(P,(X4)) )
       => aa_o_o(P,aa_nat_o(nth_o(Xs),Nb)) ) ) ).

% list_ball_nth
tff(fact_244_list__ball__nth,axiom,
    ! [Nb: nat,Xs: list_nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_nat(Xs))
     => ( ! [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),set_nat2(Xs))
           => aa_nat_o(P,X4) )
       => aa_nat_o(P,aa_nat_nat(nth_nat(Xs),Nb)) ) ) ).

% list_ball_nth
tff(fact_245_list__ball__nth,axiom,
    ! [Nb: nat,Xs: list_int,P: fun_int_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_int(Xs))
     => ( ! [X4: int] :
            ( member_int(X4,set_int2(Xs))
           => aa_int_o(P,X4) )
       => aa_int_o(P,aa_nat_int(nth_int(Xs),Nb)) ) ) ).

% list_ball_nth
tff(fact_246_nth__mem,axiom,
    ! [Nb: nat,Xs: list_complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s3451745648224563538omplex(Xs))
     => member_complex(aa_nat_complex(nth_complex(Xs),Nb),set_complex2(Xs)) ) ).

% nth_mem
tff(fact_247_nth__mem,axiom,
    ! [Nb: nat,Xs: list_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_real(Xs))
     => member_real(aa_nat_real(nth_real(Xs),Nb),set_real2(Xs)) ) ).

% nth_mem
tff(fact_248_nth__mem,axiom,
    ! [Nb: nat,Xs: list_set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s3254054031482475050et_nat(Xs))
     => member_set_nat(nth_set_nat(Xs,Nb),set_set_nat2(Xs)) ) ).

% nth_mem
tff(fact_249_nth__mem,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s6755466524823107622T_VEBT(Xs))
     => member_VEBT_VEBT(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),Nb),set_VEBT_VEBT2(Xs)) ) ).

% nth_mem
tff(fact_250_nth__mem,axiom,
    ! [Nb: nat,Xs: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_o(Xs))
     => member_o(aa_nat_o(nth_o(Xs),Nb),set_o2(Xs)) ) ).

% nth_mem
tff(fact_251_nth__mem,axiom,
    ! [Nb: nat,Xs: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_nat(Xs))
     => aa_set_nat_o(member_nat(aa_nat_nat(nth_nat(Xs),Nb)),set_nat2(Xs)) ) ).

% nth_mem
tff(fact_252_nth__mem,axiom,
    ! [Nb: nat,Xs: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_int(Xs))
     => member_int(aa_nat_int(nth_int(Xs),Nb),set_int2(Xs)) ) ).

% nth_mem
tff(fact_253_field__sum__of__halves,axiom,
    ! [X: real] : aa_real_real(plus_plus_real(aa_real_real(divide_divide_real(X),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_real_real(divide_divide_real(X),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = X ).

% field_sum_of_halves
tff(fact_254_field__sum__of__halves,axiom,
    ! [X: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(X),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_rat_rat(divide_divide_rat(X),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) = X ).

% field_sum_of_halves
tff(fact_255_both__member__options__ding,axiom,
    ! [Infoa: option4927543243414619207at_nat,Degb: nat,TreeListb: list_VEBT_VEBT,Summaryb: vEBT_VEBT,Nb: nat,X: nat] :
      ( aa_nat_o(vEBT_invar_vebt(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Infoa,Degb,TreeListb),Summaryb)),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Degb))
       => ( aa_nat_o(vEBT_V8194947554948674370ptions(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeListb),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Degb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Degb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
         => aa_nat_o(vEBT_V8194947554948674370ptions(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Infoa,Degb,TreeListb),Summaryb)),X) ) ) ) ).

% both_member_options_ding
tff(fact_256_high__inv,axiom,
    ! [X: nat,Nb: nat,Y2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),X)),Nb) = Y2 ) ) ).

% high_inv
tff(fact_257_numeral__plus__numeral,axiom,
    ! [Ma: num,Nb: num] : aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Ma)),numera6690914467698888265omplex(Nb)) = numera6690914467698888265omplex(aa_num_num(plus_plus_num(Ma),Nb)) ).

% numeral_plus_numeral
tff(fact_258_numeral__plus__numeral,axiom,
    ! [Ma: num,Nb: num] : aa_real_real(plus_plus_real(aa_num_real(numeral_numeral_real,Ma)),aa_num_real(numeral_numeral_real,Nb)) = aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(Ma),Nb)) ).

% numeral_plus_numeral
tff(fact_259_numeral__plus__numeral,axiom,
    ! [Ma: num,Nb: num] : aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,Ma)),aa_num_rat(numeral_numeral_rat,Nb)) = aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(Ma),Nb)) ).

% numeral_plus_numeral
tff(fact_260_numeral__plus__numeral,axiom,
    ! [Ma: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Ma)),aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(Ma),Nb)) ).

% numeral_plus_numeral
tff(fact_261_numeral__plus__numeral,axiom,
    ! [Ma: num,Nb: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Ma)),aa_num_int(numeral_numeral_int,Nb)) = aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(Ma),Nb)) ).

% numeral_plus_numeral
tff(fact_262_add__numeral__left,axiom,
    ! [V: num,W: num,Z2: complex] : aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(V)),aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(W)),Z2)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(aa_num_num(plus_plus_num(V),W))),Z2) ).

% add_numeral_left
tff(fact_263_add__numeral__left,axiom,
    ! [V: num,W: num,Z2: real] : aa_real_real(plus_plus_real(aa_num_real(numeral_numeral_real,V)),aa_real_real(plus_plus_real(aa_num_real(numeral_numeral_real,W)),Z2)) = aa_real_real(plus_plus_real(aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(V),W))),Z2) ).

% add_numeral_left
tff(fact_264_add__numeral__left,axiom,
    ! [V: num,W: num,Z2: rat] : aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,W)),Z2)) = aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(V),W))),Z2) ).

% add_numeral_left
tff(fact_265_add__numeral__left,axiom,
    ! [V: num,W: num,Z2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,V)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,W)),Z2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(V),W))),Z2) ).

% add_numeral_left
tff(fact_266_add__numeral__left,axiom,
    ! [V: num,W: num,Z2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,W)),Z2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(V),W))),Z2) ).

% add_numeral_left
tff(fact_267_add__less__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(C),A)),aa_real_real(plus_plus_real(C),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% add_less_cancel_left
tff(fact_268_add__less__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(C),A)),aa_rat_rat(plus_plus_rat(C),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% add_less_cancel_left
tff(fact_269_add__less__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_cancel_left
tff(fact_270_add__less__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_cancel_left
tff(fact_271_add__less__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(A),C)),aa_real_real(plus_plus_real(B),C))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% add_less_cancel_right
tff(fact_272_add__less__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),C))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% add_less_cancel_right
tff(fact_273_add__less__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_cancel_right
tff(fact_274_add__less__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_cancel_right
tff(fact_275_add__le__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(C),A)),aa_real_real(plus_plus_real(C),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% add_le_cancel_left
tff(fact_276_add__le__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(C),A)),aa_rat_rat(plus_plus_rat(C),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% add_le_cancel_left
tff(fact_277_add__le__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_cancel_left
tff(fact_278_add__le__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_cancel_left
tff(fact_279_add__le__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(A),C)),aa_real_real(plus_plus_real(B),C))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% add_le_cancel_right
tff(fact_280_add__le__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),C))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% add_le_cancel_right
tff(fact_281_add__le__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_cancel_right
tff(fact_282_add__le__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_cancel_right
tff(fact_283_numeral__less__iff,axiom,
    ! [Ma: num,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,Ma)),aa_num_real(numeral_numeral_real,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Ma),Nb) ) ).

% numeral_less_iff
tff(fact_284_numeral__less__iff,axiom,
    ! [Ma: num,Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,Ma)),aa_num_rat(numeral_numeral_rat,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Ma),Nb) ) ).

% numeral_less_iff
tff(fact_285_numeral__less__iff,axiom,
    ! [Ma: num,Nb: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Ma)),aa_num_nat(numeral_numeral_nat,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Ma),Nb) ) ).

% numeral_less_iff
tff(fact_286_numeral__less__iff,axiom,
    ! [Ma: num,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Ma)),aa_num_int(numeral_numeral_int,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Ma),Nb) ) ).

% numeral_less_iff
tff(fact_287_numeral__le__iff,axiom,
    ! [Ma: num,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Ma)),aa_num_real(numeral_numeral_real,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ma),Nb) ) ).

% numeral_le_iff
tff(fact_288_numeral__le__iff,axiom,
    ! [Ma: num,Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Ma)),aa_num_rat(numeral_numeral_rat,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ma),Nb) ) ).

% numeral_le_iff
tff(fact_289_numeral__le__iff,axiom,
    ! [Ma: num,Nb: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Ma)),aa_num_nat(numeral_numeral_nat,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ma),Nb) ) ).

% numeral_le_iff
tff(fact_290_numeral__le__iff,axiom,
    ! [Ma: num,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Ma)),aa_num_int(numeral_numeral_int,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ma),Nb) ) ).

% numeral_le_iff
tff(fact_291_VEBT__internal_Oelim__dead_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa: extended_enat,Y2: vEBT_VEBT] :
      ( ( vEBT_VEBT_elim_dead(X,Xa) = Y2 )
     => ( ! [A3: $o,B2: $o] :
            ( ( X = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B2)) )
           => ( Y2 != aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B2)) ) )
       => ( ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList),Summary) )
             => ( ( Xa = extend5688581933313929465d_enat )
               => ( Y2 != aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,map_VE8901447254227204932T_VEBT(aTP_Lamm_aa(Deg),TreeList)),vEBT_VEBT_elim_dead(Summary,extend5688581933313929465d_enat)) ) ) )
         => ~ ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList),Summary) )
               => ! [L2: nat] :
                    ( ( Xa = aa_nat_Extended_enat(extended_enat2,L2) )
                   => ( Y2 != aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,take_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,L2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),map_VE8901447254227204932T_VEBT(aTP_Lamm_aa(Deg),TreeList))),vEBT_VEBT_elim_dead(Summary,aa_nat_Extended_enat(extended_enat2,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,L2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))) ) ) ) ) ) ) ).

% VEBT_internal.elim_dead.elims
tff(fact_292_numeral__code_I2_J,axiom,
    ! [Nb: num] :
      numera6690914467698888265omplex(aa_num_num(bit0,Nb)) = $let(
        m2: complex,
        m2:= numera6690914467698888265omplex(Nb),
        aa_complex_complex(plus_plus_complex(m2),m2) ) ).

% numeral_code(2)
tff(fact_293_numeral__code_I2_J,axiom,
    ! [Nb: num] :
      aa_num_real(numeral_numeral_real,aa_num_num(bit0,Nb)) = $let(
        m2: real,
        m2:= aa_num_real(numeral_numeral_real,Nb),
        aa_real_real(plus_plus_real(m2),m2) ) ).

% numeral_code(2)
tff(fact_294_numeral__code_I2_J,axiom,
    ! [Nb: num] :
      aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,Nb)) = $let(
        m2: rat,
        m2:= aa_num_rat(numeral_numeral_rat,Nb),
        aa_rat_rat(plus_plus_rat(m2),m2) ) ).

% numeral_code(2)
tff(fact_295_numeral__code_I2_J,axiom,
    ! [Nb: num] :
      aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb)) = $let(
        m2: nat,
        m2:= aa_num_nat(numeral_numeral_nat,Nb),
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,m2),m2) ) ).

% numeral_code(2)
tff(fact_296_numeral__code_I2_J,axiom,
    ! [Nb: num] :
      aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)) = $let(
        m2: int,
        m2:= aa_num_int(numeral_numeral_int,Nb),
        aa_int_int(aa_int_fun_int_int(plus_plus_int,m2),m2) ) ).

% numeral_code(2)
tff(fact_297_numeral__eq__iff,axiom,
    ! [Ma: num,Nb: num] :
      ( ( numera6690914467698888265omplex(Ma) = numera6690914467698888265omplex(Nb) )
    <=> ( Ma = Nb ) ) ).

% numeral_eq_iff
tff(fact_298_numeral__eq__iff,axiom,
    ! [Ma: num,Nb: num] :
      ( ( aa_num_real(numeral_numeral_real,Ma) = aa_num_real(numeral_numeral_real,Nb) )
    <=> ( Ma = Nb ) ) ).

% numeral_eq_iff
tff(fact_299_numeral__eq__iff,axiom,
    ! [Ma: num,Nb: num] :
      ( ( aa_num_rat(numeral_numeral_rat,Ma) = aa_num_rat(numeral_numeral_rat,Nb) )
    <=> ( Ma = Nb ) ) ).

% numeral_eq_iff
tff(fact_300_numeral__eq__iff,axiom,
    ! [Ma: num,Nb: num] :
      ( ( aa_num_nat(numeral_numeral_nat,Ma) = aa_num_nat(numeral_numeral_nat,Nb) )
    <=> ( Ma = Nb ) ) ).

% numeral_eq_iff
tff(fact_301_numeral__eq__iff,axiom,
    ! [Ma: num,Nb: num] :
      ( ( aa_num_int(numeral_numeral_int,Ma) = aa_num_int(numeral_numeral_int,Nb) )
    <=> ( Ma = Nb ) ) ).

% numeral_eq_iff
tff(fact_302_add__right__cancel,axiom,
    ! [B: real,A: real,C: real] :
      ( ( aa_real_real(plus_plus_real(B),A) = aa_real_real(plus_plus_real(C),A) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_303_add__right__cancel,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( aa_rat_rat(plus_plus_rat(B),A) = aa_rat_rat(plus_plus_rat(C),A) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_304_add__right__cancel,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_305_add__right__cancel,axiom,
    ! [B: int,A: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_306_add__left__cancel,axiom,
    ! [A: real,B: real,C: real] :
      ( ( aa_real_real(plus_plus_real(A),B) = aa_real_real(plus_plus_real(A),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_307_add__left__cancel,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),B) = aa_rat_rat(plus_plus_rat(A),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_308_add__left__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_309_add__left__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_310_nat__add__left__cancel__le,axiom,
    ! [K2: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ).

% nat_add_left_cancel_le
tff(fact_311_VEBT_Oinject_I2_J,axiom,
    ! [X21: $o,X22: $o,Y21: $o,Y22: $o] :
      ( ( aa_o_VEBT_VEBT(vEBT_Leaf((X21)),(X22)) = aa_o_VEBT_VEBT(vEBT_Leaf((Y21)),(Y22)) )
    <=> ( ( (X21)
        <=> (Y21) )
        & ( (X22)
        <=> (Y22) ) ) ) ).

% VEBT.inject(2)
tff(fact_312_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z2: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),Z2)) = aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(times_times_num(V),W))),Z2) ).

% mult_numeral_left_semiring_numeral
tff(fact_313_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z2: real] : aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,V)),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,W)),Z2)) = aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(times_times_num(V),W))),Z2) ).

% mult_numeral_left_semiring_numeral
tff(fact_314_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z2: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),Z2)) = aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(V),W))),Z2) ).

% mult_numeral_left_semiring_numeral
tff(fact_315_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,W)),Z2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(times_times_num(V),W))),Z2) ).

% mult_numeral_left_semiring_numeral
tff(fact_316_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,W)),Z2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(V),W))),Z2) ).

% mult_numeral_left_semiring_numeral
tff(fact_317_numeral__times__numeral,axiom,
    ! [Ma: num,Nb: num] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(Ma)),numera6690914467698888265omplex(Nb)) = numera6690914467698888265omplex(aa_num_num(times_times_num(Ma),Nb)) ).

% numeral_times_numeral
tff(fact_318_numeral__times__numeral,axiom,
    ! [Ma: num,Nb: num] : aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,Ma)),aa_num_real(numeral_numeral_real,Nb)) = aa_num_real(numeral_numeral_real,aa_num_num(times_times_num(Ma),Nb)) ).

% numeral_times_numeral
tff(fact_319_numeral__times__numeral,axiom,
    ! [Ma: num,Nb: num] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,Ma)),aa_num_rat(numeral_numeral_rat,Nb)) = aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(Ma),Nb)) ).

% numeral_times_numeral
tff(fact_320_numeral__times__numeral,axiom,
    ! [Ma: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,Ma)),aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_nat(numeral_numeral_nat,aa_num_num(times_times_num(Ma),Nb)) ).

% numeral_times_numeral
tff(fact_321_numeral__times__numeral,axiom,
    ! [Ma: num,Nb: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,Ma)),aa_num_int(numeral_numeral_int,Nb)) = aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(Ma),Nb)) ).

% numeral_times_numeral
tff(fact_322_low__inv,axiom,
    ! [X: nat,Nb: nat,Y2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),X)),Nb) = X ) ) ).

% low_inv
tff(fact_323_take__all__iff,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT] :
      ( ( take_VEBT_VEBT(Nb,Xs) = Xs )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Xs)),Nb) ) ).

% take_all_iff
tff(fact_324_take__all__iff,axiom,
    ! [Nb: nat,Xs: list_o] :
      ( ( take_o(Nb,Xs) = Xs )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_o(Xs)),Nb) ) ).

% take_all_iff
tff(fact_325_take__all__iff,axiom,
    ! [Nb: nat,Xs: list_nat] :
      ( ( take_nat(Nb,Xs) = Xs )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_nat(Xs)),Nb) ) ).

% take_all_iff
tff(fact_326_take__all__iff,axiom,
    ! [Nb: nat,Xs: list_int] :
      ( ( take_int(Nb,Xs) = Xs )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_int(Xs)),Nb) ) ).

% take_all_iff
tff(fact_327_take__all,axiom,
    ! [Xs: list_VEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Xs)),Nb)
     => ( take_VEBT_VEBT(Nb,Xs) = Xs ) ) ).

% take_all
tff(fact_328_take__all,axiom,
    ! [Xs: list_o,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_o(Xs)),Nb)
     => ( take_o(Nb,Xs) = Xs ) ) ).

% take_all
tff(fact_329_take__all,axiom,
    ! [Xs: list_nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_nat(Xs)),Nb)
     => ( take_nat(Nb,Xs) = Xs ) ) ).

% take_all
tff(fact_330_take__all,axiom,
    ! [Xs: list_int,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_int(Xs)),Nb)
     => ( take_int(Nb,Xs) = Xs ) ) ).

% take_all
tff(fact_331_semiring__norm_I71_J,axiom,
    ! [Ma: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit0,Ma)),aa_num_num(bit0,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ma),Nb) ) ).

% semiring_norm(71)
tff(fact_332_semiring__norm_I68_J,axiom,
    ! [Nb: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,one),Nb) ).

% semiring_norm(68)
tff(fact_333_enat__ord__simps_I1_J,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(aa_nat_Extended_enat(extended_enat2,Ma)),aa_nat_Extended_enat(extended_enat2,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ).

% enat_ord_simps(1)
tff(fact_334_distrib__right__numeral,axiom,
    ! [A: complex,B: complex,V: num] : aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(A),B)),numera6690914467698888265omplex(V)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(V))),aa_complex_complex(times_times_complex(B),numera6690914467698888265omplex(V))) ).

% distrib_right_numeral
tff(fact_335_distrib__right__numeral,axiom,
    ! [A: real,B: real,V: num] : aa_real_real(times_times_real(aa_real_real(plus_plus_real(A),B)),aa_num_real(numeral_numeral_real,V)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),aa_num_real(numeral_numeral_real,V))),aa_real_real(times_times_real(B),aa_num_real(numeral_numeral_real,V))) ).

% distrib_right_numeral
tff(fact_336_distrib__right__numeral,axiom,
    ! [A: rat,B: rat,V: num] : aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A),B)),aa_num_rat(numeral_numeral_rat,V)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(times_times_rat(B),aa_num_rat(numeral_numeral_rat,V))) ).

% distrib_right_numeral
tff(fact_337_distrib__right__numeral,axiom,
    ! [A: nat,B: nat,V: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_num_nat(numeral_numeral_nat,V))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_num_nat(numeral_numeral_nat,V))) ).

% distrib_right_numeral
tff(fact_338_distrib__right__numeral,axiom,
    ! [A: int,B: int,V: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),aa_num_int(numeral_numeral_int,V)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_num_int(numeral_numeral_int,V))),aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_num_int(numeral_numeral_int,V))) ).

% distrib_right_numeral
tff(fact_339_distrib__left__numeral,axiom,
    ! [V: num,B: complex,C: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),aa_complex_complex(plus_plus_complex(B),C)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),B)),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),C)) ).

% distrib_left_numeral
tff(fact_340_distrib__left__numeral,axiom,
    ! [V: num,B: real,C: real] : aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,V)),aa_real_real(plus_plus_real(B),C)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,V)),B)),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,V)),C)) ).

% distrib_left_numeral
tff(fact_341_distrib__left__numeral,axiom,
    ! [V: num,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(plus_plus_rat(B),C)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),B)),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),C)) ).

% distrib_left_numeral
tff(fact_342_distrib__left__numeral,axiom,
    ! [V: num,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V)),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V)),C)) ).

% distrib_left_numeral
tff(fact_343_distrib__left__numeral,axiom,
    ! [V: num,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),B)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),C)) ).

% distrib_left_numeral
tff(fact_344_semiring__norm_I69_J,axiom,
    ! [Ma: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit0,Ma)),one) ).

% semiring_norm(69)
tff(fact_345_enat__ord__number_I1_J,axiom,
    ! [Ma: num,Nb: num] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(Ma)),numera1916890842035813515d_enat(Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Ma)),aa_num_nat(numeral_numeral_nat,Nb)) ) ).

% enat_ord_number(1)
tff(fact_346_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(divide_divide_real(B),aa_num_real(numeral_numeral_real,W)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),aa_num_real(numeral_numeral_real,W))),B) ) ).

% le_divide_eq_numeral1(1)
tff(fact_347_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,W))),B) ) ).

% le_divide_eq_numeral1(1)
tff(fact_348_divide__le__eq__numeral1_I1_J,axiom,
    ! [B: real,W: num,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(B),aa_num_real(numeral_numeral_real,W))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(times_times_real(A),aa_num_real(numeral_numeral_real,W))) ) ).

% divide_le_eq_numeral1(1)
tff(fact_349_divide__le__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W))),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,W))) ) ).

% divide_le_eq_numeral1(1)
tff(fact_350_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(divide_divide_real(B),aa_num_real(numeral_numeral_real,W)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),aa_num_real(numeral_numeral_real,W))),B) ) ).

% less_divide_eq_numeral1(1)
tff(fact_351_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,W))),B) ) ).

% less_divide_eq_numeral1(1)
tff(fact_352_divide__less__eq__numeral1_I1_J,axiom,
    ! [B: real,W: num,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(B),aa_num_real(numeral_numeral_real,W))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(times_times_real(A),aa_num_real(numeral_numeral_real,W))) ) ).

% divide_less_eq_numeral1(1)
tff(fact_353_divide__less__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W))),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,W))) ) ).

% divide_less_eq_numeral1(1)
tff(fact_354_power__add__numeral2,axiom,
    ! [A: complex,Ma: num,Nb: num,B: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),aa_num_nat(numeral_numeral_nat,Ma))),aa_complex_complex(times_times_complex(aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),aa_num_nat(numeral_numeral_nat,Nb))),B)) = aa_complex_complex(times_times_complex(aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(Ma),Nb)))),B) ).

% power_add_numeral2
tff(fact_355_power__add__numeral2,axiom,
    ! [A: real,Ma: num,Nb: num,B: real] : aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,Ma))),aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,Nb))),B)) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(Ma),Nb)))),B) ).

% power_add_numeral2
tff(fact_356_power__add__numeral2,axiom,
    ! [A: rat,Ma: num,Nb: num,B: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,Ma))),aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,Nb))),B)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(Ma),Nb)))),B) ).

% power_add_numeral2
tff(fact_357_power__add__numeral2,axiom,
    ! [A: nat,Ma: num,Nb: num,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,Ma))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,Nb))),B)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(Ma),Nb)))),B) ).

% power_add_numeral2
tff(fact_358_power__add__numeral2,axiom,
    ! [A: int,Ma: num,Nb: num,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,Ma))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,Nb))),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(Ma),Nb)))),B) ).

% power_add_numeral2
tff(fact_359_power__add__numeral,axiom,
    ! [A: complex,Ma: num,Nb: num] : aa_complex_complex(times_times_complex(aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),aa_num_nat(numeral_numeral_nat,Ma))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),aa_num_nat(numeral_numeral_nat,Nb))) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(Ma),Nb))) ).

% power_add_numeral
tff(fact_360_power__add__numeral,axiom,
    ! [A: real,Ma: num,Nb: num] : aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,Ma))),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,Nb))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(Ma),Nb))) ).

% power_add_numeral
tff(fact_361_power__add__numeral,axiom,
    ! [A: rat,Ma: num,Nb: num] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,Ma))),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,Nb))) = aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(Ma),Nb))) ).

% power_add_numeral
tff(fact_362_power__add__numeral,axiom,
    ! [A: nat,Ma: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,Ma))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(Ma),Nb))) ).

% power_add_numeral
tff(fact_363_power__add__numeral,axiom,
    ! [A: int,Ma: num,Nb: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,Ma))),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,Nb))) = aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(Ma),Nb))) ).

% power_add_numeral
tff(fact_364_numeral__le__enat__iff,axiom,
    ! [Ma: num,Nb: nat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(Ma)),aa_nat_Extended_enat(extended_enat2,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Ma)),Nb) ) ).

% numeral_le_enat_iff
tff(fact_365_enat__less__induct,axiom,
    ! [P: fun_Extended_enat_o,Nb: extended_enat] :
      ( ! [N: extended_enat] :
          ( ! [M: extended_enat] :
              ( aa_Extended_enat_o(ord_le72135733267957522d_enat(M),N)
             => aa_Extended_enat_o(P,M) )
         => aa_Extended_enat_o(P,N) )
     => aa_Extended_enat_o(P,Nb) ) ).

% enat_less_induct
tff(fact_366_mult__le__mono2,axiom,
    ! [I2: nat,J: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),I2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),J)) ) ).

% mult_le_mono2
tff(fact_367_mult__le__mono1,axiom,
    ! [I2: nat,J: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),K2)) ) ).

% mult_le_mono1
tff(fact_368_mult__le__mono,axiom,
    ! [I2: nat,J: nat,K2: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),L)) ) ) ).

% mult_le_mono
tff(fact_369_le__square,axiom,
    ! [Ma: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Ma)) ).

% le_square
tff(fact_370_le__cube,axiom,
    ! [Ma: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Ma))) ).

% le_cube
tff(fact_371_mult_Oleft__commute,axiom,
    ! [B: real,A: real,C: real] : aa_real_real(times_times_real(B),aa_real_real(times_times_real(A),C)) = aa_real_real(times_times_real(A),aa_real_real(times_times_real(B),C)) ).

% mult.left_commute
tff(fact_372_mult_Oleft__commute,axiom,
    ! [B: rat,A: rat,C: rat] : aa_rat_rat(times_times_rat(B),aa_rat_rat(times_times_rat(A),C)) = aa_rat_rat(times_times_rat(A),aa_rat_rat(times_times_rat(B),C)) ).

% mult.left_commute
tff(fact_373_mult_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% mult.left_commute
tff(fact_374_mult_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% mult.left_commute
tff(fact_375_mult_Ocommute,axiom,
    ! [A: real,B: real] : aa_real_real(times_times_real(A),B) = aa_real_real(times_times_real(B),A) ).

% mult.commute
tff(fact_376_mult_Ocommute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(times_times_rat(A),B) = aa_rat_rat(times_times_rat(B),A) ).

% mult.commute
tff(fact_377_mult_Ocommute,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A) ).

% mult.commute
tff(fact_378_mult_Ocommute,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),B) = aa_int_int(aa_int_fun_int_int(times_times_int,B),A) ).

% mult.commute
tff(fact_379_mult_Oassoc,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(times_times_real(aa_real_real(times_times_real(A),B)),C) = aa_real_real(times_times_real(A),aa_real_real(times_times_real(B),C)) ).

% mult.assoc
tff(fact_380_mult_Oassoc,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(A),B)),C) = aa_rat_rat(times_times_rat(A),aa_rat_rat(times_times_rat(B),C)) ).

% mult.assoc
tff(fact_381_mult_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% mult.assoc
tff(fact_382_mult_Oassoc,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% mult.assoc
tff(fact_383_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(times_times_real(aa_real_real(times_times_real(A),B)),C) = aa_real_real(times_times_real(A),aa_real_real(times_times_real(B),C)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_384_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(A),B)),C) = aa_rat_rat(times_times_rat(A),aa_rat_rat(times_times_rat(B),C)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_385_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_386_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_387_div__times__less__eq__dividend,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),Nb)),Nb)),Ma) ).

% div_times_less_eq_dividend
tff(fact_388_times__div__less__eq__dividend,axiom,
    ! [Nb: nat,Ma: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),Nb))),Ma) ).

% times_div_less_eq_dividend
tff(fact_389_le__num__One__iff,axiom,
    ! [X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),one)
    <=> ( X = one ) ) ).

% le_num_One_iff
tff(fact_390_set__take__subset__set__take,axiom,
    ! [Ma: nat,Nb: nat,Xs: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(take_VEBT_VEBT(Ma,Xs)),set_VEBT_VEBT2(take_VEBT_VEBT(Nb,Xs))) ) ).

% set_take_subset_set_take
tff(fact_391_set__take__subset__set__take,axiom,
    ! [Ma: nat,Nb: nat,Xs: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => aa_set_nat_o(ord_less_eq_set_nat(set_nat2(take_nat(Ma,Xs))),set_nat2(take_nat(Nb,Xs))) ) ).

% set_take_subset_set_take
tff(fact_392_set__take__subset__set__take,axiom,
    ! [Ma: nat,Nb: nat,Xs: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => ord_less_eq_set_int(set_int2(take_int(Ma,Xs)),set_int2(take_int(Nb,Xs))) ) ).

% set_take_subset_set_take
tff(fact_393_power__commuting__commutes,axiom,
    ! [X: complex,Y2: complex,Nb: nat] :
      ( ( aa_complex_complex(times_times_complex(X),Y2) = aa_complex_complex(times_times_complex(Y2),X) )
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,X),Nb)),Y2) = aa_complex_complex(times_times_complex(Y2),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,X),Nb)) ) ) ).

% power_commuting_commutes
tff(fact_394_power__commuting__commutes,axiom,
    ! [X: real,Y2: real,Nb: nat] :
      ( ( aa_real_real(times_times_real(X),Y2) = aa_real_real(times_times_real(Y2),X) )
     => ( aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Nb)),Y2) = aa_real_real(times_times_real(Y2),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Nb)) ) ) ).

% power_commuting_commutes
tff(fact_395_power__commuting__commutes,axiom,
    ! [X: rat,Y2: rat,Nb: nat] :
      ( ( aa_rat_rat(times_times_rat(X),Y2) = aa_rat_rat(times_times_rat(Y2),X) )
     => ( aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(X),Nb)),Y2) = aa_rat_rat(times_times_rat(Y2),aa_nat_rat(power_power_rat(X),Nb)) ) ) ).

% power_commuting_commutes
tff(fact_396_power__commuting__commutes,axiom,
    ! [X: nat,Y2: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Y2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y2),X) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),Nb)),Y2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),Nb)) ) ) ).

% power_commuting_commutes
tff(fact_397_power__commuting__commutes,axiom,
    ! [X: int,Y2: int,Nb: nat] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,X),Y2) = aa_int_int(aa_int_fun_int_int(times_times_int,Y2),X) )
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X),Nb)),Y2) = aa_int_int(aa_int_fun_int_int(times_times_int,Y2),aa_nat_int(aa_int_fun_nat_int(power_power_int,X),Nb)) ) ) ).

% power_commuting_commutes
tff(fact_398_power__mult__distrib,axiom,
    ! [A: complex,B: complex,Nb: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(times_times_complex(A),B)),Nb) = aa_complex_complex(times_times_complex(aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),Nb)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,B),Nb)) ).

% power_mult_distrib
tff(fact_399_power__mult__distrib,axiom,
    ! [A: real,B: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(times_times_real(A),B)),Nb) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)) ).

% power_mult_distrib
tff(fact_400_power__mult__distrib,axiom,
    ! [A: rat,B: rat,Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(times_times_rat(A),B)),Nb) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb)) ).

% power_mult_distrib
tff(fact_401_power__mult__distrib,axiom,
    ! [A: nat,B: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)) ).

% power_mult_distrib
tff(fact_402_power__mult__distrib,axiom,
    ! [A: int,B: int,Nb: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),Nb) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Nb)) ).

% power_mult_distrib
tff(fact_403_power__commutes,axiom,
    ! [A: complex,Nb: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),Nb)),A) = aa_complex_complex(times_times_complex(A),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),Nb)) ).

% power_commutes
tff(fact_404_power__commutes,axiom,
    ! [A: real,Nb: nat] : aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),A) = aa_real_real(times_times_real(A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ).

% power_commutes
tff(fact_405_power__commutes,axiom,
    ! [A: rat,Nb: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),Nb)),A) = aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),Nb)) ).

% power_commutes
tff(fact_406_power__commutes,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ).

% power_commutes
tff(fact_407_power__commutes,axiom,
    ! [A: int,Nb: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)),A) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)) ).

% power_commutes
tff(fact_408_power__mult,axiom,
    ! [A: nat,Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Ma)),Nb) ).

% power_mult
tff(fact_409_power__mult,axiom,
    ! [A: real,Ma: nat,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Ma)),Nb) ).

% power_mult
tff(fact_410_power__mult,axiom,
    ! [A: int,Ma: nat,Nb: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Nb)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Ma)),Nb) ).

% power_mult
tff(fact_411_power__mult,axiom,
    ! [A: complex,Ma: nat,Nb: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Nb)) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),Ma)),Nb) ).

% power_mult
tff(fact_412_VEBT_Oexhaust,axiom,
    ! [Y2: vEBT_VEBT] :
      ( ! [X112: option4927543243414619207at_nat,X122: nat,X132: list_VEBT_VEBT,X142: vEBT_VEBT] : Y2 != aa_VEB5832953974749077948T_VEBT(vEBT_Node(X112,X122,X132),X142)
     => ~ ! [X212: $o,X222: $o] : Y2 != aa_o_VEBT_VEBT(vEBT_Leaf((X212)),(X222)) ) ).

% VEBT.exhaust
tff(fact_413_VEBT_Odistinct_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,X21: $o,X22: $o] : aa_VEB5832953974749077948T_VEBT(vEBT_Node(X11,X12,X13),X14) != aa_o_VEBT_VEBT(vEBT_Leaf((X21)),(X22)) ).

% VEBT.distinct(1)
tff(fact_414_less__mono__imp__le__mono,axiom,
    ! [F: fun_nat_nat,I2: nat,J: nat] :
      ( ! [I3: nat,J2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),J2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,I3)),aa_nat_nat(F,J2)) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,I2)),aa_nat_nat(F,J)) ) ) ).

% less_mono_imp_le_mono
tff(fact_415_le__neq__implies__less,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => ( ( Ma != Nb )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb) ) ) ).

% le_neq_implies_less
tff(fact_416_less__or__eq__imp__le,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
        | ( Ma = Nb ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ).

% less_or_eq_imp_le
tff(fact_417_le__eq__less__or__eq,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
        | ( Ma = Nb ) ) ) ).

% le_eq_less_or_eq
tff(fact_418_less__imp__le__nat,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ).

% less_imp_le_nat
tff(fact_419_nat__less__le,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
        & ( Ma != Nb ) ) ) ).

% nat_less_le
tff(fact_420_left__add__mult__distrib,axiom,
    ! [I2: nat,U: nat,J: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),U)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),J)),U)),K2) ).

% left_add_mult_distrib
tff(fact_421_add__mult__distrib2,axiom,
    ! [K2: nat,Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Nb)) ).

% add_mult_distrib2
tff(fact_422_add__mult__distrib,axiom,
    ! [Ma: nat,Nb: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb)),K2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),K2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K2)) ).

% add_mult_distrib
tff(fact_423_subset__code_I1_J,axiom,
    ! [Xs: list_complex,B3: set_complex] :
      ( ord_le211207098394363844omplex(set_complex2(Xs),B3)
    <=> ! [X3: complex] :
          ( member_complex(X3,set_complex2(Xs))
         => member_complex(X3,B3) ) ) ).

% subset_code(1)
tff(fact_424_subset__code_I1_J,axiom,
    ! [Xs: list_real,B3: set_real] :
      ( ord_less_eq_set_real(set_real2(Xs),B3)
    <=> ! [X3: real] :
          ( member_real(X3,set_real2(Xs))
         => member_real(X3,B3) ) ) ).

% subset_code(1)
tff(fact_425_subset__code_I1_J,axiom,
    ! [Xs: list_set_nat,B3: set_set_nat] :
      ( ord_le6893508408891458716et_nat(set_set_nat2(Xs),B3)
    <=> ! [X3: set_nat] :
          ( member_set_nat(X3,set_set_nat2(Xs))
         => member_set_nat(X3,B3) ) ) ).

% subset_code(1)
tff(fact_426_subset__code_I1_J,axiom,
    ! [Xs: list_VEBT_VEBT,B3: set_VEBT_VEBT] :
      ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Xs),B3)
    <=> ! [X3: vEBT_VEBT] :
          ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(Xs))
         => member_VEBT_VEBT(X3,B3) ) ) ).

% subset_code(1)
tff(fact_427_subset__code_I1_J,axiom,
    ! [Xs: list_nat,B3: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(set_nat2(Xs)),B3)
    <=> ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),set_nat2(Xs))
         => aa_set_nat_o(member_nat(X3),B3) ) ) ).

% subset_code(1)
tff(fact_428_subset__code_I1_J,axiom,
    ! [Xs: list_int,B3: set_int] :
      ( ord_less_eq_set_int(set_int2(Xs),B3)
    <=> ! [X3: int] :
          ( member_int(X3,set_int2(Xs))
         => member_int(X3,B3) ) ) ).

% subset_code(1)
tff(fact_429_nat__le__iff__add,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
    <=> ? [K3: nat] : Nb = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),K3) ) ).

% nat_le_iff_add
tff(fact_430_trans__le__add2,axiom,
    ! [I2: nat,J: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),J)) ) ).

% trans_le_add2
tff(fact_431_trans__le__add1,axiom,
    ! [I2: nat,J: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),Ma)) ) ).

% trans_le_add1
tff(fact_432_add__le__mono1,axiom,
    ! [I2: nat,J: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K2)) ) ).

% add_le_mono1
tff(fact_433_add__le__mono,axiom,
    ! [I2: nat,J: nat,K2: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ) ).

% add_le_mono
tff(fact_434_le__Suc__ex,axiom,
    ! [K2: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),L)
     => ? [N: nat] : L = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),N) ) ).

% le_Suc_ex
tff(fact_435_add__leD2,axiom,
    ! [Ma: nat,K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),K2)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb) ) ).

% add_leD2
tff(fact_436_add__leD1,axiom,
    ! [Ma: nat,K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),K2)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ).

% add_leD1
tff(fact_437_le__add2,axiom,
    ! [Nb: nat,Ma: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb)) ).

% le_add2
tff(fact_438_le__add1,axiom,
    ! [Nb: nat,Ma: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Ma)) ).

% le_add1
tff(fact_439_add__leE,axiom,
    ! [Ma: nat,K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),K2)),Nb)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb) ) ) ).

% add_leE
tff(fact_440_div__mult2__eq,axiom,
    ! [Ma: nat,Nb: nat,Q: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),Nb)),Q) ).

% div_mult2_eq
tff(fact_441_div__le__dividend,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),Nb)),Ma) ).

% div_le_dividend
tff(fact_442_div__le__mono,axiom,
    ! [Ma: nat,Nb: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),K2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),K2)) ) ).

% div_le_mono
tff(fact_443_mult__numeral__1__right,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(one)) = A ).

% mult_numeral_1_right
tff(fact_444_mult__numeral__1__right,axiom,
    ! [A: real] : aa_real_real(times_times_real(A),aa_num_real(numeral_numeral_real,one)) = A ).

% mult_numeral_1_right
tff(fact_445_mult__numeral__1__right,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,one)) = A ).

% mult_numeral_1_right
tff(fact_446_mult__numeral__1__right,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_num_nat(numeral_numeral_nat,one)) = A ).

% mult_numeral_1_right
tff(fact_447_mult__numeral__1__right,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_num_int(numeral_numeral_int,one)) = A ).

% mult_numeral_1_right
tff(fact_448_mult__numeral__1,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(one)),A) = A ).

% mult_numeral_1
tff(fact_449_mult__numeral__1,axiom,
    ! [A: real] : aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,one)),A) = A ).

% mult_numeral_1
tff(fact_450_mult__numeral__1,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,one)),A) = A ).

% mult_numeral_1
tff(fact_451_mult__numeral__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,one)),A) = A ).

% mult_numeral_1
tff(fact_452_mult__numeral__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,one)),A) = A ).

% mult_numeral_1
tff(fact_453_VEBT__internal_Oelim__dead_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,Uu: extended_enat] : vEBT_VEBT_elim_dead(aa_o_VEBT_VEBT(vEBT_Leaf((A)),(B)),Uu) = aa_o_VEBT_VEBT(vEBT_Leaf((A)),(B)) ).

% VEBT_internal.elim_dead.simps(1)
tff(fact_454_less__eq__enat__def,axiom,
    ! [Ma: extended_enat,Nb: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(Ma),Nb)
    <=> aa_Extended_enat_o(aa_o_f2257658052187763793enat_o(aa_fun2142289715968585979enat_o(extended_case_enat_o,aTP_Lamm_an(Ma)),$true),Nb) ) ).

% less_eq_enat_def
tff(fact_455_power__add,axiom,
    ! [A: complex,Ma: nat,Nb: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb)) = aa_complex_complex(times_times_complex(aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),Ma)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),Nb)) ).

% power_add
tff(fact_456_power__add,axiom,
    ! [A: real,Ma: nat,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb)) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Ma)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ).

% power_add
tff(fact_457_power__add,axiom,
    ! [A: rat,Ma: nat,Nb: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),Ma)),aa_nat_rat(power_power_rat(A),Nb)) ).

% power_add
tff(fact_458_power__add,axiom,
    ! [A: nat,Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ).

% power_add
tff(fact_459_power__add,axiom,
    ! [A: int,Ma: nat,Nb: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Ma)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)) ).

% power_add
tff(fact_460_mono__nat__linear__lb,axiom,
    ! [F: fun_nat_nat,Ma: nat,K2: nat] :
      ( ! [M2: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,M2)),aa_nat_nat(F,N)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(F,Ma)),K2)),aa_nat_nat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),K2))) ) ).

% mono_nat_linear_lb
tff(fact_461_less__mult__imp__div__less,axiom,
    ! [Ma: nat,I2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),Nb)),I2) ) ).

% less_mult_imp_div_less
tff(fact_462_set__take__subset,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT] : ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(take_VEBT_VEBT(Nb,Xs)),set_VEBT_VEBT2(Xs)) ).

% set_take_subset
tff(fact_463_set__take__subset,axiom,
    ! [Nb: nat,Xs: list_nat] : aa_set_nat_o(ord_less_eq_set_nat(set_nat2(take_nat(Nb,Xs))),set_nat2(Xs)) ).

% set_take_subset
tff(fact_464_set__take__subset,axiom,
    ! [Nb: nat,Xs: list_int] : ord_less_eq_set_int(set_int2(take_int(Nb,Xs)),set_int2(Xs)) ).

% set_take_subset
tff(fact_465_left__add__twice,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(plus_plus_complex(A),aa_complex_complex(plus_plus_complex(A),B)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),A)),B) ).

% left_add_twice
tff(fact_466_left__add__twice,axiom,
    ! [A: real,B: real] : aa_real_real(plus_plus_real(A),aa_real_real(plus_plus_real(A),B)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),A)),B) ).

% left_add_twice
tff(fact_467_left__add__twice,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(A),B)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),A)),B) ).

% left_add_twice
tff(fact_468_left__add__twice,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)),B) ).

% left_add_twice
tff(fact_469_left__add__twice,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),B) ).

% left_add_twice
tff(fact_470_mult__2__right,axiom,
    ! [Z2: complex] : aa_complex_complex(times_times_complex(Z2),numera6690914467698888265omplex(aa_num_num(bit0,one))) = aa_complex_complex(plus_plus_complex(Z2),Z2) ).

% mult_2_right
tff(fact_471_mult__2__right,axiom,
    ! [Z2: real] : aa_real_real(times_times_real(Z2),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) = aa_real_real(plus_plus_real(Z2),Z2) ).

% mult_2_right
tff(fact_472_mult__2__right,axiom,
    ! [Z2: rat] : aa_rat_rat(times_times_rat(Z2),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))) = aa_rat_rat(plus_plus_rat(Z2),Z2) ).

% mult_2_right
tff(fact_473_mult__2__right,axiom,
    ! [Z2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Z2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z2),Z2) ).

% mult_2_right
tff(fact_474_mult__2__right,axiom,
    ! [Z2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,Z2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Z2),Z2) ).

% mult_2_right
tff(fact_475_mult__2,axiom,
    ! [Z2: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Z2) = aa_complex_complex(plus_plus_complex(Z2),Z2) ).

% mult_2
tff(fact_476_mult__2,axiom,
    ! [Z2: real] : aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Z2) = aa_real_real(plus_plus_real(Z2),Z2) ).

% mult_2
tff(fact_477_mult__2,axiom,
    ! [Z2: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Z2) = aa_rat_rat(plus_plus_rat(Z2),Z2) ).

% mult_2
tff(fact_478_mult__2,axiom,
    ! [Z2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Z2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z2),Z2) ).

% mult_2
tff(fact_479_mult__2,axiom,
    ! [Z2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Z2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Z2),Z2) ).

% mult_2
tff(fact_480_add__right__imp__eq,axiom,
    ! [B: real,A: real,C: real] :
      ( ( aa_real_real(plus_plus_real(B),A) = aa_real_real(plus_plus_real(C),A) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_481_add__right__imp__eq,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( aa_rat_rat(plus_plus_rat(B),A) = aa_rat_rat(plus_plus_rat(C),A) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_482_add__right__imp__eq,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_483_add__right__imp__eq,axiom,
    ! [B: int,A: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_484_add__left__imp__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( aa_real_real(plus_plus_real(A),B) = aa_real_real(plus_plus_real(A),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_485_add__left__imp__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),B) = aa_rat_rat(plus_plus_rat(A),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_486_add__left__imp__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_487_add__left__imp__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_488_add_Oleft__commute,axiom,
    ! [B: real,A: real,C: real] : aa_real_real(plus_plus_real(B),aa_real_real(plus_plus_real(A),C)) = aa_real_real(plus_plus_real(A),aa_real_real(plus_plus_real(B),C)) ).

% add.left_commute
tff(fact_489_add_Oleft__commute,axiom,
    ! [B: rat,A: rat,C: rat] : aa_rat_rat(plus_plus_rat(B),aa_rat_rat(plus_plus_rat(A),C)) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(B),C)) ).

% add.left_commute
tff(fact_490_add_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ).

% add.left_commute
tff(fact_491_add_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ).

% add.left_commute
tff(fact_492_add_Ocommute,axiom,
    ! [A: real,B: real] : aa_real_real(plus_plus_real(A),B) = aa_real_real(plus_plus_real(B),A) ).

% add.commute
tff(fact_493_add_Ocommute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(plus_plus_rat(A),B) = aa_rat_rat(plus_plus_rat(B),A) ).

% add.commute
tff(fact_494_add_Ocommute,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) ).

% add.commute
tff(fact_495_add_Ocommute,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) ).

% add.commute
tff(fact_496_add_Oright__cancel,axiom,
    ! [B: real,A: real,C: real] :
      ( ( aa_real_real(plus_plus_real(B),A) = aa_real_real(plus_plus_real(C),A) )
    <=> ( B = C ) ) ).

% add.right_cancel
tff(fact_497_add_Oright__cancel,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( aa_rat_rat(plus_plus_rat(B),A) = aa_rat_rat(plus_plus_rat(C),A) )
    <=> ( B = C ) ) ).

% add.right_cancel
tff(fact_498_add_Oright__cancel,axiom,
    ! [B: int,A: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A) )
    <=> ( B = C ) ) ).

% add.right_cancel
tff(fact_499_add_Oleft__cancel,axiom,
    ! [A: real,B: real,C: real] :
      ( ( aa_real_real(plus_plus_real(A),B) = aa_real_real(plus_plus_real(A),C) )
    <=> ( B = C ) ) ).

% add.left_cancel
tff(fact_500_add_Oleft__cancel,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),B) = aa_rat_rat(plus_plus_rat(A),C) )
    <=> ( B = C ) ) ).

% add.left_cancel
tff(fact_501_add_Oleft__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) )
    <=> ( B = C ) ) ).

% add.left_cancel
tff(fact_502_add_Oassoc,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(A),B)),C) = aa_real_real(plus_plus_real(A),aa_real_real(plus_plus_real(B),C)) ).

% add.assoc
tff(fact_503_add_Oassoc,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(A),B)),C) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(B),C)) ).

% add.assoc
tff(fact_504_add_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ).

% add.assoc
tff(fact_505_add_Oassoc,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ).

% add.assoc
tff(fact_506_group__cancel_Oadd2,axiom,
    ! [B3: real,K2: real,B: real,A: real] :
      ( ( B3 = aa_real_real(plus_plus_real(K2),B) )
     => ( aa_real_real(plus_plus_real(A),B3) = aa_real_real(plus_plus_real(K2),aa_real_real(plus_plus_real(A),B)) ) ) ).

% group_cancel.add2
tff(fact_507_group__cancel_Oadd2,axiom,
    ! [B3: rat,K2: rat,B: rat,A: rat] :
      ( ( B3 = aa_rat_rat(plus_plus_rat(K2),B) )
     => ( aa_rat_rat(plus_plus_rat(A),B3) = aa_rat_rat(plus_plus_rat(K2),aa_rat_rat(plus_plus_rat(A),B)) ) ) ).

% group_cancel.add2
tff(fact_508_group__cancel_Oadd2,axiom,
    ! [B3: nat,K2: nat,B: nat,A: nat] :
      ( ( B3 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),B) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% group_cancel.add2
tff(fact_509_group__cancel_Oadd2,axiom,
    ! [B3: int,K2: int,B: int,A: int] :
      ( ( B3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),B) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B3) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% group_cancel.add2
tff(fact_510_group__cancel_Oadd1,axiom,
    ! [A2: real,K2: real,A: real,B: real] :
      ( ( A2 = aa_real_real(plus_plus_real(K2),A) )
     => ( aa_real_real(plus_plus_real(A2),B) = aa_real_real(plus_plus_real(K2),aa_real_real(plus_plus_real(A),B)) ) ) ).

% group_cancel.add1
tff(fact_511_group__cancel_Oadd1,axiom,
    ! [A2: rat,K2: rat,A: rat,B: rat] :
      ( ( A2 = aa_rat_rat(plus_plus_rat(K2),A) )
     => ( aa_rat_rat(plus_plus_rat(A2),B) = aa_rat_rat(plus_plus_rat(K2),aa_rat_rat(plus_plus_rat(A),B)) ) ) ).

% group_cancel.add1
tff(fact_512_group__cancel_Oadd1,axiom,
    ! [A2: nat,K2: nat,A: nat,B: nat] :
      ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),A) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% group_cancel.add1
tff(fact_513_group__cancel_Oadd1,axiom,
    ! [A2: int,K2: int,A: int,B: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),A) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% group_cancel.add1
tff(fact_514_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I2: real,J: real,K2: real,L: real] :
      ( ( ( I2 = J )
        & ( K2 = L ) )
     => ( aa_real_real(plus_plus_real(I2),K2) = aa_real_real(plus_plus_real(J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_515_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I2: rat,J: rat,K2: rat,L: rat] :
      ( ( ( I2 = J )
        & ( K2 = L ) )
     => ( aa_rat_rat(plus_plus_rat(I2),K2) = aa_rat_rat(plus_plus_rat(J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_516_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I2: nat,J: nat,K2: nat,L: nat] :
      ( ( ( I2 = J )
        & ( K2 = L ) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_517_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I2: int,J: int,K2: int,L: int] :
      ( ( ( I2 = J )
        & ( K2 = L ) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),K2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_518_is__num__normalize_I1_J,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(A),B)),C) = aa_real_real(plus_plus_real(A),aa_real_real(plus_plus_real(B),C)) ).

% is_num_normalize(1)
tff(fact_519_is__num__normalize_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(A),B)),C) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(B),C)) ).

% is_num_normalize(1)
tff(fact_520_is__num__normalize_I1_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ).

% is_num_normalize(1)
tff(fact_521_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(A),B)),C) = aa_real_real(plus_plus_real(A),aa_real_real(plus_plus_real(B),C)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_522_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(A),B)),C) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(B),C)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_523_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_524_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_525_add__One__commute,axiom,
    ! [Nb: num] : aa_num_num(plus_plus_num(one),Nb) = aa_num_num(plus_plus_num(Nb),one) ).

% add_One_commute
tff(fact_526_power__numeral__even,axiom,
    ! [Z2: complex,W: num] :
      aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Z2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = $let(
        w: complex,
        w:= aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Z2),aa_num_nat(numeral_numeral_nat,W)),
        aa_complex_complex(times_times_complex(w),w) ) ).

% power_numeral_even
tff(fact_527_power__numeral__even,axiom,
    ! [Z2: real,W: num] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,Z2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = $let(
        w: real,
        w:= aa_nat_real(aa_real_fun_nat_real(power_power_real,Z2),aa_num_nat(numeral_numeral_nat,W)),
        aa_real_real(times_times_real(w),w) ) ).

% power_numeral_even
tff(fact_528_power__numeral__even,axiom,
    ! [Z2: rat,W: num] :
      aa_nat_rat(power_power_rat(Z2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = $let(
        w: rat,
        w:= aa_nat_rat(power_power_rat(Z2),aa_num_nat(numeral_numeral_nat,W)),
        aa_rat_rat(times_times_rat(w),w) ) ).

% power_numeral_even
tff(fact_529_power__numeral__even,axiom,
    ! [Z2: nat,W: num] :
      aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = $let(
        w: nat,
        w:= aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z2),aa_num_nat(numeral_numeral_nat,W)),
        aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,w),w) ) ).

% power_numeral_even
tff(fact_530_power__numeral__even,axiom,
    ! [Z2: int,W: num] :
      aa_nat_int(aa_int_fun_nat_int(power_power_int,Z2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = $let(
        w: int,
        w:= aa_nat_int(aa_int_fun_nat_int(power_power_int,Z2),aa_num_nat(numeral_numeral_nat,W)),
        aa_int_int(aa_int_fun_int_int(times_times_int,w),w) ) ).

% power_numeral_even
tff(fact_531_iadd__le__enat__iff,axiom,
    ! [X: extended_enat,Y2: extended_enat,Nb: nat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(plus_p3455044024723400733d_enat(X,Y2)),aa_nat_Extended_enat(extended_enat2,Nb))
    <=> ? [Y3: nat,X5: nat] :
          ( ( X = aa_nat_Extended_enat(extended_enat2,X5) )
          & ( Y2 = aa_nat_Extended_enat(extended_enat2,Y3) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X5),Y3)),Nb) ) ) ).

% iadd_le_enat_iff
tff(fact_532_power2__eq__square,axiom,
    ! [A: complex] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(times_times_complex(A),A) ).

% power2_eq_square
tff(fact_533_power2__eq__square,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(times_times_real(A),A) ).

% power2_eq_square
tff(fact_534_power2__eq__square,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(times_times_rat(A),A) ).

% power2_eq_square
tff(fact_535_power2__eq__square,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),A) ).

% power2_eq_square
tff(fact_536_power2__eq__square,axiom,
    ! [A: int] : aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(times_times_int,A),A) ).

% power2_eq_square
tff(fact_537_power4__eq__xxxx,axiom,
    ! [X: complex] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(X),X)),X)),X) ).

% power4_eq_xxxx
tff(fact_538_power4__eq__xxxx,axiom,
    ! [X: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(times_times_real(X),X)),X)),X) ).

% power4_eq_xxxx
tff(fact_539_power4__eq__xxxx,axiom,
    ! [X: rat] : aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(X),X)),X)),X) ).

% power4_eq_xxxx
tff(fact_540_power4__eq__xxxx,axiom,
    ! [X: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),X)),X)),X) ).

% power4_eq_xxxx
tff(fact_541_power4__eq__xxxx,axiom,
    ! [X: int] : aa_nat_int(aa_int_fun_nat_int(power_power_int,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),X)),X) ).

% power4_eq_xxxx
tff(fact_542_power__even__eq,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_543_power__even__eq,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_544_power__even__eq,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_545_power__even__eq,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_546_power2__nat__le__imp__le,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ma),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ).

% power2_nat_le_imp_le
tff(fact_547_power2__nat__le__eq__le,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ma),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ).

% power2_nat_le_eq_le
tff(fact_548_self__le__ge2__pow,axiom,
    ! [K2: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K2),Ma)) ) ).

% self_le_ge2_pow
tff(fact_549_nth__take__lemma,axiom,
    ! [K2: nat,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),size_s6755466524823107622T_VEBT(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),size_s6755466524823107622T_VEBT(Ys))
       => ( ! [I3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),K2)
             => ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I3) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Ys),I3) ) )
         => ( take_VEBT_VEBT(K2,Xs) = take_VEBT_VEBT(K2,Ys) ) ) ) ) ).

% nth_take_lemma
tff(fact_550_nth__take__lemma,axiom,
    ! [K2: nat,Xs: list_o,Ys: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),size_size_list_o(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),size_size_list_o(Ys))
       => ( ! [I3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),K2)
             => ( aa_nat_o(nth_o(Xs),I3)
              <=> aa_nat_o(nth_o(Ys),I3) ) )
         => ( take_o(K2,Xs) = take_o(K2,Ys) ) ) ) ) ).

% nth_take_lemma
tff(fact_551_nth__take__lemma,axiom,
    ! [K2: nat,Xs: list_nat,Ys: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),size_size_list_nat(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),size_size_list_nat(Ys))
       => ( ! [I3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),K2)
             => ( aa_nat_nat(nth_nat(Xs),I3) = aa_nat_nat(nth_nat(Ys),I3) ) )
         => ( take_nat(K2,Xs) = take_nat(K2,Ys) ) ) ) ) ).

% nth_take_lemma
tff(fact_552_nth__take__lemma,axiom,
    ! [K2: nat,Xs: list_int,Ys: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),size_size_list_int(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),size_size_list_int(Ys))
       => ( ! [I3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),K2)
             => ( aa_nat_int(nth_int(Xs),I3) = aa_nat_int(nth_int(Ys),I3) ) )
         => ( take_int(K2,Xs) = take_int(K2,Ys) ) ) ) ) ).

% nth_take_lemma
tff(fact_553_power2__sum,axiom,
    ! [X: complex,Y2: complex] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(plus_plus_complex(X),Y2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),X)),Y2)) ).

% power2_sum
tff(fact_554_power2__sum,axiom,
    ! [X: real,Y2: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(plus_plus_real(X),Y2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X)),Y2)) ).

% power2_sum
tff(fact_555_power2__sum,axiom,
    ! [X: rat,Y2: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(X),Y2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),X)),Y2)) ).

% power2_sum
tff(fact_556_power2__sum,axiom,
    ! [X: nat,Y2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X)),Y2)) ).

% power2_sum
tff(fact_557_power2__sum,axiom,
    ! [X: int,Y2: int] : aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X)),Y2)) ).

% power2_sum
tff(fact_558_add__le__imp__le__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(A),C)),aa_real_real(plus_plus_real(B),C))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% add_le_imp_le_right
tff(fact_559_add__le__imp__le__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),C))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% add_le_imp_le_right
tff(fact_560_add__le__imp__le__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_imp_le_right
tff(fact_561_add__le__imp__le__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_imp_le_right
tff(fact_562_add__le__imp__le__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(C),A)),aa_real_real(plus_plus_real(C),B))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% add_le_imp_le_left
tff(fact_563_add__le__imp__le__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(C),A)),aa_rat_rat(plus_plus_rat(C),B))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% add_le_imp_le_left
tff(fact_564_add__le__imp__le__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_imp_le_left
tff(fact_565_add__le__imp__le__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_imp_le_left
tff(fact_566_le__iff__add,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> ? [C2: nat] : B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2) ) ).

% le_iff_add
tff(fact_567_add__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(A),C)),aa_real_real(plus_plus_real(B),C)) ) ).

% add_right_mono
tff(fact_568_add__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),C)) ) ).

% add_right_mono
tff(fact_569_add__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ) ).

% add_right_mono
tff(fact_570_add__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ) ).

% add_right_mono
tff(fact_571_less__eqE,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ~ ! [C3: nat] : B != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C3) ) ).

% less_eqE
tff(fact_572_add__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(C),A)),aa_real_real(plus_plus_real(C),B)) ) ).

% add_left_mono
tff(fact_573_add__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(C),A)),aa_rat_rat(plus_plus_rat(C),B)) ) ).

% add_left_mono
tff(fact_574_add__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)) ) ).

% add_left_mono
tff(fact_575_add__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)) ) ).

% add_left_mono
tff(fact_576_add__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(A),C)),aa_real_real(plus_plus_real(B),D)) ) ) ).

% add_mono
tff(fact_577_add__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),D)) ) ) ).

% add_mono
tff(fact_578_add__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D)) ) ) ).

% add_mono
tff(fact_579_add__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D)) ) ) ).

% add_mono
tff(fact_580_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I2: real,J: real,K2: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I2),J)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,K2),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(I2),K2)),aa_real_real(plus_plus_real(J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_581_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I2: rat,J: rat,K2: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I2),J)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K2),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I2),K2)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_582_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I2: nat,J: nat,K2: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_583_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I2: int,J: int,K2: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),K2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_584_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I2: real,J: real,K2: real,L: real] :
      ( ( ( I2 = J )
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,K2),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(I2),K2)),aa_real_real(plus_plus_real(J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_585_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I2: rat,J: rat,K2: rat,L: rat] :
      ( ( ( I2 = J )
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K2),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I2),K2)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_586_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I2: nat,J: nat,K2: nat,L: nat] :
      ( ( ( I2 = J )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_587_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I2: int,J: int,K2: int,L: int] :
      ( ( ( I2 = J )
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),K2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_588_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I2: real,J: real,K2: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I2),J)
        & ( K2 = L ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(I2),K2)),aa_real_real(plus_plus_real(J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_589_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I2: rat,J: rat,K2: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I2),J)
        & ( K2 = L ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I2),K2)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_590_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I2: nat,J: nat,K2: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
        & ( K2 = L ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_591_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I2: int,J: int,K2: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),J)
        & ( K2 = L ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),K2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_592_add__less__imp__less__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(A),C)),aa_real_real(plus_plus_real(B),C))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% add_less_imp_less_right
tff(fact_593_add__less__imp__less__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),C))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% add_less_imp_less_right
tff(fact_594_add__less__imp__less__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_imp_less_right
tff(fact_595_add__less__imp__less__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))
     => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_imp_less_right
tff(fact_596_add__less__imp__less__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(C),A)),aa_real_real(plus_plus_real(C),B))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% add_less_imp_less_left
tff(fact_597_add__less__imp__less__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(C),A)),aa_rat_rat(plus_plus_rat(C),B))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% add_less_imp_less_left
tff(fact_598_add__less__imp__less__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_imp_less_left
tff(fact_599_add__less__imp__less__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))
     => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_imp_less_left
tff(fact_600_add__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(A),C)),aa_real_real(plus_plus_real(B),C)) ) ).

% add_strict_right_mono
tff(fact_601_add__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),C)) ) ).

% add_strict_right_mono
tff(fact_602_add__strict__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ) ).

% add_strict_right_mono
tff(fact_603_add__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ) ).

% add_strict_right_mono
tff(fact_604_add__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(C),A)),aa_real_real(plus_plus_real(C),B)) ) ).

% add_strict_left_mono
tff(fact_605_add__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(C),A)),aa_rat_rat(plus_plus_rat(C),B)) ) ).

% add_strict_left_mono
tff(fact_606_add__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)) ) ).

% add_strict_left_mono
tff(fact_607_add__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)) ) ).

% add_strict_left_mono
tff(fact_608_add__strict__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(A),C)),aa_real_real(plus_plus_real(B),D)) ) ) ).

% add_strict_mono
tff(fact_609_add__strict__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),D)) ) ) ).

% add_strict_mono
tff(fact_610_add__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D)) ) ) ).

% add_strict_mono
tff(fact_611_add__strict__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D)) ) ) ).

% add_strict_mono
tff(fact_612_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I2: real,J: real,K2: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,I2),J)
        & ( K2 = L ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(I2),K2)),aa_real_real(plus_plus_real(J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_613_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I2: rat,J: rat,K2: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I2),J)
        & ( K2 = L ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(I2),K2)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_614_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I2: nat,J: nat,K2: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
        & ( K2 = L ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_615_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I2: int,J: int,K2: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I2),J)
        & ( K2 = L ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),K2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_616_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I2: real,J: real,K2: real,L: real] :
      ( ( ( I2 = J )
        & aa_real_o(aa_real_fun_real_o(ord_less_real,K2),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(I2),K2)),aa_real_real(plus_plus_real(J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_617_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I2: rat,J: rat,K2: rat,L: rat] :
      ( ( ( I2 = J )
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,K2),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(I2),K2)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_618_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I2: nat,J: nat,K2: nat,L: nat] :
      ( ( ( I2 = J )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_619_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I2: int,J: int,K2: int,L: int] :
      ( ( ( I2 = J )
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K2),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),K2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_620_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I2: real,J: real,K2: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,I2),J)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,K2),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(I2),K2)),aa_real_real(plus_plus_real(J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_621_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I2: rat,J: rat,K2: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I2),J)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,K2),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(I2),K2)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_622_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I2: nat,J: nat,K2: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_623_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I2: int,J: int,K2: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I2),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K2),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),K2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_624_add__less__le__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(A),C)),aa_real_real(plus_plus_real(B),D)) ) ) ).

% add_less_le_mono
tff(fact_625_add__less__le__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),D)) ) ) ).

% add_less_le_mono
tff(fact_626_add__less__le__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D)) ) ) ).

% add_less_le_mono
tff(fact_627_add__less__le__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D)) ) ) ).

% add_less_le_mono
tff(fact_628_add__le__less__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(A),C)),aa_real_real(plus_plus_real(B),D)) ) ) ).

% add_le_less_mono
tff(fact_629_add__le__less__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),D)) ) ) ).

% add_le_less_mono
tff(fact_630_add__le__less__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D)) ) ) ).

% add_le_less_mono
tff(fact_631_add__le__less__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D)) ) ) ).

% add_le_less_mono
tff(fact_632_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I2: real,J: real,K2: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,I2),J)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,K2),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(I2),K2)),aa_real_real(plus_plus_real(J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_633_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I2: rat,J: rat,K2: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I2),J)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K2),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(I2),K2)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_634_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I2: nat,J: nat,K2: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_635_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I2: int,J: int,K2: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I2),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),K2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_636_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I2: real,J: real,K2: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I2),J)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,K2),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(I2),K2)),aa_real_real(plus_plus_real(J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_637_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I2: rat,J: rat,K2: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I2),J)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,K2),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(I2),K2)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_638_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I2: nat,J: nat,K2: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_639_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I2: int,J: int,K2: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K2),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),K2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_640_numeral__Bit0,axiom,
    ! [Nb: num] : numera6690914467698888265omplex(aa_num_num(bit0,Nb)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Nb)),numera6690914467698888265omplex(Nb)) ).

% numeral_Bit0
tff(fact_641_numeral__Bit0,axiom,
    ! [Nb: num] : aa_num_real(numeral_numeral_real,aa_num_num(bit0,Nb)) = aa_real_real(plus_plus_real(aa_num_real(numeral_numeral_real,Nb)),aa_num_real(numeral_numeral_real,Nb)) ).

% numeral_Bit0
tff(fact_642_numeral__Bit0,axiom,
    ! [Nb: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,Nb)) = aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,Nb)),aa_num_rat(numeral_numeral_rat,Nb)) ).

% numeral_Bit0
tff(fact_643_numeral__Bit0,axiom,
    ! [Nb: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,Nb)) ).

% numeral_Bit0
tff(fact_644_numeral__Bit0,axiom,
    ! [Nb: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,Nb)) ).

% numeral_Bit0
tff(fact_645_divide__numeral__1,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),numera6690914467698888265omplex(one)) = A ).

% divide_numeral_1
tff(fact_646_divide__numeral__1,axiom,
    ! [A: real] : aa_real_real(divide_divide_real(A),aa_num_real(numeral_numeral_real,one)) = A ).

% divide_numeral_1
tff(fact_647_divide__numeral__1,axiom,
    ! [A: rat] : aa_rat_rat(divide_divide_rat(A),aa_num_rat(numeral_numeral_rat,one)) = A ).

% divide_numeral_1
tff(fact_648_in__children__def,axiom,
    ! [Nb: nat,TreeListb: list_VEBT_VEBT,X: nat] :
      ( aa_nat_o(vEBT_V5917875025757280293ildren(Nb,TreeListb),X)
    <=> aa_nat_o(vEBT_V8194947554948674370ptions(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeListb),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),Nb)) ) ).

% in_children_def
tff(fact_649_sum__squares__bound,axiom,
    ! [X: real,Y2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X)),Y2)),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_squares_bound
tff(fact_650_sum__squares__bound,axiom,
    ! [X: rat,Y2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),X)),Y2)),aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_squares_bound
tff(fact_651_set__n__deg__not__0,axiom,
    ! [TreeListb: list_VEBT_VEBT,Nb: nat,Ma: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeListb))
         => aa_nat_o(vEBT_invar_vebt(X4),Nb) )
     => ( ( size_s6755466524823107622T_VEBT(TreeListb) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb) ) ) ).

% set_n_deg_not_0
tff(fact_652_times__divide__eq__right,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(times_times_complex(A),aa_complex_complex(divide1717551699836669952omplex(B),C)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),B)),C) ).

% times_divide_eq_right
tff(fact_653_times__divide__eq__right,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(times_times_real(A),aa_real_real(divide_divide_real(B),C)) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(A),B)),C) ).

% times_divide_eq_right
tff(fact_654_times__divide__eq__right,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),aa_rat_rat(divide_divide_rat(B),C)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A),B)),C) ).

% times_divide_eq_right
tff(fact_655_divide__divide__eq__right,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(divide1717551699836669952omplex(B),C)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),C)),B) ).

% divide_divide_eq_right
tff(fact_656_divide__divide__eq__right,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(divide_divide_real(A),aa_real_real(divide_divide_real(B),C)) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(A),C)),B) ).

% divide_divide_eq_right
tff(fact_657_divide__divide__eq__right,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(divide_divide_rat(A),aa_rat_rat(divide_divide_rat(B),C)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A),C)),B) ).

% divide_divide_eq_right
tff(fact_658_divide__divide__eq__left,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(divide1717551699836669952omplex(A),B)),C) = aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(times_times_complex(B),C)) ).

% divide_divide_eq_left
tff(fact_659_divide__divide__eq__left,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(divide_divide_real(aa_real_real(divide_divide_real(A),B)),C) = aa_real_real(divide_divide_real(A),aa_real_real(times_times_real(B),C)) ).

% divide_divide_eq_left
tff(fact_660_divide__divide__eq__left,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(divide_divide_rat(A),B)),C) = aa_rat_rat(divide_divide_rat(A),aa_rat_rat(times_times_rat(B),C)) ).

% divide_divide_eq_left
tff(fact_661_times__divide__eq__left,axiom,
    ! [B: complex,C: complex,A: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(B),C)),A) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(B),A)),C) ).

% times_divide_eq_left
tff(fact_662_times__divide__eq__left,axiom,
    ! [B: real,C: real,A: real] : aa_real_real(times_times_real(aa_real_real(divide_divide_real(B),C)),A) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(B),A)),C) ).

% times_divide_eq_left
tff(fact_663_times__divide__eq__left,axiom,
    ! [B: rat,C: rat,A: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(divide_divide_rat(B),C)),A) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(B),A)),C) ).

% times_divide_eq_left
tff(fact_664_low__def,axiom,
    ! [X: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% low_def
tff(fact_665_set__vebt__def,axiom,
    ! [T: vEBT_VEBT] : vEBT_set_vebt(T) = collect_nat(vEBT_V8194947554948674370ptions(T)) ).

% set_vebt_def
tff(fact_666_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeListb: list_VEBT_VEBT,Nb: nat,Summaryb: vEBT_VEBT,Ma: nat,Degb: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeListb))
         => aa_nat_o(vEBT_invar_vebt(X4),Nb) )
     => ( aa_nat_o(vEBT_invar_vebt(Summaryb),Ma)
       => ( ( size_s6755466524823107622T_VEBT(TreeListb) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma) )
         => ( ( Ma = Nb )
           => ( ( Degb = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Ma) )
             => ( ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summaryb),X_13)
               => ( ! [X4: vEBT_VEBT] :
                      ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeListb))
                     => ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X_13) )
                 => aa_nat_o(vEBT_invar_vebt(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Degb,TreeListb),Summaryb)),Degb) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
tff(fact_667_power__numeral,axiom,
    ! [K2: num,L: num] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,numera6690914467698888265omplex(K2)),aa_num_nat(numeral_numeral_nat,L)) = numera6690914467698888265omplex(aa_num_num(pow(K2),L)) ).

% power_numeral
tff(fact_668_power__numeral,axiom,
    ! [K2: num,L: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,K2)),aa_num_nat(numeral_numeral_nat,L)) = aa_num_real(numeral_numeral_real,aa_num_num(pow(K2),L)) ).

% power_numeral
tff(fact_669_power__numeral,axiom,
    ! [K2: num,L: num] : aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,K2)),aa_num_nat(numeral_numeral_nat,L)) = aa_num_rat(numeral_numeral_rat,aa_num_num(pow(K2),L)) ).

% power_numeral
tff(fact_670_power__numeral,axiom,
    ! [K2: num,L: num] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,K2)),aa_num_nat(numeral_numeral_nat,L)) = aa_num_nat(numeral_numeral_nat,aa_num_num(pow(K2),L)) ).

% power_numeral
tff(fact_671_power__numeral,axiom,
    ! [K2: num,L: num] : aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,K2)),aa_num_nat(numeral_numeral_nat,L)) = aa_num_int(numeral_numeral_int,aa_num_num(pow(K2),L)) ).

% power_numeral
tff(fact_672_verit__eq__simplify_I8_J,axiom,
    ! [X23: num,Y23: num] :
      ( ( aa_num_num(bit0,X23) = aa_num_num(bit0,Y23) )
    <=> ( X23 = Y23 ) ) ).

% verit_eq_simplify(8)
tff(fact_673_mod__mod__trivial,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mod_trivial
tff(fact_674_mod__mod__trivial,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_mod_trivial
tff(fact_675_mod__mod__trivial,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(modulo364778990260209775nteger(A,B),B) = modulo364778990260209775nteger(A,B) ).

% mod_mod_trivial
tff(fact_676_times__enat__simps_I2_J,axiom,
    aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(extend5688581933313929465d_enat),extend5688581933313929465d_enat) = extend5688581933313929465d_enat ).

% times_enat_simps(2)
tff(fact_677_mult_Oright__neutral,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),one_one_complex) = A ).

% mult.right_neutral
tff(fact_678_mult_Oright__neutral,axiom,
    ! [A: real] : aa_real_real(times_times_real(A),one_one_real) = A ).

% mult.right_neutral
tff(fact_679_mult_Oright__neutral,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(A),one_one_rat) = A ).

% mult.right_neutral
tff(fact_680_mult_Oright__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),one_one_nat) = A ).

% mult.right_neutral
tff(fact_681_mult_Oright__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),one_one_int) = A ).

% mult.right_neutral
tff(fact_682_mult__1,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(one_one_complex),A) = A ).

% mult_1
tff(fact_683_mult__1,axiom,
    ! [A: real] : aa_real_real(times_times_real(one_one_real),A) = A ).

% mult_1
tff(fact_684_mult__1,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(one_one_rat),A) = A ).

% mult_1
tff(fact_685_mult__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),A) = A ).

% mult_1
tff(fact_686_mult__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,one_one_int),A) = A ).

% mult_1
tff(fact_687_bits__div__by__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),one_one_nat) = A ).

% bits_div_by_1
tff(fact_688_bits__div__by__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),one_one_int) = A ).

% bits_div_by_1
tff(fact_689_power__one,axiom,
    ! [Nb: nat] : aa_nat_rat(power_power_rat(one_one_rat),Nb) = one_one_rat ).

% power_one
tff(fact_690_power__one,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,one_one_nat),Nb) = one_one_nat ).

% power_one
tff(fact_691_power__one,axiom,
    ! [Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,one_one_real),Nb) = one_one_real ).

% power_one
tff(fact_692_power__one,axiom,
    ! [Nb: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,one_one_int),Nb) = one_one_int ).

% power_one
tff(fact_693_power__one,axiom,
    ! [Nb: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,one_one_complex),Nb) = one_one_complex ).

% power_one
tff(fact_694_mod__add__self1,axiom,
    ! [B: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_add_self1
tff(fact_695_mod__add__self1,axiom,
    ! [B: int,A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_add_self1
tff(fact_696_mod__add__self1,axiom,
    ! [B: code_integer,A: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(B,A),B) = modulo364778990260209775nteger(A,B) ).

% mod_add_self1
tff(fact_697_mod__add__self2,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_add_self2
tff(fact_698_mod__add__self2,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_add_self2
tff(fact_699_mod__add__self2,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,B),B) = modulo364778990260209775nteger(A,B) ).

% mod_add_self2
tff(fact_700_power__one__right,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),one_one_nat) = A ).

% power_one_right
tff(fact_701_power__one__right,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),one_one_nat) = A ).

% power_one_right
tff(fact_702_power__one__right,axiom,
    ! [A: int] : aa_nat_int(aa_int_fun_nat_int(power_power_int,A),one_one_nat) = A ).

% power_one_right
tff(fact_703_power__one__right,axiom,
    ! [A: complex] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),one_one_nat) = A ).

% power_one_right
tff(fact_704_nat__mult__eq__1__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Nb) = one_one_nat )
    <=> ( ( Ma = one_one_nat )
        & ( Nb = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
tff(fact_705_nat__1__eq__mult__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( one_one_nat = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Nb) )
    <=> ( ( Ma = one_one_nat )
        & ( Nb = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
tff(fact_706_mod__less,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),Nb) = Ma ) ) ).

% mod_less
tff(fact_707_times__enat__simps_I1_J,axiom,
    ! [Ma: nat,Nb: nat] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(extended_enat2,Ma)),aa_nat_Extended_enat(extended_enat2,Nb)) = aa_nat_Extended_enat(extended_enat2,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Nb)) ).

% times_enat_simps(1)
tff(fact_708_semiring__norm_I13_J,axiom,
    ! [Ma: num,Nb: num] : aa_num_num(times_times_num(aa_num_num(bit0,Ma)),aa_num_num(bit0,Nb)) = aa_num_num(bit0,aa_num_num(bit0,aa_num_num(times_times_num(Ma),Nb))) ).

% semiring_norm(13)
tff(fact_709_semiring__norm_I12_J,axiom,
    ! [Nb: num] : aa_num_num(times_times_num(one),Nb) = Nb ).

% semiring_norm(12)
tff(fact_710_semiring__norm_I11_J,axiom,
    ! [Ma: num] : aa_num_num(times_times_num(Ma),one) = Ma ).

% semiring_norm(11)
tff(fact_711_numeral__eq__one__iff,axiom,
    ! [Nb: num] :
      ( ( numera6690914467698888265omplex(Nb) = one_one_complex )
    <=> ( Nb = one ) ) ).

% numeral_eq_one_iff
tff(fact_712_numeral__eq__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_num_real(numeral_numeral_real,Nb) = one_one_real )
    <=> ( Nb = one ) ) ).

% numeral_eq_one_iff
tff(fact_713_numeral__eq__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_num_rat(numeral_numeral_rat,Nb) = one_one_rat )
    <=> ( Nb = one ) ) ).

% numeral_eq_one_iff
tff(fact_714_numeral__eq__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_num_nat(numeral_numeral_nat,Nb) = one_one_nat )
    <=> ( Nb = one ) ) ).

% numeral_eq_one_iff
tff(fact_715_numeral__eq__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_num_int(numeral_numeral_int,Nb) = one_one_int )
    <=> ( Nb = one ) ) ).

% numeral_eq_one_iff
tff(fact_716_one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( one_one_complex = numera6690914467698888265omplex(Nb) )
    <=> ( one = Nb ) ) ).

% one_eq_numeral_iff
tff(fact_717_one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( one_one_real = aa_num_real(numeral_numeral_real,Nb) )
    <=> ( one = Nb ) ) ).

% one_eq_numeral_iff
tff(fact_718_one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( one_one_rat = aa_num_rat(numeral_numeral_rat,Nb) )
    <=> ( one = Nb ) ) ).

% one_eq_numeral_iff
tff(fact_719_one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( one_one_nat = aa_num_nat(numeral_numeral_nat,Nb) )
    <=> ( one = Nb ) ) ).

% one_eq_numeral_iff
tff(fact_720_one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( one_one_int = aa_num_int(numeral_numeral_int,Nb) )
    <=> ( one = Nb ) ) ).

% one_eq_numeral_iff
tff(fact_721_power__inject__exp,axiom,
    ! [A: real,Ma: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Ma) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb) )
      <=> ( Ma = Nb ) ) ) ).

% power_inject_exp
tff(fact_722_power__inject__exp,axiom,
    ! [A: rat,Ma: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => ( ( aa_nat_rat(power_power_rat(A),Ma) = aa_nat_rat(power_power_rat(A),Nb) )
      <=> ( Ma = Nb ) ) ) ).

% power_inject_exp
tff(fact_723_power__inject__exp,axiom,
    ! [A: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Ma) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb) )
      <=> ( Ma = Nb ) ) ) ).

% power_inject_exp
tff(fact_724_power__inject__exp,axiom,
    ! [A: int,Ma: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Ma) = aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb) )
      <=> ( Ma = Nb ) ) ) ).

% power_inject_exp
tff(fact_725_mod__mult__self4,axiom,
    ! [B: nat,C: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self4
tff(fact_726_mod__mult__self4,axiom,
    ! [B: int,C: int,A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),C)),A)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_mult_self4
tff(fact_727_mod__mult__self4,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C),A),B) = modulo364778990260209775nteger(A,B) ).

% mod_mult_self4
tff(fact_728_mod__mult__self3,axiom,
    ! [C: nat,B: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self3
tff(fact_729_mod__mult__self3,axiom,
    ! [C: int,B: int,A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),B)),A)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_mult_self3
tff(fact_730_mod__mult__self3,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B),A),B) = modulo364778990260209775nteger(A,B) ).

% mod_mult_self3
tff(fact_731_mod__mult__self2,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self2
tff(fact_732_mod__mult__self2,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_mult_self2
tff(fact_733_mod__mult__self2,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)),B) = modulo364778990260209775nteger(A,B) ).

% mod_mult_self2
tff(fact_734_mod__mult__self1,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self1
tff(fact_735_mod__mult__self1,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_mult_self1
tff(fact_736_mod__mult__self1,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)),B) = modulo364778990260209775nteger(A,B) ).

% mod_mult_self1
tff(fact_737_num__double,axiom,
    ! [Nb: num] : aa_num_num(times_times_num(aa_num_num(bit0,one)),Nb) = aa_num_num(bit0,Nb) ).

% num_double
tff(fact_738_power__mult__numeral,axiom,
    ! [A: nat,Ma: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,Ma))),aa_num_nat(numeral_numeral_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(times_times_num(Ma),Nb))) ).

% power_mult_numeral
tff(fact_739_power__mult__numeral,axiom,
    ! [A: real,Ma: num,Nb: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,Ma))),aa_num_nat(numeral_numeral_nat,Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(times_times_num(Ma),Nb))) ).

% power_mult_numeral
tff(fact_740_power__mult__numeral,axiom,
    ! [A: int,Ma: num,Nb: num] : aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,Ma))),aa_num_nat(numeral_numeral_nat,Nb)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,aa_num_num(times_times_num(Ma),Nb))) ).

% power_mult_numeral
tff(fact_741_power__mult__numeral,axiom,
    ! [A: complex,Ma: num,Nb: num] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),aa_num_nat(numeral_numeral_nat,Ma))),aa_num_nat(numeral_numeral_nat,Nb)) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),aa_num_nat(numeral_numeral_nat,aa_num_num(times_times_num(Ma),Nb))) ).

% power_mult_numeral
tff(fact_742_power__strict__increasing__iff,axiom,
    ! [B: real,X: nat,Y2: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Y2))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y2) ) ) ).

% power_strict_increasing_iff
tff(fact_743_power__strict__increasing__iff,axiom,
    ! [B: rat,X: nat,Y2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(B),X)),aa_nat_rat(power_power_rat(B),Y2))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y2) ) ) ).

% power_strict_increasing_iff
tff(fact_744_power__strict__increasing__iff,axiom,
    ! [B: nat,X: nat,Y2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Y2))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y2) ) ) ).

% power_strict_increasing_iff
tff(fact_745_power__strict__increasing__iff,axiom,
    ! [B: int,X: nat,Y2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,B),X)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Y2))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y2) ) ) ).

% power_strict_increasing_iff
tff(fact_746_one__add__one,axiom,
    aa_complex_complex(plus_plus_complex(one_one_complex),one_one_complex) = numera6690914467698888265omplex(aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_747_one__add__one,axiom,
    aa_real_real(plus_plus_real(one_one_real),one_one_real) = aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_748_one__add__one,axiom,
    aa_rat_rat(plus_plus_rat(one_one_rat),one_one_rat) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_749_one__add__one,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_750_one__add__one,axiom,
    aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_751_power__increasing__iff,axiom,
    ! [B: real,X: nat,Y2: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Y2))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y2) ) ) ).

% power_increasing_iff
tff(fact_752_power__increasing__iff,axiom,
    ! [B: rat,X: nat,Y2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(B),X)),aa_nat_rat(power_power_rat(B),Y2))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y2) ) ) ).

% power_increasing_iff
tff(fact_753_power__increasing__iff,axiom,
    ! [B: nat,X: nat,Y2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Y2))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y2) ) ) ).

% power_increasing_iff
tff(fact_754_power__increasing__iff,axiom,
    ! [B: int,X: nat,Y2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,B),X)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Y2))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y2) ) ) ).

% power_increasing_iff
tff(fact_755_bits__one__mod__two__eq__one,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ).

% bits_one_mod_two_eq_one
tff(fact_756_bits__one__mod__two__eq__one,axiom,
    aa_int_int(aa_int_fun_int_int(modulo_modulo_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ).

% bits_one_mod_two_eq_one
tff(fact_757_bits__one__mod__two__eq__one,axiom,
    modulo364778990260209775nteger(one_one_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))) = one_one_Code_integer ).

% bits_one_mod_two_eq_one
tff(fact_758_one__mod__two__eq__one,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ).

% one_mod_two_eq_one
tff(fact_759_one__mod__two__eq__one,axiom,
    aa_int_int(aa_int_fun_int_int(modulo_modulo_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ).

% one_mod_two_eq_one
tff(fact_760_one__mod__two__eq__one,axiom,
    modulo364778990260209775nteger(one_one_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))) = one_one_Code_integer ).

% one_mod_two_eq_one
tff(fact_761_numeral__plus__one,axiom,
    ! [Nb: num] : aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Nb)),one_one_complex) = numera6690914467698888265omplex(aa_num_num(plus_plus_num(Nb),one)) ).

% numeral_plus_one
tff(fact_762_numeral__plus__one,axiom,
    ! [Nb: num] : aa_real_real(plus_plus_real(aa_num_real(numeral_numeral_real,Nb)),one_one_real) = aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(Nb),one)) ).

% numeral_plus_one
tff(fact_763_numeral__plus__one,axiom,
    ! [Nb: num] : aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,Nb)),one_one_rat) = aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(Nb),one)) ).

% numeral_plus_one
tff(fact_764_numeral__plus__one,axiom,
    ! [Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Nb)),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(Nb),one)) ).

% numeral_plus_one
tff(fact_765_numeral__plus__one,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Nb)),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(Nb),one)) ).

% numeral_plus_one
tff(fact_766_one__plus__numeral,axiom,
    ! [Nb: num] : aa_complex_complex(plus_plus_complex(one_one_complex),numera6690914467698888265omplex(Nb)) = numera6690914467698888265omplex(aa_num_num(plus_plus_num(one),Nb)) ).

% one_plus_numeral
tff(fact_767_one__plus__numeral,axiom,
    ! [Nb: num] : aa_real_real(plus_plus_real(one_one_real),aa_num_real(numeral_numeral_real,Nb)) = aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(one),Nb)) ).

% one_plus_numeral
tff(fact_768_one__plus__numeral,axiom,
    ! [Nb: num] : aa_rat_rat(plus_plus_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,Nb)) = aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(one),Nb)) ).

% one_plus_numeral
tff(fact_769_one__plus__numeral,axiom,
    ! [Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(one),Nb)) ).

% one_plus_numeral
tff(fact_770_one__plus__numeral,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_num_int(numeral_numeral_int,Nb)) = aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(one),Nb)) ).

% one_plus_numeral
tff(fact_771_numeral__le__one__iff,axiom,
    ! [Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Nb)),one_one_real)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),one) ) ).

% numeral_le_one_iff
tff(fact_772_numeral__le__one__iff,axiom,
    ! [Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Nb)),one_one_rat)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),one) ) ).

% numeral_le_one_iff
tff(fact_773_numeral__le__one__iff,axiom,
    ! [Nb: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Nb)),one_one_nat)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),one) ) ).

% numeral_le_one_iff
tff(fact_774_numeral__le__one__iff,axiom,
    ! [Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Nb)),one_one_int)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),one) ) ).

% numeral_le_one_iff
tff(fact_775_one__less__numeral__iff,axiom,
    ! [Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_num_real(numeral_numeral_real,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),Nb) ) ).

% one_less_numeral_iff
tff(fact_776_one__less__numeral__iff,axiom,
    ! [Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),Nb) ) ).

% one_less_numeral_iff
tff(fact_777_one__less__numeral__iff,axiom,
    ! [Nb: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),Nb) ) ).

% one_less_numeral_iff
tff(fact_778_one__less__numeral__iff,axiom,
    ! [Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_num_int(numeral_numeral_int,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),Nb) ) ).

% one_less_numeral_iff
tff(fact_779_mod__less__eq__dividend,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),Nb)),Ma) ).

% mod_less_eq_dividend
tff(fact_780_Nat_Oex__has__greatest__nat,axiom,
    ! [P: fun_nat_o,K2: nat,B: nat] :
      ( aa_nat_o(P,K2)
     => ( ! [Y4: nat] :
            ( aa_nat_o(P,Y4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y4),B) )
       => ? [X4: nat] :
            ( aa_nat_o(P,X4)
            & ! [Y5: nat] :
                ( aa_nat_o(P,Y5)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y5),X4) ) ) ) ) ).

% Nat.ex_has_greatest_nat
tff(fact_781_nat__le__linear,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma) ) ).

% nat_le_linear
tff(fact_782_le__antisym,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma)
       => ( Ma = Nb ) ) ) ).

% le_antisym
tff(fact_783_eq__imp__le,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( Ma = Nb )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ).

% eq_imp_le
tff(fact_784_le__trans,axiom,
    ! [I2: nat,J: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),K2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),K2) ) ) ).

% le_trans
tff(fact_785_le__refl,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Nb) ).

% le_refl
tff(fact_786_one__reorient,axiom,
    ! [X: complex] :
      ( ( one_one_complex = X )
    <=> ( X = one_one_complex ) ) ).

% one_reorient
tff(fact_787_one__reorient,axiom,
    ! [X: real] :
      ( ( one_one_real = X )
    <=> ( X = one_one_real ) ) ).

% one_reorient
tff(fact_788_one__reorient,axiom,
    ! [X: rat] :
      ( ( one_one_rat = X )
    <=> ( X = one_one_rat ) ) ).

% one_reorient
tff(fact_789_one__reorient,axiom,
    ! [X: nat] :
      ( ( one_one_nat = X )
    <=> ( X = one_one_nat ) ) ).

% one_reorient
tff(fact_790_one__reorient,axiom,
    ! [X: int] :
      ( ( one_one_int = X )
    <=> ( X = one_one_int ) ) ).

% one_reorient
tff(fact_791_mod__mult__right__eq,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ).

% mod_mult_right_eq
tff(fact_792_mod__mult__right__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) ).

% mod_mult_right_eq
tff(fact_793_mod__mult__right__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),C) ).

% mod_mult_right_eq
tff(fact_794_mod__mult__left__eq,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C)),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ).

% mod_mult_left_eq
tff(fact_795_mod__mult__left__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C)),B)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) ).

% mod_mult_left_eq
tff(fact_796_mod__mult__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,modulo364778990260209775nteger(A,C)),B),C) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),C) ).

% mod_mult_left_eq
tff(fact_797_mult__mod__right,axiom,
    ! [C: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ).

% mult_mod_right
tff(fact_798_mult__mod__right,axiom,
    ! [C: int,A: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,C),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ).

% mult_mod_right
tff(fact_799_mult__mod__right,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),modulo364778990260209775nteger(A,B)) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) ).

% mult_mod_right
tff(fact_800_mod__mult__mult2,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),C) ).

% mod_mult_mult2
tff(fact_801_mod__mult__mult2,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),C) ).

% mod_mult_mult2
tff(fact_802_mod__mult__mult2,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,modulo364778990260209775nteger(A,B)),C) ).

% mod_mult_mult2
tff(fact_803_mod__mult__cong,axiom,
    ! [A: nat,C: nat,A4: nat,B: nat,B4: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A4),C) )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B4),C) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A4),B4)),C) ) ) ) ).

% mod_mult_cong
tff(fact_804_mod__mult__cong,axiom,
    ! [A: int,C: int,A4: int,B: int,B4: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A4),C) )
     => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B4),C) )
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A4),B4)),C) ) ) ) ).

% mod_mult_cong
tff(fact_805_mod__mult__cong,axiom,
    ! [A: code_integer,C: code_integer,A4: code_integer,B: code_integer,B4: code_integer] :
      ( ( modulo364778990260209775nteger(A,C) = modulo364778990260209775nteger(A4,C) )
     => ( ( modulo364778990260209775nteger(B,C) = modulo364778990260209775nteger(B4,C) )
       => ( modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),C) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A4),B4),C) ) ) ) ).

% mod_mult_cong
tff(fact_806_mod__mult__eq,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ).

% mod_mult_eq
tff(fact_807_mod__mult__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) ).

% mod_mult_eq
tff(fact_808_mod__mult__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,modulo364778990260209775nteger(A,C)),modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),C) ).

% mod_mult_eq
tff(fact_809_mod__add__eq,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) ).

% mod_add_eq
tff(fact_810_mod__add__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ).

% mod_add_eq
tff(fact_811_mod__add__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,C),modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,B),C) ).

% mod_add_eq
tff(fact_812_mod__add__cong,axiom,
    ! [A: nat,C: nat,A4: nat,B: nat,B4: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A4),C) )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B4),C) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A4),B4)),C) ) ) ) ).

% mod_add_cong
tff(fact_813_mod__add__cong,axiom,
    ! [A: int,C: int,A4: int,B: int,B4: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A4),C) )
     => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B4),C) )
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A4),B4)),C) ) ) ) ).

% mod_add_cong
tff(fact_814_mod__add__cong,axiom,
    ! [A: code_integer,C: code_integer,A4: code_integer,B: code_integer,B4: code_integer] :
      ( ( modulo364778990260209775nteger(A,C) = modulo364778990260209775nteger(A4,C) )
     => ( ( modulo364778990260209775nteger(B,C) = modulo364778990260209775nteger(B4,C) )
       => ( modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,B),C) = modulo364778990260209775nteger(plus_p5714425477246183910nteger(A4,B4),C) ) ) ) ).

% mod_add_cong
tff(fact_815_mod__add__left__eq,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C)),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) ).

% mod_add_left_eq
tff(fact_816_mod__add__left__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C)),B)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ).

% mod_add_left_eq
tff(fact_817_mod__add__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,C),B),C) = modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,B),C) ).

% mod_add_left_eq
tff(fact_818_mod__add__right__eq,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) ).

% mod_add_right_eq
tff(fact_819_mod__add__right__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ).

% mod_add_right_eq
tff(fact_820_mod__add__right__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,B),C) ).

% mod_add_right_eq
tff(fact_821_power__mod,axiom,
    ! [A: nat,B: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),Nb)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),B) ).

% power_mod
tff(fact_822_power__mod,axiom,
    ! [A: int,B: int,Nb: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),Nb)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)),B) ).

% power_mod
tff(fact_823_power__mod,axiom,
    ! [A: code_integer,B: code_integer,Nb: nat] : modulo364778990260209775nteger(aa_nat_Code_integer(power_8256067586552552935nteger(modulo364778990260209775nteger(A,B)),Nb),B) = modulo364778990260209775nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb),B) ).

% power_mod
tff(fact_824_le__numeral__extra_I4_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),one_one_real) ).

% le_numeral_extra(4)
tff(fact_825_le__numeral__extra_I4_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),one_one_rat) ).

% le_numeral_extra(4)
tff(fact_826_le__numeral__extra_I4_J,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),one_one_nat) ).

% le_numeral_extra(4)
tff(fact_827_le__numeral__extra_I4_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),one_one_int) ).

% le_numeral_extra(4)
tff(fact_828_less__numeral__extra_I4_J,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),one_one_real) ).

% less_numeral_extra(4)
tff(fact_829_less__numeral__extra_I4_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),one_one_rat) ).

% less_numeral_extra(4)
tff(fact_830_less__numeral__extra_I4_J,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),one_one_nat) ).

% less_numeral_extra(4)
tff(fact_831_less__numeral__extra_I4_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),one_one_int) ).

% less_numeral_extra(4)
tff(fact_832_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(one_one_complex),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_833_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: real] : aa_real_real(times_times_real(one_one_real),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_834_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(one_one_rat),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_835_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_836_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,one_one_int),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_837_mult_Ocomm__neutral,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),one_one_complex) = A ).

% mult.comm_neutral
tff(fact_838_mult_Ocomm__neutral,axiom,
    ! [A: real] : aa_real_real(times_times_real(A),one_one_real) = A ).

% mult.comm_neutral
tff(fact_839_mult_Ocomm__neutral,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(A),one_one_rat) = A ).

% mult.comm_neutral
tff(fact_840_mult_Ocomm__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),one_one_nat) = A ).

% mult.comm_neutral
tff(fact_841_mult_Ocomm__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),one_one_int) = A ).

% mult.comm_neutral
tff(fact_842_nat__mult__1__right,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),one_one_nat) = Nb ).

% nat_mult_1_right
tff(fact_843_nat__mult__1,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),Nb) = Nb ).

% nat_mult_1
tff(fact_844_mod__eqE,axiom,
    ! [A: int,C: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C) )
     => ~ ! [D2: int] : B != aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),D2)) ) ).

% mod_eqE
tff(fact_845_mod__eqE,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger(A,C) = modulo364778990260209775nteger(B,C) )
     => ~ ! [D2: code_integer] : B != plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),D2)) ) ).

% mod_eqE
tff(fact_846_div__add1__eq,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C)) ).

% div_add1_eq
tff(fact_847_div__add1__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C)) ).

% div_add1_eq
tff(fact_848_div__add1__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,B),C) = plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(divide6298287555418463151nteger(A,C),divide6298287555418463151nteger(B,C)),divide6298287555418463151nteger(plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,C),modulo364778990260209775nteger(B,C)),C)) ).

% div_add1_eq
tff(fact_849_four__x__squared,axiom,
    ! [X: real] : aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% four_x_squared
tff(fact_850_L2__set__mult__ineq__lemma,axiom,
    ! [A: real,C: real,B: real,D: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(times_times_real(A),C))),aa_real_real(times_times_real(B),D))),aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,D),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,C),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% L2_set_mult_ineq_lemma
tff(fact_851_div__mod__decomp,axiom,
    ! [A2: nat,Nb: nat] : A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),Nb)),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),Nb)) ).

% div_mod_decomp
tff(fact_852_div__mult2__numeral__eq,axiom,
    ! [A: nat,K2: num,L: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,K2))),aa_num_nat(numeral_numeral_nat,L)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(times_times_num(K2),L))) ).

% div_mult2_numeral_eq
tff(fact_853_div__mult2__numeral__eq,axiom,
    ! [A: int,K2: num,L: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,K2))),aa_num_int(numeral_numeral_int,L)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(K2),L))) ).

% div_mult2_numeral_eq
tff(fact_854_gt__half__sum,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(A),B)),aa_real_real(plus_plus_real(one_one_real),one_one_real))),B) ) ).

% gt_half_sum
tff(fact_855_gt__half__sum,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(A),B)),aa_rat_rat(plus_plus_rat(one_one_rat),one_one_rat))),B) ) ).

% gt_half_sum
tff(fact_856_less__half__sum,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(A),B)),aa_real_real(plus_plus_real(one_one_real),one_one_real))) ) ).

% less_half_sum
tff(fact_857_less__half__sum,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(A),B)),aa_rat_rat(plus_plus_rat(one_one_rat),one_one_rat))) ) ).

% less_half_sum
tff(fact_858_one__le__numeral,axiom,
    ! [Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_num_real(numeral_numeral_real,Nb)) ).

% one_le_numeral
tff(fact_859_one__le__numeral,axiom,
    ! [Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,Nb)) ).

% one_le_numeral
tff(fact_860_one__le__numeral,axiom,
    ! [Nb: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Nb)) ).

% one_le_numeral
tff(fact_861_one__le__numeral,axiom,
    ! [Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_num_int(numeral_numeral_int,Nb)) ).

% one_le_numeral
tff(fact_862_not__numeral__less__one,axiom,
    ! [Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,Nb)),one_one_real) ).

% not_numeral_less_one
tff(fact_863_not__numeral__less__one,axiom,
    ! [Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,Nb)),one_one_rat) ).

% not_numeral_less_one
tff(fact_864_not__numeral__less__one,axiom,
    ! [Nb: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Nb)),one_one_nat) ).

% not_numeral_less_one
tff(fact_865_not__numeral__less__one,axiom,
    ! [Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Nb)),one_one_int) ).

% not_numeral_less_one
tff(fact_866_one__plus__numeral__commute,axiom,
    ! [X: num] : aa_complex_complex(plus_plus_complex(one_one_complex),numera6690914467698888265omplex(X)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(X)),one_one_complex) ).

% one_plus_numeral_commute
tff(fact_867_one__plus__numeral__commute,axiom,
    ! [X: num] : aa_real_real(plus_plus_real(one_one_real),aa_num_real(numeral_numeral_real,X)) = aa_real_real(plus_plus_real(aa_num_real(numeral_numeral_real,X)),one_one_real) ).

% one_plus_numeral_commute
tff(fact_868_one__plus__numeral__commute,axiom,
    ! [X: num] : aa_rat_rat(plus_plus_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,X)) = aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,X)),one_one_rat) ).

% one_plus_numeral_commute
tff(fact_869_one__plus__numeral__commute,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,X)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,X)),one_one_nat) ).

% one_plus_numeral_commute
tff(fact_870_one__plus__numeral__commute,axiom,
    ! [X: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_num_int(numeral_numeral_int,X)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,X)),one_one_int) ).

% one_plus_numeral_commute
tff(fact_871_numeral__One,axiom,
    numera6690914467698888265omplex(one) = one_one_complex ).

% numeral_One
tff(fact_872_numeral__One,axiom,
    aa_num_real(numeral_numeral_real,one) = one_one_real ).

% numeral_One
tff(fact_873_numeral__One,axiom,
    aa_num_rat(numeral_numeral_rat,one) = one_one_rat ).

% numeral_One
tff(fact_874_numeral__One,axiom,
    aa_num_nat(numeral_numeral_nat,one) = one_one_nat ).

% numeral_One
tff(fact_875_numeral__One,axiom,
    aa_num_int(numeral_numeral_int,one) = one_one_int ).

% numeral_One
tff(fact_876_one__le__power,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ).

% one_le_power
tff(fact_877_one__le__power,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(power_power_rat(A),Nb)) ) ).

% one_le_power
tff(fact_878_one__le__power,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ).

% one_le_power
tff(fact_879_one__le__power,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)) ) ).

% one_le_power
tff(fact_880_power__increasing,axiom,
    ! [Nb: nat,N2: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N2)) ) ) ).

% power_increasing
tff(fact_881_power__increasing,axiom,
    ! [Nb: nat,N2: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(A),N2)) ) ) ).

% power_increasing
tff(fact_882_power__increasing,axiom,
    ! [Nb: nat,N2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2)) ) ) ).

% power_increasing
tff(fact_883_power__increasing,axiom,
    ! [Nb: nat,N2: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N2)) ) ) ).

% power_increasing
tff(fact_884_left__right__inverse__power,axiom,
    ! [X: complex,Y2: complex,Nb: nat] :
      ( ( aa_complex_complex(times_times_complex(X),Y2) = one_one_complex )
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,X),Nb)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Y2),Nb)) = one_one_complex ) ) ).

% left_right_inverse_power
tff(fact_885_left__right__inverse__power,axiom,
    ! [X: real,Y2: real,Nb: nat] :
      ( ( aa_real_real(times_times_real(X),Y2) = one_one_real )
     => ( aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),Nb)) = one_one_real ) ) ).

% left_right_inverse_power
tff(fact_886_left__right__inverse__power,axiom,
    ! [X: rat,Y2: rat,Nb: nat] :
      ( ( aa_rat_rat(times_times_rat(X),Y2) = one_one_rat )
     => ( aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(X),Nb)),aa_nat_rat(power_power_rat(Y2),Nb)) = one_one_rat ) ) ).

% left_right_inverse_power
tff(fact_887_left__right__inverse__power,axiom,
    ! [X: nat,Y2: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Y2) = one_one_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y2),Nb)) = one_one_nat ) ) ).

% left_right_inverse_power
tff(fact_888_left__right__inverse__power,axiom,
    ! [X: int,Y2: int,Nb: nat] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,X),Y2) = one_one_int )
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X),Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y2),Nb)) = one_one_int ) ) ).

% left_right_inverse_power
tff(fact_889_power__one__over,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(divide1717551699836669952omplex(one_one_complex),A)),Nb) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),Nb)) ).

% power_one_over
tff(fact_890_power__one__over,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(divide_divide_real(one_one_real),A)),Nb) = aa_real_real(divide_divide_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ).

% power_one_over
tff(fact_891_power__one__over,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(divide_divide_rat(one_one_rat),A)),Nb) = aa_rat_rat(divide_divide_rat(one_one_rat),aa_nat_rat(power_power_rat(A),Nb)) ).

% power_one_over
tff(fact_892_numerals_I1_J,axiom,
    aa_num_nat(numeral_numeral_nat,one) = one_one_nat ).

% numerals(1)
tff(fact_893_pow_Osimps_I1_J,axiom,
    ! [X: num] : aa_num_num(pow(X),one) = X ).

% pow.simps(1)
tff(fact_894_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o,D: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(aa_o_VEBT_VEBT(vEBT_Leaf((Uu)),(Uv))),D)
    <=> ( D = one_one_nat ) ) ).

% VEBT_internal.valid'.simps(1)
tff(fact_895_div__mult1__eq,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C)) ).

% div_mult1_eq
tff(fact_896_div__mult1__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C)) ).

% div_mult1_eq
tff(fact_897_div__mult1__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),C) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),divide6298287555418463151nteger(B,C)),divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),modulo364778990260209775nteger(B,C)),C)) ).

% div_mult1_eq
tff(fact_898_mod__mult2__eq,axiom,
    ! [Ma: nat,Nb: nat,Q: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),Nb)),Q))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),Nb)) ).

% mod_mult2_eq
tff(fact_899_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_int] : ord_less_eq_set_int(A,A) ).

% verit_comp_simplify1(2)
tff(fact_900_verit__comp__simplify1_I2_J,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),A) ).

% verit_comp_simplify1(2)
tff(fact_901_verit__comp__simplify1_I2_J,axiom,
    ! [A: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),A) ).

% verit_comp_simplify1(2)
tff(fact_902_verit__comp__simplify1_I2_J,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),A) ).

% verit_comp_simplify1(2)
tff(fact_903_verit__comp__simplify1_I2_J,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),A) ).

% verit_comp_simplify1(2)
tff(fact_904_verit__la__disequality,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
      | ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
      | ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ).

% verit_la_disequality
tff(fact_905_verit__la__disequality,axiom,
    ! [A: num,B: num] :
      ( ( A = B )
      | ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
      | ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A) ) ).

% verit_la_disequality
tff(fact_906_verit__la__disequality,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
      | ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
      | ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ).

% verit_la_disequality
tff(fact_907_verit__la__disequality,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
      | ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
      | ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ).

% verit_la_disequality
tff(fact_908_verit__comp__simplify1_I1_J,axiom,
    ! [A: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),A) ).

% verit_comp_simplify1(1)
tff(fact_909_verit__comp__simplify1_I1_J,axiom,
    ! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),A) ).

% verit_comp_simplify1(1)
tff(fact_910_verit__comp__simplify1_I1_J,axiom,
    ! [A: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A),A) ).

% verit_comp_simplify1(1)
tff(fact_911_verit__comp__simplify1_I1_J,axiom,
    ! [A: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),A) ).

% verit_comp_simplify1(1)
tff(fact_912_verit__comp__simplify1_I1_J,axiom,
    ! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),A) ).

% verit_comp_simplify1(1)
tff(fact_913_linordered__field__no__lb,axiom,
    ! [X2: real] :
    ? [Y4: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y4),X2) ).

% linordered_field_no_lb
tff(fact_914_linordered__field__no__lb,axiom,
    ! [X2: rat] :
    ? [Y4: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y4),X2) ).

% linordered_field_no_lb
tff(fact_915_linordered__field__no__ub,axiom,
    ! [X2: real] :
    ? [X_13: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,X2),X_13) ).

% linordered_field_no_ub
tff(fact_916_linordered__field__no__ub,axiom,
    ! [X2: rat] :
    ? [X_13: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),X_13) ).

% linordered_field_no_ub
tff(fact_917_power__le__imp__le__exp,axiom,
    ! [A: real,Ma: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Ma)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ) ).

% power_le_imp_le_exp
tff(fact_918_power__le__imp__le__exp,axiom,
    ! [A: rat,Ma: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Ma)),aa_nat_rat(power_power_rat(A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ) ).

% power_le_imp_le_exp
tff(fact_919_power__le__imp__le__exp,axiom,
    ! [A: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ) ).

% power_le_imp_le_exp
tff(fact_920_power__le__imp__le__exp,axiom,
    ! [A: int,Ma: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Ma)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ) ).

% power_le_imp_le_exp
tff(fact_921_power__gt1__lemma,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(times_times_real(A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))) ) ).

% power_gt1_lemma
tff(fact_922_power__gt1__lemma,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),Nb))) ) ).

% power_gt1_lemma
tff(fact_923_power__gt1__lemma,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb))) ) ).

% power_gt1_lemma
tff(fact_924_power__gt1__lemma,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb))) ) ).

% power_gt1_lemma
tff(fact_925_power__less__power__Suc,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_real_real(times_times_real(A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))) ) ).

% power_less_power_Suc
tff(fact_926_power__less__power__Suc,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),Nb))) ) ).

% power_less_power_Suc
tff(fact_927_power__less__power__Suc,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb))) ) ).

% power_less_power_Suc
tff(fact_928_power__less__power__Suc,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)),aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb))) ) ).

% power_less_power_Suc
tff(fact_929_power__less__imp__less__exp,axiom,
    ! [A: real,Ma: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Ma)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb) ) ) ).

% power_less_imp_less_exp
tff(fact_930_power__less__imp__less__exp,axiom,
    ! [A: rat,Ma: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Ma)),aa_nat_rat(power_power_rat(A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb) ) ) ).

% power_less_imp_less_exp
tff(fact_931_power__less__imp__less__exp,axiom,
    ! [A: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb) ) ) ).

% power_less_imp_less_exp
tff(fact_932_power__less__imp__less__exp,axiom,
    ! [A: int,Ma: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Ma)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb) ) ) ).

% power_less_imp_less_exp
tff(fact_933_power__strict__increasing,axiom,
    ! [Nb: nat,N2: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N2)) ) ) ).

% power_strict_increasing
tff(fact_934_power__strict__increasing,axiom,
    ! [Nb: nat,N2: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(A),N2)) ) ) ).

% power_strict_increasing
tff(fact_935_power__strict__increasing,axiom,
    ! [Nb: nat,N2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2)) ) ) ).

% power_strict_increasing
tff(fact_936_power__strict__increasing,axiom,
    ! [Nb: nat,N2: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N2)) ) ) ).

% power_strict_increasing
tff(fact_937_one__power2,axiom,
    aa_nat_rat(power_power_rat(one_one_rat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_rat ).

% one_power2
tff(fact_938_one__power2,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ).

% one_power2
tff(fact_939_one__power2,axiom,
    aa_nat_real(aa_real_fun_nat_real(power_power_real,one_one_real),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_real ).

% one_power2
tff(fact_940_one__power2,axiom,
    aa_nat_int(aa_int_fun_nat_int(power_power_int,one_one_int),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_int ).

% one_power2
tff(fact_941_one__power2,axiom,
    aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,one_one_complex),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_complex ).

% one_power2
tff(fact_942_nat__1__add__1,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% nat_1_add_1
tff(fact_943_div__exp__mod__exp__eq,axiom,
    ! [A: nat,Nb: nat,Ma: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Ma)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% div_exp_mod_exp_eq
tff(fact_944_div__exp__mod__exp__eq,axiom,
    ! [A: int,Nb: nat,Ma: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Ma)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Ma)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% div_exp_mod_exp_eq
tff(fact_945_div__exp__mod__exp__eq,axiom,
    ! [A: code_integer,Nb: nat,Ma: nat] : modulo364778990260209775nteger(divide6298287555418463151nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Ma)) = divide6298287555418463151nteger(modulo364778990260209775nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Ma))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)) ).

% div_exp_mod_exp_eq
tff(fact_946_ex__power__ivl1,axiom,
    ! [B: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),K2)
       => ? [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)),K2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))) ) ) ) ).

% ex_power_ivl1
tff(fact_947_ex__power__ivl2,axiom,
    ! [B: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K2)
       => ? [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)),K2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))) ) ) ) ).

% ex_power_ivl2
tff(fact_948_verit__comp__simplify1_I3_J,axiom,
    ! [B4: real,A4: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B4),A4)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A4),B4) ) ).

% verit_comp_simplify1(3)
tff(fact_949_verit__comp__simplify1_I3_J,axiom,
    ! [B4: rat,A4: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B4),A4)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A4),B4) ) ).

% verit_comp_simplify1(3)
tff(fact_950_verit__comp__simplify1_I3_J,axiom,
    ! [B4: num,A4: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B4),A4)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,A4),B4) ) ).

% verit_comp_simplify1(3)
tff(fact_951_verit__comp__simplify1_I3_J,axiom,
    ! [B4: nat,A4: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B4),A4)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A4),B4) ) ).

% verit_comp_simplify1(3)
tff(fact_952_verit__comp__simplify1_I3_J,axiom,
    ! [B4: int,A4: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B4),A4)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A4),B4) ) ).

% verit_comp_simplify1(3)
tff(fact_953_verit__eq__simplify_I10_J,axiom,
    ! [X23: num] : one != aa_num_num(bit0,X23) ).

% verit_eq_simplify(10)
tff(fact_954_times__divide__times__eq,axiom,
    ! [X: complex,Y2: complex,Z2: complex,W: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(X),Y2)),aa_complex_complex(divide1717551699836669952omplex(Z2),W)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(X),Z2)),aa_complex_complex(times_times_complex(Y2),W)) ).

% times_divide_times_eq
tff(fact_955_times__divide__times__eq,axiom,
    ! [X: real,Y2: real,Z2: real,W: real] : aa_real_real(times_times_real(aa_real_real(divide_divide_real(X),Y2)),aa_real_real(divide_divide_real(Z2),W)) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(X),Z2)),aa_real_real(times_times_real(Y2),W)) ).

% times_divide_times_eq
tff(fact_956_times__divide__times__eq,axiom,
    ! [X: rat,Y2: rat,Z2: rat,W: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(divide_divide_rat(X),Y2)),aa_rat_rat(divide_divide_rat(Z2),W)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(X),Z2)),aa_rat_rat(times_times_rat(Y2),W)) ).

% times_divide_times_eq
tff(fact_957_divide__divide__times__eq,axiom,
    ! [X: complex,Y2: complex,Z2: complex,W: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(divide1717551699836669952omplex(X),Y2)),aa_complex_complex(divide1717551699836669952omplex(Z2),W)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(X),W)),aa_complex_complex(times_times_complex(Y2),Z2)) ).

% divide_divide_times_eq
tff(fact_958_divide__divide__times__eq,axiom,
    ! [X: real,Y2: real,Z2: real,W: real] : aa_real_real(divide_divide_real(aa_real_real(divide_divide_real(X),Y2)),aa_real_real(divide_divide_real(Z2),W)) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(X),W)),aa_real_real(times_times_real(Y2),Z2)) ).

% divide_divide_times_eq
tff(fact_959_divide__divide__times__eq,axiom,
    ! [X: rat,Y2: rat,Z2: rat,W: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(divide_divide_rat(X),Y2)),aa_rat_rat(divide_divide_rat(Z2),W)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(X),W)),aa_rat_rat(times_times_rat(Y2),Z2)) ).

% divide_divide_times_eq
tff(fact_960_divide__divide__eq__left_H,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(divide1717551699836669952omplex(A),B)),C) = aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(times_times_complex(C),B)) ).

% divide_divide_eq_left'
tff(fact_961_divide__divide__eq__left_H,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(divide_divide_real(aa_real_real(divide_divide_real(A),B)),C) = aa_real_real(divide_divide_real(A),aa_real_real(times_times_real(C),B)) ).

% divide_divide_eq_left'
tff(fact_962_divide__divide__eq__left_H,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(divide_divide_rat(A),B)),C) = aa_rat_rat(divide_divide_rat(A),aa_rat_rat(times_times_rat(C),B)) ).

% divide_divide_eq_left'
tff(fact_963_add__divide__distrib,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(A),B)),C) = aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(A),C)),aa_complex_complex(divide1717551699836669952omplex(B),C)) ).

% add_divide_distrib
tff(fact_964_add__divide__distrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(A),B)),C) = aa_real_real(plus_plus_real(aa_real_real(divide_divide_real(A),C)),aa_real_real(divide_divide_real(B),C)) ).

% add_divide_distrib
tff(fact_965_add__divide__distrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(A),B)),C) = aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C)) ).

% add_divide_distrib
tff(fact_966_div__by__1,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),one_one_complex) = A ).

% div_by_1
tff(fact_967_div__by__1,axiom,
    ! [A: real] : aa_real_real(divide_divide_real(A),one_one_real) = A ).

% div_by_1
tff(fact_968_div__by__1,axiom,
    ! [A: rat] : aa_rat_rat(divide_divide_rat(A),one_one_rat) = A ).

% div_by_1
tff(fact_969_div__by__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),one_one_nat) = A ).

% div_by_1
tff(fact_970_div__by__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),one_one_int) = A ).

% div_by_1
tff(fact_971_mod__eq__nat1E,axiom,
    ! [Ma: nat,Q: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),Q) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),Q) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma)
       => ~ ! [S2: nat] : Ma != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q),S2)) ) ) ).

% mod_eq_nat1E
tff(fact_972_mod__eq__nat2E,axiom,
    ! [Ma: nat,Q: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),Q) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),Q) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
       => ~ ! [S2: nat] : Nb != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q),S2)) ) ) ).

% mod_eq_nat2E
tff(fact_973_nat__mod__eq__lemma,axiom,
    ! [X: nat,Nb: nat,Y2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Y2),Nb) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y2),X)
       => ? [Q3: nat] : X = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q3)) ) ) ).

% nat_mod_eq_lemma
tff(fact_974_cancel__div__mod__rules_I2_J,axiom,
    ! [B: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) ).

% cancel_div_mod_rules(2)
tff(fact_975_cancel__div__mod__rules_I2_J,axiom,
    ! [B: int,A: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) ).

% cancel_div_mod_rules(2)
tff(fact_976_cancel__div__mod__rules_I2_J,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] : plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(A,B)),modulo364778990260209775nteger(A,B)),C) = plus_p5714425477246183910nteger(A,C) ).

% cancel_div_mod_rules(2)
tff(fact_977_cancel__div__mod__rules_I1_J,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) ).

% cancel_div_mod_rules(1)
tff(fact_978_cancel__div__mod__rules_I1_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) ).

% cancel_div_mod_rules(1)
tff(fact_979_cancel__div__mod__rules_I1_J,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),B),modulo364778990260209775nteger(A,B)),C) = plus_p5714425477246183910nteger(A,C) ).

% cancel_div_mod_rules(1)
tff(fact_980_mod__div__decomp,axiom,
    ! [A: nat,B: nat] : A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) ).

% mod_div_decomp
tff(fact_981_mod__div__decomp,axiom,
    ! [A: int,B: int] : A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ).

% mod_div_decomp
tff(fact_982_mod__div__decomp,axiom,
    ! [A: code_integer,B: code_integer] : A = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),B),modulo364778990260209775nteger(A,B)) ).

% mod_div_decomp
tff(fact_983_div__mult__mod__eq,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = A ).

% div_mult_mod_eq
tff(fact_984_div__mult__mod__eq,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) = A ).

% div_mult_mod_eq
tff(fact_985_div__mult__mod__eq,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),B),modulo364778990260209775nteger(A,B)) = A ).

% div_mult_mod_eq
tff(fact_986_mod__div__mult__eq,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)) = A ).

% mod_div_mult_eq
tff(fact_987_mod__div__mult__eq,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B)) = A ).

% mod_div_mult_eq
tff(fact_988_mod__div__mult__eq,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),B)) = A ).

% mod_div_mult_eq
tff(fact_989_mod__mult__div__eq,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B))) = A ).

% mod_mult_div_eq
tff(fact_990_mod__mult__div__eq,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))) = A ).

% mod_mult_div_eq
tff(fact_991_mod__mult__div__eq,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(A,B))) = A ).

% mod_mult_div_eq
tff(fact_992_mult__div__mod__eq,axiom,
    ! [B: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = A ).

% mult_div_mod_eq
tff(fact_993_mult__div__mod__eq,axiom,
    ! [B: int,A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) = A ).

% mult_div_mod_eq
tff(fact_994_mult__div__mod__eq,axiom,
    ! [B: code_integer,A: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(A,B)),modulo364778990260209775nteger(A,B)) = A ).

% mult_div_mod_eq
tff(fact_995_zdiv__numeral__Bit0,axiom,
    ! [V: num,W: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,V))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,W)) ).

% zdiv_numeral_Bit0
tff(fact_996_real__divide__square__eq,axiom,
    ! [R: real,A: real] : aa_real_real(divide_divide_real(aa_real_real(times_times_real(R),A)),aa_real_real(times_times_real(R),R)) = aa_real_real(divide_divide_real(A),R) ).

% real_divide_square_eq
tff(fact_997_zmod__numeral__Bit0,axiom,
    ! [V: num,W: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,V))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,W))) ).

% zmod_numeral_Bit0
tff(fact_998_less__eq__real__def,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y2)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2)
        | ( X = Y2 ) ) ) ).

% less_eq_real_def
tff(fact_999_real__arch__pow,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X)
     => ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)) ) ).

% real_arch_pow
tff(fact_1000_complete__real,axiom,
    ! [S3: set_real] :
      ( ? [X2: real] : member_real(X2,S3)
     => ( ? [Z3: real] :
          ! [X4: real] :
            ( member_real(X4,S3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),Z3) )
       => ? [Y4: real] :
            ( ! [X2: real] :
                ( member_real(X2,S3)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),Y4) )
            & ! [Z3: real] :
                ( ! [X4: real] :
                    ( member_real(X4,S3)
                   => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),Z3) )
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y4),Z3) ) ) ) ) ).

% complete_real
tff(fact_1001_infinity__ne__i1,axiom,
    extend5688581933313929465d_enat != one_on7984719198319812577d_enat ).

% infinity_ne_i1
tff(fact_1002_two__realpow__ge__one,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Nb)) ).

% two_realpow_ge_one
tff(fact_1003_one__enat__def,axiom,
    one_on7984719198319812577d_enat = aa_nat_Extended_enat(extended_enat2,one_one_nat) ).

% one_enat_def
tff(fact_1004_enat__1__iff_I1_J,axiom,
    ! [X: nat] :
      ( ( aa_nat_Extended_enat(extended_enat2,X) = one_on7984719198319812577d_enat )
    <=> ( X = one_one_nat ) ) ).

% enat_1_iff(1)
tff(fact_1005_enat__1__iff_I2_J,axiom,
    ! [X: nat] :
      ( ( one_on7984719198319812577d_enat = aa_nat_Extended_enat(extended_enat2,X) )
    <=> ( X = one_one_nat ) ) ).

% enat_1_iff(2)
tff(fact_1006_linorder__neqE__linordered__idom,axiom,
    ! [X: real,Y2: real] :
      ( ( X != Y2 )
     => ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),X) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_1007_linorder__neqE__linordered__idom,axiom,
    ! [X: rat,Y2: rat] :
      ( ( X != Y2 )
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y2),X) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_1008_linorder__neqE__linordered__idom,axiom,
    ! [X: int,Y2: int] :
      ( ( X != Y2 )
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Y2),X) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_1009_combine__common__factor,axiom,
    ! [A: real,E: real,B: real,C: real] : aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),E)),aa_real_real(plus_plus_real(aa_real_real(times_times_real(B),E)),C)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(plus_plus_real(A),B)),E)),C) ).

% combine_common_factor
tff(fact_1010_combine__common__factor,axiom,
    ! [A: rat,E: rat,B: rat,C: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E)),C)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A),B)),E)),C) ).

% combine_common_factor
tff(fact_1011_combine__common__factor,axiom,
    ! [A: nat,E: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),E)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),E)),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),E)),C) ).

% combine_common_factor
tff(fact_1012_combine__common__factor,axiom,
    ! [A: int,E: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),E)),C) ).

% combine_common_factor
tff(fact_1013_distrib__right,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(times_times_real(aa_real_real(plus_plus_real(A),B)),C) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),C)) ).

% distrib_right
tff(fact_1014_distrib__right,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A),B)),C) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) ).

% distrib_right
tff(fact_1015_distrib__right,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% distrib_right
tff(fact_1016_distrib__right,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% distrib_right
tff(fact_1017_distrib__left,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(times_times_real(A),aa_real_real(plus_plus_real(B),C)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),B)),aa_real_real(times_times_real(A),C)) ).

% distrib_left
tff(fact_1018_distrib__left,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),aa_rat_rat(plus_plus_rat(B),C)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),B)),aa_rat_rat(times_times_rat(A),C)) ).

% distrib_left
tff(fact_1019_distrib__left,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)) ).

% distrib_left
tff(fact_1020_distrib__left,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ).

% distrib_left
tff(fact_1021_comm__semiring__class_Odistrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(times_times_real(aa_real_real(plus_plus_real(A),B)),C) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),C)) ).

% comm_semiring_class.distrib
tff(fact_1022_comm__semiring__class_Odistrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A),B)),C) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) ).

% comm_semiring_class.distrib
tff(fact_1023_comm__semiring__class_Odistrib,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% comm_semiring_class.distrib
tff(fact_1024_comm__semiring__class_Odistrib,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% comm_semiring_class.distrib
tff(fact_1025_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(times_times_real(A),aa_real_real(plus_plus_real(B),C)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),B)),aa_real_real(times_times_real(A),C)) ).

% ring_class.ring_distribs(1)
tff(fact_1026_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),aa_rat_rat(plus_plus_rat(B),C)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),B)),aa_rat_rat(times_times_rat(A),C)) ).

% ring_class.ring_distribs(1)
tff(fact_1027_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ).

% ring_class.ring_distribs(1)
tff(fact_1028_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(times_times_real(aa_real_real(plus_plus_real(A),B)),C) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),C)) ).

% ring_class.ring_distribs(2)
tff(fact_1029_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A),B)),C) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) ).

% ring_class.ring_distribs(2)
tff(fact_1030_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% ring_class.ring_distribs(2)
tff(fact_1031_lambda__one,axiom,
    aTP_Lamm_ao = times_times_complex(one_one_complex) ).

% lambda_one
tff(fact_1032_lambda__one,axiom,
    aTP_Lamm_ap = times_times_real(one_one_real) ).

% lambda_one
tff(fact_1033_lambda__one,axiom,
    aTP_Lamm_aq = times_times_rat(one_one_rat) ).

% lambda_one
tff(fact_1034_lambda__one,axiom,
    aTP_Lamm_ai = aa_nat_fun_nat_nat(times_times_nat,one_one_nat) ).

% lambda_one
tff(fact_1035_lambda__one,axiom,
    aTP_Lamm_ar = aa_int_fun_int_int(times_times_int,one_one_int) ).

% lambda_one
tff(fact_1036_less__1__mult,axiom,
    ! [Ma: real,Nb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Ma)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Nb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(times_times_real(Ma),Nb)) ) ) ).

% less_1_mult
tff(fact_1037_less__1__mult,axiom,
    ! [Ma: rat,Nb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Ma)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Nb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(times_times_rat(Ma),Nb)) ) ) ).

% less_1_mult
tff(fact_1038_less__1__mult,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Ma)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Nb)) ) ) ).

% less_1_mult
tff(fact_1039_less__1__mult,axiom,
    ! [Ma: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Ma)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,Ma),Nb)) ) ) ).

% less_1_mult
tff(fact_1040_add__mono1,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(A),one_one_real)),aa_real_real(plus_plus_real(B),one_one_real)) ) ).

% add_mono1
tff(fact_1041_add__mono1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),one_one_rat)),aa_rat_rat(plus_plus_rat(B),one_one_rat)) ) ).

% add_mono1
tff(fact_1042_add__mono1,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),one_one_nat)) ) ).

% add_mono1
tff(fact_1043_add__mono1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),one_one_int)) ) ).

% add_mono1
tff(fact_1044_less__add__one,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(plus_plus_real(A),one_one_real)) ).

% less_add_one
tff(fact_1045_less__add__one,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(plus_plus_rat(A),one_one_rat)) ).

% less_add_one
tff(fact_1046_less__add__one,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)) ).

% less_add_one
tff(fact_1047_less__add__one,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)) ).

% less_add_one
tff(fact_1048_cong__exp__iff__simps_I9_J,axiom,
    ! [Ma: num,Q: num,Nb: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Ma))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Ma)),aa_num_nat(numeral_numeral_nat,Q)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,Q)) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_1049_cong__exp__iff__simps_I9_J,axiom,
    ! [Ma: num,Q: num,Nb: num] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Ma))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q))) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q))) )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Ma)),aa_num_int(numeral_numeral_int,Q)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,Q)) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_1050_cong__exp__iff__simps_I9_J,axiom,
    ! [Ma: num,Q: num,Nb: num] :
      ( ( modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,Ma)),numera6620942414471956472nteger(aa_num_num(bit0,Q))) = modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,Nb)),numera6620942414471956472nteger(aa_num_num(bit0,Q))) )
    <=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(Ma),numera6620942414471956472nteger(Q)) = modulo364778990260209775nteger(numera6620942414471956472nteger(Nb),numera6620942414471956472nteger(Q)) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_1051_cong__exp__iff__simps_I4_J,axiom,
    ! [Ma: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Ma)),aa_num_nat(numeral_numeral_nat,one)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,one)) ).

% cong_exp_iff_simps(4)
tff(fact_1052_cong__exp__iff__simps_I4_J,axiom,
    ! [Ma: num,Nb: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Ma)),aa_num_int(numeral_numeral_int,one)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,one)) ).

% cong_exp_iff_simps(4)
tff(fact_1053_cong__exp__iff__simps_I4_J,axiom,
    ! [Ma: num,Nb: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(Ma),numera6620942414471956472nteger(one)) = modulo364778990260209775nteger(numera6620942414471956472nteger(Nb),numera6620942414471956472nteger(one)) ).

% cong_exp_iff_simps(4)
tff(fact_1054_nat__mod__eq__iff,axiom,
    ! [X: nat,Nb: nat,Y2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Y2),Nb) )
    <=> ? [Q1: nat,Q22: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q1)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q22)) ) ).

% nat_mod_eq_iff
tff(fact_1055_discrete,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),B) ) ).

% discrete
tff(fact_1056_discrete,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),B) ) ).

% discrete
tff(fact_1057_cong__exp__iff__simps_I6_J,axiom,
    ! [Q: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q))) ).

% cong_exp_iff_simps(6)
tff(fact_1058_cong__exp__iff__simps_I6_J,axiom,
    ! [Q: num,Nb: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q))) != aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q))) ).

% cong_exp_iff_simps(6)
tff(fact_1059_cong__exp__iff__simps_I6_J,axiom,
    ! [Q: num,Nb: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(one),numera6620942414471956472nteger(aa_num_num(bit0,Q))) != modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,Nb)),numera6620942414471956472nteger(aa_num_num(bit0,Q))) ).

% cong_exp_iff_simps(6)
tff(fact_1060_cong__exp__iff__simps_I8_J,axiom,
    ! [Ma: num,Q: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Ma))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q))) ).

% cong_exp_iff_simps(8)
tff(fact_1061_cong__exp__iff__simps_I8_J,axiom,
    ! [Ma: num,Q: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Ma))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q))) != aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q))) ).

% cong_exp_iff_simps(8)
tff(fact_1062_cong__exp__iff__simps_I8_J,axiom,
    ! [Ma: num,Q: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,Ma)),numera6620942414471956472nteger(aa_num_num(bit0,Q))) != modulo364778990260209775nteger(numera6620942414471956472nteger(one),numera6620942414471956472nteger(aa_num_num(bit0,Q))) ).

% cong_exp_iff_simps(8)
tff(fact_1063_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeListb: list_VEBT_VEBT,Nb: nat,Summaryb: vEBT_VEBT,Ma: nat,Degb: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeListb))
         => aa_nat_o(vEBT_invar_vebt(X4),Nb) )
     => ( aa_nat_o(vEBT_invar_vebt(Summaryb),Ma)
       => ( ( size_s6755466524823107622T_VEBT(TreeListb) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma) )
         => ( ( Ma = aa_nat_nat(suc,Nb) )
           => ( ( Degb = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Ma) )
             => ( ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summaryb),X_13)
               => ( ! [X4: vEBT_VEBT] :
                      ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeListb))
                     => ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X_13) )
                 => aa_nat_o(vEBT_invar_vebt(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Degb,TreeListb),Summaryb)),Degb) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
tff(fact_1064_dbl__simps_I3_J,axiom,
    aa_complex_complex(neg_nu7009210354673126013omplex,one_one_complex) = numera6690914467698888265omplex(aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_1065_dbl__simps_I3_J,axiom,
    aa_real_real(neg_numeral_dbl_real,one_one_real) = aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_1066_dbl__simps_I3_J,axiom,
    aa_rat_rat(neg_numeral_dbl_rat,one_one_rat) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_1067_dbl__simps_I3_J,axiom,
    aa_int_int(neg_numeral_dbl_int,one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_1068_divmod__digit__1_I1_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B)))
         => ( plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B))),one_one_Code_integer) = divide6298287555418463151nteger(A,B) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_1069_divmod__digit__1_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)))),one_one_nat) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_1070_divmod__digit__1_I1_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)))
         => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)))),one_one_int) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_1071_arith__geo__mean,axiom,
    ! [U: real,X: real,Y2: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,U),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(times_times_real(X),Y2) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,U),aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(X),Y2)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).

% arith_geo_mean
tff(fact_1072_arith__geo__mean,axiom,
    ! [U: rat,X: rat,Y2: rat] :
      ( ( aa_nat_rat(power_power_rat(U),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(times_times_rat(X),Y2) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,U),aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(X),Y2)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ) ) ) ).

% arith_geo_mean
tff(fact_1073_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [X: nat,Degb: nat,TreeListb: list_VEBT_VEBT,Mi: nat,Ma: nat,Summaryb: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Degb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeListb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Degb)
       => ( aa_nat_o(vEBT_V8194947554948674370ptions(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeListb),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Degb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Degb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
         => aa_nat_o(vEBT_V8194947554948674370ptions(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Degb,TreeListb),Summaryb)),X) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
tff(fact_1074_subset__antisym,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ord_less_eq_set_int(A2,B3)
     => ( ord_less_eq_set_int(B3,A2)
       => ( A2 = B3 ) ) ) ).

% subset_antisym
tff(fact_1075_subsetI,axiom,
    ! [A2: set_complex,B3: set_complex] :
      ( ! [X4: complex] :
          ( member_complex(X4,A2)
         => member_complex(X4,B3) )
     => ord_le211207098394363844omplex(A2,B3) ) ).

% subsetI
tff(fact_1076_subsetI,axiom,
    ! [A2: set_real,B3: set_real] :
      ( ! [X4: real] :
          ( member_real(X4,A2)
         => member_real(X4,B3) )
     => ord_less_eq_set_real(A2,B3) ) ).

% subsetI
tff(fact_1077_subsetI,axiom,
    ! [A2: set_set_nat,B3: set_set_nat] :
      ( ! [X4: set_nat] :
          ( member_set_nat(X4,A2)
         => member_set_nat(X4,B3) )
     => ord_le6893508408891458716et_nat(A2,B3) ) ).

% subsetI
tff(fact_1078_subsetI,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),A2)
         => aa_set_nat_o(member_nat(X4),B3) )
     => aa_set_nat_o(ord_less_eq_set_nat(A2),B3) ) ).

% subsetI
tff(fact_1079_subsetI,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ! [X4: int] :
          ( member_int(X4,A2)
         => member_int(X4,B3) )
     => ord_less_eq_set_int(A2,B3) ) ).

% subsetI
tff(fact_1080_psubsetI,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ord_less_eq_set_int(A2,B3)
     => ( ( A2 != B3 )
       => ord_less_set_int(A2,B3) ) ) ).

% psubsetI
tff(fact_1081_mod__double__modulus,axiom,
    ! [Ma: code_integer,X: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Ma)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X)
       => ( ( modulo364778990260209775nteger(X,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Ma)) = modulo364778990260209775nteger(X,Ma) )
          | ( modulo364778990260209775nteger(X,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Ma)) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(X,Ma),Ma) ) ) ) ) ).

% mod_double_modulus
tff(fact_1082_mod__double__modulus,axiom,
    ! [Ma: nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ma)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),X)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Ma) )
          | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Ma)),Ma) ) ) ) ) ).

% mod_double_modulus
tff(fact_1083_mod__double__modulus,axiom,
    ! [Ma: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ma)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
       => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Ma)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X),Ma) )
          | ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Ma)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X),Ma)),Ma) ) ) ) ) ).

% mod_double_modulus
tff(fact_1084_even__odd__cases,axiom,
    ! [X: nat] :
      ( ! [N: nat] : X != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),N)
     => ~ ! [N: nat] : X != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(suc,N)) ) ).

% even_odd_cases
tff(fact_1085_deg__not__0,axiom,
    ! [T: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(T),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% deg_not_0
tff(fact_1086_old_Onat_Oinject,axiom,
    ! [Nat: nat,Nat3: nat] :
      ( ( aa_nat_nat(suc,Nat) = aa_nat_nat(suc,Nat3) )
    <=> ( Nat = Nat3 ) ) ).

% old.nat.inject
tff(fact_1087_nat_Oinject,axiom,
    ! [X23: nat,Y23: nat] :
      ( ( aa_nat_nat(suc,X23) = aa_nat_nat(suc,Y23) )
    <=> ( X23 = Y23 ) ) ).

% nat.inject
tff(fact_1088_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A) ) ).

% bot_nat_0.not_eq_extremum
tff(fact_1089_neq0__conv,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% neq0_conv
tff(fact_1090_less__nat__zero__code,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),zero_zero_nat) ).

% less_nat_zero_code
tff(fact_1091_le0,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Nb) ).

% le0
tff(fact_1092_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A) ).

% bot_nat_0.extremum
tff(fact_1093_add__is__0,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb) = zero_zero_nat )
    <=> ( ( Ma = zero_zero_nat )
        & ( Nb = zero_zero_nat ) ) ) ).

% add_is_0
tff(fact_1094_Nat_Oadd__0__right,axiom,
    ! [Ma: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),zero_zero_nat) = Ma ).

% Nat.add_0_right
tff(fact_1095_mult__cancel2,axiom,
    ! [Ma: nat,K2: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),K2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K2) )
    <=> ( ( Ma = Nb )
        | ( K2 = zero_zero_nat ) ) ) ).

% mult_cancel2
tff(fact_1096_mult__cancel1,axiom,
    ! [K2: nat,Ma: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Ma) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Nb) )
    <=> ( ( Ma = Nb )
        | ( K2 = zero_zero_nat ) ) ) ).

% mult_cancel1
tff(fact_1097_mult__0__right,axiom,
    ! [Ma: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),zero_zero_nat) = zero_zero_nat ).

% mult_0_right
tff(fact_1098_mult__is__0,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Nb) = zero_zero_nat )
    <=> ( ( Ma = zero_zero_nat )
        | ( Nb = zero_zero_nat ) ) ) ).

% mult_is_0
tff(fact_1099_div__pos__pos__trivial,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),L)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),L) = zero_zero_int ) ) ) ).

% div_pos_pos_trivial
tff(fact_1100_div__neg__neg__trivial,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),K2)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),L) = zero_zero_int ) ) ) ).

% div_neg_neg_trivial
tff(fact_1101_mod__pos__pos__trivial,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),L)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),L) = K2 ) ) ) ).

% mod_pos_pos_trivial
tff(fact_1102_mod__neg__neg__trivial,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),K2)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),L) = K2 ) ) ) ).

% mod_neg_neg_trivial
tff(fact_1103_i0__less,axiom,
    ! [Nb: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),Nb)
    <=> ( Nb != zero_z5237406670263579293d_enat ) ) ).

% i0_less
tff(fact_1104_le__zero__eq,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),zero_zero_nat)
    <=> ( Nb = zero_zero_nat ) ) ).

% le_zero_eq
tff(fact_1105_not__gr__zero,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
    <=> ( Nb = zero_zero_nat ) ) ).

% not_gr_zero
tff(fact_1106_mult__zero__left,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(zero_zero_complex),A) = zero_zero_complex ).

% mult_zero_left
tff(fact_1107_mult__zero__left,axiom,
    ! [A: real] : aa_real_real(times_times_real(zero_zero_real),A) = zero_zero_real ).

% mult_zero_left
tff(fact_1108_mult__zero__left,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(zero_zero_rat),A) = zero_zero_rat ).

% mult_zero_left
tff(fact_1109_mult__zero__left,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat),A) = zero_zero_nat ).

% mult_zero_left
tff(fact_1110_mult__zero__left,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,zero_zero_int),A) = zero_zero_int ).

% mult_zero_left
tff(fact_1111_mult__zero__right,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),zero_zero_complex) = zero_zero_complex ).

% mult_zero_right
tff(fact_1112_mult__zero__right,axiom,
    ! [A: real] : aa_real_real(times_times_real(A),zero_zero_real) = zero_zero_real ).

% mult_zero_right
tff(fact_1113_mult__zero__right,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(A),zero_zero_rat) = zero_zero_rat ).

% mult_zero_right
tff(fact_1114_mult__zero__right,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),zero_zero_nat) = zero_zero_nat ).

% mult_zero_right
tff(fact_1115_mult__zero__right,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),zero_zero_int) = zero_zero_int ).

% mult_zero_right
tff(fact_1116_mult__eq__0__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(A),B) = zero_zero_complex )
    <=> ( ( A = zero_zero_complex )
        | ( B = zero_zero_complex ) ) ) ).

% mult_eq_0_iff
tff(fact_1117_mult__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(times_times_real(A),B) = zero_zero_real )
    <=> ( ( A = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% mult_eq_0_iff
tff(fact_1118_mult__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(A),B) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% mult_eq_0_iff
tff(fact_1119_mult__eq__0__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        | ( B = zero_zero_nat ) ) ) ).

% mult_eq_0_iff
tff(fact_1120_mult__eq__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) = zero_zero_int )
    <=> ( ( A = zero_zero_int )
        | ( B = zero_zero_int ) ) ) ).

% mult_eq_0_iff
tff(fact_1121_mult__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(C),A) = aa_complex_complex(times_times_complex(C),B) )
    <=> ( ( C = zero_zero_complex )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_1122_mult__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( aa_real_real(times_times_real(C),A) = aa_real_real(times_times_real(C),B) )
    <=> ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_1123_mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(C),A) = aa_rat_rat(times_times_rat(C),B) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_1124_mult__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B) )
    <=> ( ( C = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_1125_mult__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C),A) = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) )
    <=> ( ( C = zero_zero_int )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_1126_mult__cancel__right,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(A),C) = aa_complex_complex(times_times_complex(B),C) )
    <=> ( ( C = zero_zero_complex )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_1127_mult__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( aa_real_real(times_times_real(A),C) = aa_real_real(times_times_real(B),C) )
    <=> ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_1128_mult__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(A),C) = aa_rat_rat(times_times_rat(B),C) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_1129_mult__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C) )
    <=> ( ( C = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_1130_mult__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),C) = aa_int_int(aa_int_fun_int_int(times_times_int,B),C) )
    <=> ( ( C = zero_zero_int )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_1131_add_Oright__neutral,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(A),zero_zero_complex) = A ).

% add.right_neutral
tff(fact_1132_add_Oright__neutral,axiom,
    ! [A: real] : aa_real_real(plus_plus_real(A),zero_zero_real) = A ).

% add.right_neutral
tff(fact_1133_add_Oright__neutral,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(A),zero_zero_rat) = A ).

% add.right_neutral
tff(fact_1134_add_Oright__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),zero_zero_nat) = A ).

% add.right_neutral
tff(fact_1135_add_Oright__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),zero_zero_int) = A ).

% add.right_neutral
tff(fact_1136_double__zero__sym,axiom,
    ! [A: real] :
      ( ( zero_zero_real = aa_real_real(plus_plus_real(A),A) )
    <=> ( A = zero_zero_real ) ) ).

% double_zero_sym
tff(fact_1137_double__zero__sym,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = aa_rat_rat(plus_plus_rat(A),A) )
    <=> ( A = zero_zero_rat ) ) ).

% double_zero_sym
tff(fact_1138_double__zero__sym,axiom,
    ! [A: int] :
      ( ( zero_zero_int = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A) )
    <=> ( A = zero_zero_int ) ) ).

% double_zero_sym
tff(fact_1139_add__cancel__left__left,axiom,
    ! [B: complex,A: complex] :
      ( ( aa_complex_complex(plus_plus_complex(B),A) = A )
    <=> ( B = zero_zero_complex ) ) ).

% add_cancel_left_left
tff(fact_1140_add__cancel__left__left,axiom,
    ! [B: real,A: real] :
      ( ( aa_real_real(plus_plus_real(B),A) = A )
    <=> ( B = zero_zero_real ) ) ).

% add_cancel_left_left
tff(fact_1141_add__cancel__left__left,axiom,
    ! [B: rat,A: rat] :
      ( ( aa_rat_rat(plus_plus_rat(B),A) = A )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_left_left
tff(fact_1142_add__cancel__left__left,axiom,
    ! [B: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) = A )
    <=> ( B = zero_zero_nat ) ) ).

% add_cancel_left_left
tff(fact_1143_add__cancel__left__left,axiom,
    ! [B: int,A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) = A )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_left_left
tff(fact_1144_add__cancel__left__right,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(plus_plus_complex(A),B) = A )
    <=> ( B = zero_zero_complex ) ) ).

% add_cancel_left_right
tff(fact_1145_add__cancel__left__right,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(plus_plus_real(A),B) = A )
    <=> ( B = zero_zero_real ) ) ).

% add_cancel_left_right
tff(fact_1146_add__cancel__left__right,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),B) = A )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_left_right
tff(fact_1147_add__cancel__left__right,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = A )
    <=> ( B = zero_zero_nat ) ) ).

% add_cancel_left_right
tff(fact_1148_add__cancel__left__right,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = A )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_left_right
tff(fact_1149_add__cancel__right__left,axiom,
    ! [A: complex,B: complex] :
      ( ( A = aa_complex_complex(plus_plus_complex(B),A) )
    <=> ( B = zero_zero_complex ) ) ).

% add_cancel_right_left
tff(fact_1150_add__cancel__right__left,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(plus_plus_real(B),A) )
    <=> ( B = zero_zero_real ) ) ).

% add_cancel_right_left
tff(fact_1151_add__cancel__right__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(plus_plus_rat(B),A) )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_right_left
tff(fact_1152_add__cancel__right__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) )
    <=> ( B = zero_zero_nat ) ) ).

% add_cancel_right_left
tff(fact_1153_add__cancel__right__left,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_right_left
tff(fact_1154_add__cancel__right__right,axiom,
    ! [A: complex,B: complex] :
      ( ( A = aa_complex_complex(plus_plus_complex(A),B) )
    <=> ( B = zero_zero_complex ) ) ).

% add_cancel_right_right
tff(fact_1155_add__cancel__right__right,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(plus_plus_real(A),B) )
    <=> ( B = zero_zero_real ) ) ).

% add_cancel_right_right
tff(fact_1156_add__cancel__right__right,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(plus_plus_rat(A),B) )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_right_right
tff(fact_1157_add__cancel__right__right,axiom,
    ! [A: nat,B: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) )
    <=> ( B = zero_zero_nat ) ) ).

% add_cancel_right_right
tff(fact_1158_add__cancel__right__right,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_right_right
tff(fact_1159_add__eq__0__iff__both__eq__0,axiom,
    ! [X: nat,Y2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y2) = zero_zero_nat )
    <=> ( ( X = zero_zero_nat )
        & ( Y2 = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
tff(fact_1160_zero__eq__add__iff__both__eq__0,axiom,
    ! [X: nat,Y2: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y2) )
    <=> ( ( X = zero_zero_nat )
        & ( Y2 = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
tff(fact_1161_add__0,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(zero_zero_complex),A) = A ).

% add_0
tff(fact_1162_add__0,axiom,
    ! [A: real] : aa_real_real(plus_plus_real(zero_zero_real),A) = A ).

% add_0
tff(fact_1163_add__0,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(zero_zero_rat),A) = A ).

% add_0
tff(fact_1164_add__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),A) = A ).

% add_0
tff(fact_1165_add__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),A) = A ).

% add_0
tff(fact_1166_divide__eq__0__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A),B) = zero_zero_complex )
    <=> ( ( A = zero_zero_complex )
        | ( B = zero_zero_complex ) ) ) ).

% divide_eq_0_iff
tff(fact_1167_divide__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(divide_divide_real(A),B) = zero_zero_real )
    <=> ( ( A = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% divide_eq_0_iff
tff(fact_1168_divide__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(divide_divide_rat(A),B) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% divide_eq_0_iff
tff(fact_1169_divide__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(C),A) = aa_complex_complex(divide1717551699836669952omplex(C),B) )
    <=> ( ( C = zero_zero_complex )
        | ( A = B ) ) ) ).

% divide_cancel_left
tff(fact_1170_divide__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( aa_real_real(divide_divide_real(C),A) = aa_real_real(divide_divide_real(C),B) )
    <=> ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% divide_cancel_left
tff(fact_1171_divide__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( aa_rat_rat(divide_divide_rat(C),A) = aa_rat_rat(divide_divide_rat(C),B) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% divide_cancel_left
tff(fact_1172_divide__cancel__right,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A),C) = aa_complex_complex(divide1717551699836669952omplex(B),C) )
    <=> ( ( C = zero_zero_complex )
        | ( A = B ) ) ) ).

% divide_cancel_right
tff(fact_1173_divide__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( aa_real_real(divide_divide_real(A),C) = aa_real_real(divide_divide_real(B),C) )
    <=> ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% divide_cancel_right
tff(fact_1174_divide__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( aa_rat_rat(divide_divide_rat(A),C) = aa_rat_rat(divide_divide_rat(B),C) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% divide_cancel_right
tff(fact_1175_division__ring__divide__zero,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),zero_zero_complex) = zero_zero_complex ).

% division_ring_divide_zero
tff(fact_1176_division__ring__divide__zero,axiom,
    ! [A: real] : aa_real_real(divide_divide_real(A),zero_zero_real) = zero_zero_real ).

% division_ring_divide_zero
tff(fact_1177_division__ring__divide__zero,axiom,
    ! [A: rat] : aa_rat_rat(divide_divide_rat(A),zero_zero_rat) = zero_zero_rat ).

% division_ring_divide_zero
tff(fact_1178_bits__div__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,zero_zero_nat),A) = zero_zero_nat ).

% bits_div_0
tff(fact_1179_bits__div__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,zero_zero_int),A) = zero_zero_int ).

% bits_div_0
tff(fact_1180_bits__div__by__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),zero_zero_nat) = zero_zero_nat ).

% bits_div_by_0
tff(fact_1181_bits__div__by__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),zero_zero_int) = zero_zero_int ).

% bits_div_by_0
tff(fact_1182_div__0,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(zero_zero_complex),A) = zero_zero_complex ).

% div_0
tff(fact_1183_div__0,axiom,
    ! [A: real] : aa_real_real(divide_divide_real(zero_zero_real),A) = zero_zero_real ).

% div_0
tff(fact_1184_div__0,axiom,
    ! [A: rat] : aa_rat_rat(divide_divide_rat(zero_zero_rat),A) = zero_zero_rat ).

% div_0
tff(fact_1185_div__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,zero_zero_nat),A) = zero_zero_nat ).

% div_0
tff(fact_1186_div__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,zero_zero_int),A) = zero_zero_int ).

% div_0
tff(fact_1187_div__by__0,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),zero_zero_complex) = zero_zero_complex ).

% div_by_0
tff(fact_1188_div__by__0,axiom,
    ! [A: real] : aa_real_real(divide_divide_real(A),zero_zero_real) = zero_zero_real ).

% div_by_0
tff(fact_1189_div__by__0,axiom,
    ! [A: rat] : aa_rat_rat(divide_divide_rat(A),zero_zero_rat) = zero_zero_rat ).

% div_by_0
tff(fact_1190_div__by__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),zero_zero_nat) = zero_zero_nat ).

% div_by_0
tff(fact_1191_div__by__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),zero_zero_int) = zero_zero_int ).

% div_by_0
tff(fact_1192_bits__mod__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,zero_zero_nat),A) = zero_zero_nat ).

% bits_mod_0
tff(fact_1193_bits__mod__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,zero_zero_int),A) = zero_zero_int ).

% bits_mod_0
tff(fact_1194_bits__mod__0,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(zero_z3403309356797280102nteger,A) = zero_z3403309356797280102nteger ).

% bits_mod_0
tff(fact_1195_mod__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,zero_zero_nat),A) = zero_zero_nat ).

% mod_0
tff(fact_1196_mod__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,zero_zero_int),A) = zero_zero_int ).

% mod_0
tff(fact_1197_mod__0,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(zero_z3403309356797280102nteger,A) = zero_z3403309356797280102nteger ).

% mod_0
tff(fact_1198_mod__by__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),zero_zero_nat) = A ).

% mod_by_0
tff(fact_1199_mod__by__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),zero_zero_int) = A ).

% mod_by_0
tff(fact_1200_mod__by__0,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,zero_z3403309356797280102nteger) = A ).

% mod_by_0
tff(fact_1201_mod__self,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),A) = zero_zero_nat ).

% mod_self
tff(fact_1202_mod__self,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),A) = zero_zero_int ).

% mod_self
tff(fact_1203_mod__self,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,A) = zero_z3403309356797280102nteger ).

% mod_self
tff(fact_1204_power__Suc0__right,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,zero_zero_nat)) = A ).

% power_Suc0_right
tff(fact_1205_power__Suc0__right,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,zero_zero_nat)) = A ).

% power_Suc0_right
tff(fact_1206_power__Suc0__right,axiom,
    ! [A: int] : aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(suc,zero_zero_nat)) = A ).

% power_Suc0_right
tff(fact_1207_power__Suc0__right,axiom,
    ! [A: complex] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),aa_nat_nat(suc,zero_zero_nat)) = A ).

% power_Suc0_right
tff(fact_1208_zero__less__Suc,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,Nb)) ).

% zero_less_Suc
tff(fact_1209_less__Suc0,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,zero_zero_nat))
    <=> ( Nb = zero_zero_nat ) ) ).

% less_Suc0
tff(fact_1210_lessI,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Nb)) ).

% lessI
tff(fact_1211_Suc__mono,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Ma)),aa_nat_nat(suc,Nb)) ) ).

% Suc_mono
tff(fact_1212_Suc__less__eq,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Ma)),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb) ) ).

% Suc_less_eq
tff(fact_1213_Suc__le__mono,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),aa_nat_nat(suc,Ma))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma) ) ).

% Suc_le_mono
tff(fact_1214_add__Suc__right,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),aa_nat_nat(suc,Nb)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb)) ).

% add_Suc_right
tff(fact_1215_add__gr__0,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ma)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% add_gr_0
tff(fact_1216_one__eq__mult__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Nb) )
    <=> ( ( Ma = aa_nat_nat(suc,zero_zero_nat) )
        & ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% one_eq_mult_iff
tff(fact_1217_mult__eq__1__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Nb) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( Ma = aa_nat_nat(suc,zero_zero_nat) )
        & ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% mult_eq_1_iff
tff(fact_1218_nat__mult__less__cancel__disj,axiom,
    ! [K2: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb) ) ) ).

% nat_mult_less_cancel_disj
tff(fact_1219_nat__0__less__mult__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ma)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% nat_0_less_mult_iff
tff(fact_1220_mult__less__cancel2,axiom,
    ! [Ma: nat,K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),K2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb) ) ) ).

% mult_less_cancel2
tff(fact_1221_not__real__square__gt__zero,axiom,
    ! [X: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(times_times_real(X),X))
    <=> ( X = zero_zero_real ) ) ).

% not_real_square_gt_zero
tff(fact_1222_div__by__Suc__0,axiom,
    ! [Ma: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),aa_nat_nat(suc,zero_zero_nat)) = Ma ).

% div_by_Suc_0
tff(fact_1223_less__one,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),one_one_nat)
    <=> ( Nb = zero_zero_nat ) ) ).

% less_one
tff(fact_1224_nat__power__eq__Suc__0__iff,axiom,
    ! [X: nat,Ma: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),Ma) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( Ma = zero_zero_nat )
        | ( X = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% nat_power_eq_Suc_0_iff
tff(fact_1225_power__Suc__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = aa_nat_nat(suc,zero_zero_nat) ).

% power_Suc_0
tff(fact_1226_div__less,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),Nb) = zero_zero_nat ) ) ).

% div_less
tff(fact_1227_nat__zero__less__power__iff,axiom,
    ! [X: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),X)
        | ( Nb = zero_zero_nat ) ) ) ).

% nat_zero_less_power_iff
tff(fact_1228_mod__by__Suc__0,axiom,
    ! [Ma: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),aa_nat_nat(suc,zero_zero_nat)) = zero_zero_nat ).

% mod_by_Suc_0
tff(fact_1229_nat__mult__div__cancel__disj,axiom,
    ! [K2: nat,Ma: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Nb)) = $ite(K2 = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),Nb)) ).

% nat_mult_div_cancel_disj
tff(fact_1230_dbl__simps_I2_J,axiom,
    aa_complex_complex(neg_nu7009210354673126013omplex,zero_zero_complex) = zero_zero_complex ).

% dbl_simps(2)
tff(fact_1231_dbl__simps_I2_J,axiom,
    aa_real_real(neg_numeral_dbl_real,zero_zero_real) = zero_zero_real ).

% dbl_simps(2)
tff(fact_1232_dbl__simps_I2_J,axiom,
    aa_rat_rat(neg_numeral_dbl_rat,zero_zero_rat) = zero_zero_rat ).

% dbl_simps(2)
tff(fact_1233_dbl__simps_I2_J,axiom,
    aa_int_int(neg_numeral_dbl_int,zero_zero_int) = zero_zero_int ).

% dbl_simps(2)
tff(fact_1234_mi__ma__2__deg,axiom,
    ! [Mi: nat,Ma: nat,Degb: nat,TreeListb: list_VEBT_VEBT,Summaryb: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Degb,TreeListb),Summaryb)),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi),Ma)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Degb)) ) ) ).

% mi_ma_2_deg
tff(fact_1235_add__le__same__cancel1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(B),A)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% add_le_same_cancel1
tff(fact_1236_add__le__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(B),A)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% add_le_same_cancel1
tff(fact_1237_add__le__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat) ) ).

% add_le_same_cancel1
tff(fact_1238_add__le__same__cancel1,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% add_le_same_cancel1
tff(fact_1239_add__le__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(A),B)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% add_le_same_cancel2
tff(fact_1240_add__le__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),B)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% add_le_same_cancel2
tff(fact_1241_add__le__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat) ) ).

% add_le_same_cancel2
tff(fact_1242_add__le__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% add_le_same_cancel2
tff(fact_1243_le__add__same__cancel1,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(plus_plus_real(A),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) ).

% le_add_same_cancel1
tff(fact_1244_le__add__same__cancel1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(plus_plus_rat(A),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) ).

% le_add_same_cancel1
tff(fact_1245_le__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B) ) ).

% le_add_same_cancel1
tff(fact_1246_le__add__same__cancel1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) ) ).

% le_add_same_cancel1
tff(fact_1247_le__add__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(plus_plus_real(B),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) ).

% le_add_same_cancel2
tff(fact_1248_le__add__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(plus_plus_rat(B),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) ).

% le_add_same_cancel2
tff(fact_1249_le__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B) ) ).

% le_add_same_cancel2
tff(fact_1250_le__add__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) ) ).

% le_add_same_cancel2
tff(fact_1251_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(A),A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_1252_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_1253_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_1254_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(plus_plus_real(A),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_1255_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_1256_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_1257_add__less__same__cancel1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(B),A)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% add_less_same_cancel1
tff(fact_1258_add__less__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(B),A)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% add_less_same_cancel1
tff(fact_1259_add__less__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat) ) ).

% add_less_same_cancel1
tff(fact_1260_add__less__same__cancel1,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% add_less_same_cancel1
tff(fact_1261_add__less__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(A),B)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% add_less_same_cancel2
tff(fact_1262_add__less__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),B)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% add_less_same_cancel2
tff(fact_1263_add__less__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat) ) ).

% add_less_same_cancel2
tff(fact_1264_add__less__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% add_less_same_cancel2
tff(fact_1265_less__add__same__cancel1,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(plus_plus_real(A),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ).

% less_add_same_cancel1
tff(fact_1266_less__add__same__cancel1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(plus_plus_rat(A),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ).

% less_add_same_cancel1
tff(fact_1267_less__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ).

% less_add_same_cancel1
tff(fact_1268_less__add__same__cancel1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ).

% less_add_same_cancel1
tff(fact_1269_less__add__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(plus_plus_real(B),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ).

% less_add_same_cancel2
tff(fact_1270_less__add__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(plus_plus_rat(B),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ).

% less_add_same_cancel2
tff(fact_1271_less__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ).

% less_add_same_cancel2
tff(fact_1272_less__add__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ).

% less_add_same_cancel2
tff(fact_1273_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(A),A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_1274_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_1275_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_1276_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(plus_plus_real(A),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_1277_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_1278_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_1279_mult__cancel__right2,axiom,
    ! [A: complex,C: complex] :
      ( ( aa_complex_complex(times_times_complex(A),C) = C )
    <=> ( ( C = zero_zero_complex )
        | ( A = one_one_complex ) ) ) ).

% mult_cancel_right2
tff(fact_1280_mult__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( ( aa_real_real(times_times_real(A),C) = C )
    <=> ( ( C = zero_zero_real )
        | ( A = one_one_real ) ) ) ).

% mult_cancel_right2
tff(fact_1281_mult__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( ( aa_rat_rat(times_times_rat(A),C) = C )
    <=> ( ( C = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_right2
tff(fact_1282_mult__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),C) = C )
    <=> ( ( C = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_right2
tff(fact_1283_mult__cancel__right1,axiom,
    ! [C: complex,B: complex] :
      ( ( C = aa_complex_complex(times_times_complex(B),C) )
    <=> ( ( C = zero_zero_complex )
        | ( B = one_one_complex ) ) ) ).

% mult_cancel_right1
tff(fact_1284_mult__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( ( C = aa_real_real(times_times_real(B),C) )
    <=> ( ( C = zero_zero_real )
        | ( B = one_one_real ) ) ) ).

% mult_cancel_right1
tff(fact_1285_mult__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( ( C = aa_rat_rat(times_times_rat(B),C) )
    <=> ( ( C = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_right1
tff(fact_1286_mult__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( ( C = aa_int_int(aa_int_fun_int_int(times_times_int,B),C) )
    <=> ( ( C = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_right1
tff(fact_1287_mult__cancel__left2,axiom,
    ! [C: complex,A: complex] :
      ( ( aa_complex_complex(times_times_complex(C),A) = C )
    <=> ( ( C = zero_zero_complex )
        | ( A = one_one_complex ) ) ) ).

% mult_cancel_left2
tff(fact_1288_mult__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( ( aa_real_real(times_times_real(C),A) = C )
    <=> ( ( C = zero_zero_real )
        | ( A = one_one_real ) ) ) ).

% mult_cancel_left2
tff(fact_1289_mult__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( ( aa_rat_rat(times_times_rat(C),A) = C )
    <=> ( ( C = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_left2
tff(fact_1290_mult__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C),A) = C )
    <=> ( ( C = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_left2
tff(fact_1291_mult__cancel__left1,axiom,
    ! [C: complex,B: complex] :
      ( ( C = aa_complex_complex(times_times_complex(C),B) )
    <=> ( ( C = zero_zero_complex )
        | ( B = one_one_complex ) ) ) ).

% mult_cancel_left1
tff(fact_1292_mult__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( ( C = aa_real_real(times_times_real(C),B) )
    <=> ( ( C = zero_zero_real )
        | ( B = one_one_real ) ) ) ).

% mult_cancel_left1
tff(fact_1293_mult__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( ( C = aa_rat_rat(times_times_rat(C),B) )
    <=> ( ( C = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_left1
tff(fact_1294_mult__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( ( C = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) )
    <=> ( ( C = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_left1
tff(fact_1295_sum__squares__eq__zero__iff,axiom,
    ! [X: real,Y2: real] :
      ( ( aa_real_real(plus_plus_real(aa_real_real(times_times_real(X),X)),aa_real_real(times_times_real(Y2),Y2)) = zero_zero_real )
    <=> ( ( X = zero_zero_real )
        & ( Y2 = zero_zero_real ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_1296_sum__squares__eq__zero__iff,axiom,
    ! [X: rat,Y2: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(X),X)),aa_rat_rat(times_times_rat(Y2),Y2)) = zero_zero_rat )
    <=> ( ( X = zero_zero_rat )
        & ( Y2 = zero_zero_rat ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_1297_sum__squares__eq__zero__iff,axiom,
    ! [X: int,Y2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y2),Y2)) = zero_zero_int )
    <=> ( ( X = zero_zero_int )
        & ( Y2 = zero_zero_int ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_1298_mult__divide__mult__cancel__left__if,axiom,
    ! [C: complex,A: complex,B: complex] :
      aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C),A)),aa_complex_complex(times_times_complex(C),B)) = $ite(C = zero_zero_complex,zero_zero_complex,aa_complex_complex(divide1717551699836669952omplex(A),B)) ).

% mult_divide_mult_cancel_left_if
tff(fact_1299_mult__divide__mult__cancel__left__if,axiom,
    ! [C: real,A: real,B: real] :
      aa_real_real(divide_divide_real(aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B)) = $ite(C = zero_zero_real,zero_zero_real,aa_real_real(divide_divide_real(A),B)) ).

% mult_divide_mult_cancel_left_if
tff(fact_1300_mult__divide__mult__cancel__left__if,axiom,
    ! [C: rat,A: rat,B: rat] :
      aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)) = $ite(C = zero_zero_rat,zero_zero_rat,aa_rat_rat(divide_divide_rat(A),B)) ).

% mult_divide_mult_cancel_left_if
tff(fact_1301_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C),A)),aa_complex_complex(times_times_complex(C),B)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_1302_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(divide_divide_real(aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B)) = aa_real_real(divide_divide_real(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_1303_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_1304_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C),A)),aa_complex_complex(times_times_complex(B),C)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_1305_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(divide_divide_real(aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(B),C)) = aa_real_real(divide_divide_real(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_1306_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(B),C)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_1307_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),C)),aa_complex_complex(times_times_complex(B),C)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_1308_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(divide_divide_real(aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),C)) = aa_real_real(divide_divide_real(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_1309_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_1310_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),C)),aa_complex_complex(times_times_complex(C),B)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_1311_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(divide_divide_real(aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(C),B)) = aa_real_real(divide_divide_real(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_1312_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(C),B)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_1313_div__mult__mult1__if,axiom,
    ! [C: nat,A: nat,B: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) = $ite(C = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ).

% div_mult_mult1_if
tff(fact_1314_div__mult__mult1__if,axiom,
    ! [C: int,A: int,B: int] :
      aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) = $ite(C = zero_zero_int,zero_zero_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ).

% div_mult_mult1_if
tff(fact_1315_div__mult__mult2,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ).

% div_mult_mult2
tff(fact_1316_div__mult__mult2,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ).

% div_mult_mult2
tff(fact_1317_div__mult__mult1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ).

% div_mult_mult1
tff(fact_1318_div__mult__mult1,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ).

% div_mult_mult1
tff(fact_1319_nonzero__mult__div__cancel__left,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_1320_nonzero__mult__div__cancel__left,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(divide_divide_real(aa_real_real(times_times_real(A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_1321_nonzero__mult__div__cancel__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_1322_nonzero__mult__div__cancel__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_1323_nonzero__mult__div__cancel__left,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_1324_nonzero__mult__div__cancel__right,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_1325_nonzero__mult__div__cancel__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(divide_divide_real(aa_real_real(times_times_real(A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_1326_nonzero__mult__div__cancel__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_1327_nonzero__mult__div__cancel__right,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_1328_nonzero__mult__div__cancel__right,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_1329_divide__eq__1__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A),B) = one_one_complex )
    <=> ( ( B != zero_zero_complex )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
tff(fact_1330_divide__eq__1__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(divide_divide_real(A),B) = one_one_real )
    <=> ( ( B != zero_zero_real )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
tff(fact_1331_divide__eq__1__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(divide_divide_rat(A),B) = one_one_rat )
    <=> ( ( B != zero_zero_rat )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
tff(fact_1332_one__eq__divide__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( one_one_complex = aa_complex_complex(divide1717551699836669952omplex(A),B) )
    <=> ( ( B != zero_zero_complex )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
tff(fact_1333_one__eq__divide__iff,axiom,
    ! [A: real,B: real] :
      ( ( one_one_real = aa_real_real(divide_divide_real(A),B) )
    <=> ( ( B != zero_zero_real )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
tff(fact_1334_one__eq__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( one_one_rat = aa_rat_rat(divide_divide_rat(A),B) )
    <=> ( ( B != zero_zero_rat )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
tff(fact_1335_divide__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(A),A) = one_one_complex ) ) ).

% divide_self
tff(fact_1336_divide__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(divide_divide_real(A),A) = one_one_real ) ) ).

% divide_self
tff(fact_1337_divide__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(A),A) = one_one_rat ) ) ).

% divide_self
tff(fact_1338_divide__self__if,axiom,
    ! [A: complex] :
      aa_complex_complex(divide1717551699836669952omplex(A),A) = $ite(A = zero_zero_complex,zero_zero_complex,one_one_complex) ).

% divide_self_if
tff(fact_1339_divide__self__if,axiom,
    ! [A: real] :
      aa_real_real(divide_divide_real(A),A) = $ite(A = zero_zero_real,zero_zero_real,one_one_real) ).

% divide_self_if
tff(fact_1340_divide__self__if,axiom,
    ! [A: rat] :
      aa_rat_rat(divide_divide_rat(A),A) = $ite(A = zero_zero_rat,zero_zero_rat,one_one_rat) ).

% divide_self_if
tff(fact_1341_divide__eq__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( aa_real_real(divide_divide_real(B),A) = one_one_real )
    <=> ( ( A != zero_zero_real )
        & ( A = B ) ) ) ).

% divide_eq_eq_1
tff(fact_1342_divide__eq__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( aa_rat_rat(divide_divide_rat(B),A) = one_one_rat )
    <=> ( ( A != zero_zero_rat )
        & ( A = B ) ) ) ).

% divide_eq_eq_1
tff(fact_1343_eq__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( one_one_real = aa_real_real(divide_divide_real(B),A) )
    <=> ( ( A != zero_zero_real )
        & ( A = B ) ) ) ).

% eq_divide_eq_1
tff(fact_1344_eq__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( one_one_rat = aa_rat_rat(divide_divide_rat(B),A) )
    <=> ( ( A != zero_zero_rat )
        & ( A = B ) ) ) ).

% eq_divide_eq_1
tff(fact_1345_one__divide__eq__0__iff,axiom,
    ! [A: real] :
      ( ( aa_real_real(divide_divide_real(one_one_real),A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% one_divide_eq_0_iff
tff(fact_1346_one__divide__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(divide_divide_rat(one_one_rat),A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% one_divide_eq_0_iff
tff(fact_1347_zero__eq__1__divide__iff,axiom,
    ! [A: real] :
      ( ( zero_zero_real = aa_real_real(divide_divide_real(one_one_real),A) )
    <=> ( A = zero_zero_real ) ) ).

% zero_eq_1_divide_iff
tff(fact_1348_zero__eq__1__divide__iff,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = aa_rat_rat(divide_divide_rat(one_one_rat),A) )
    <=> ( A = zero_zero_rat ) ) ).

% zero_eq_1_divide_iff
tff(fact_1349_div__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(A),A) = one_one_complex ) ) ).

% div_self
tff(fact_1350_div__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(divide_divide_real(A),A) = one_one_real ) ) ).

% div_self
tff(fact_1351_div__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(A),A) = one_one_rat ) ) ).

% div_self
tff(fact_1352_div__self,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),A) = one_one_nat ) ) ).

% div_self
tff(fact_1353_div__self,axiom,
    ! [A: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),A) = one_one_int ) ) ).

% div_self
tff(fact_1354_power__0__Suc,axiom,
    ! [Nb: nat] : aa_nat_rat(power_power_rat(zero_zero_rat),aa_nat_nat(suc,Nb)) = zero_zero_rat ).

% power_0_Suc
tff(fact_1355_power__0__Suc,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),aa_nat_nat(suc,Nb)) = zero_zero_nat ).

% power_0_Suc
tff(fact_1356_power__0__Suc,axiom,
    ! [Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),aa_nat_nat(suc,Nb)) = zero_zero_real ).

% power_0_Suc
tff(fact_1357_power__0__Suc,axiom,
    ! [Nb: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,zero_zero_int),aa_nat_nat(suc,Nb)) = zero_zero_int ).

% power_0_Suc
tff(fact_1358_power__0__Suc,axiom,
    ! [Nb: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,zero_zero_complex),aa_nat_nat(suc,Nb)) = zero_zero_complex ).

% power_0_Suc
tff(fact_1359_power__zero__numeral,axiom,
    ! [K2: num] : aa_nat_rat(power_power_rat(zero_zero_rat),aa_num_nat(numeral_numeral_nat,K2)) = zero_zero_rat ).

% power_zero_numeral
tff(fact_1360_power__zero__numeral,axiom,
    ! [K2: num] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,K2)) = zero_zero_nat ).

% power_zero_numeral
tff(fact_1361_power__zero__numeral,axiom,
    ! [K2: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),aa_num_nat(numeral_numeral_nat,K2)) = zero_zero_real ).

% power_zero_numeral
tff(fact_1362_power__zero__numeral,axiom,
    ! [K2: num] : aa_nat_int(aa_int_fun_nat_int(power_power_int,zero_zero_int),aa_num_nat(numeral_numeral_nat,K2)) = zero_zero_int ).

% power_zero_numeral
tff(fact_1363_power__zero__numeral,axiom,
    ! [K2: num] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,zero_zero_complex),aa_num_nat(numeral_numeral_nat,K2)) = zero_zero_complex ).

% power_zero_numeral
tff(fact_1364_power__eq__0__iff,axiom,
    ! [A: rat,Nb: nat] :
      ( ( aa_nat_rat(power_power_rat(A),Nb) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% power_eq_0_iff
tff(fact_1365_power__eq__0__iff,axiom,
    ! [A: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% power_eq_0_iff
tff(fact_1366_power__eq__0__iff,axiom,
    ! [A: real,Nb: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb) = zero_zero_real )
    <=> ( ( A = zero_zero_real )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% power_eq_0_iff
tff(fact_1367_power__eq__0__iff,axiom,
    ! [A: int,Nb: nat] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb) = zero_zero_int )
    <=> ( ( A = zero_zero_int )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% power_eq_0_iff
tff(fact_1368_power__eq__0__iff,axiom,
    ! [A: complex,Nb: nat] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),Nb) = zero_zero_complex )
    <=> ( ( A = zero_zero_complex )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% power_eq_0_iff
tff(fact_1369_mod__mult__self2__is__0,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),B) = zero_zero_nat ).

% mod_mult_self2_is_0
tff(fact_1370_mod__mult__self2__is__0,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),B) = zero_zero_int ).

% mod_mult_self2_is_0
tff(fact_1371_mod__mult__self2__is__0,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),B) = zero_z3403309356797280102nteger ).

% mod_mult_self2_is_0
tff(fact_1372_mod__mult__self1__is__0,axiom,
    ! [B: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),B) = zero_zero_nat ).

% mod_mult_self1_is_0
tff(fact_1373_mod__mult__self1__is__0,axiom,
    ! [B: int,A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),B) = zero_zero_int ).

% mod_mult_self1_is_0
tff(fact_1374_mod__mult__self1__is__0,axiom,
    ! [B: code_integer,A: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A),B) = zero_z3403309356797280102nteger ).

% mod_mult_self1_is_0
tff(fact_1375_bits__mod__by__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),one_one_nat) = zero_zero_nat ).

% bits_mod_by_1
tff(fact_1376_bits__mod__by__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),one_one_int) = zero_zero_int ).

% bits_mod_by_1
tff(fact_1377_bits__mod__by__1,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,one_one_Code_integer) = zero_z3403309356797280102nteger ).

% bits_mod_by_1
tff(fact_1378_mod__by__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),one_one_nat) = zero_zero_nat ).

% mod_by_1
tff(fact_1379_mod__by__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),one_one_int) = zero_zero_int ).

% mod_by_1
tff(fact_1380_mod__by__1,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,one_one_Code_integer) = zero_z3403309356797280102nteger ).

% mod_by_1
tff(fact_1381_mod__div__trivial,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),B) = zero_zero_nat ).

% mod_div_trivial
tff(fact_1382_mod__div__trivial,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),B) = zero_zero_int ).

% mod_div_trivial
tff(fact_1383_mod__div__trivial,axiom,
    ! [A: code_integer,B: code_integer] : divide6298287555418463151nteger(modulo364778990260209775nteger(A,B),B) = zero_z3403309356797280102nteger ).

% mod_div_trivial
tff(fact_1384_bits__mod__div__trivial,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),B) = zero_zero_nat ).

% bits_mod_div_trivial
tff(fact_1385_bits__mod__div__trivial,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),B) = zero_zero_int ).

% bits_mod_div_trivial
tff(fact_1386_bits__mod__div__trivial,axiom,
    ! [A: code_integer,B: code_integer] : divide6298287555418463151nteger(modulo364778990260209775nteger(A,B),B) = zero_z3403309356797280102nteger ).

% bits_mod_div_trivial
tff(fact_1387_one__le__mult__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),Ma)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) ) ) ).

% one_le_mult_iff
tff(fact_1388_mult__Suc__right,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Nb)) ).

% mult_Suc_right
tff(fact_1389_mult__le__cancel2,axiom,
    ! [Ma: nat,K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),K2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ) ).

% mult_le_cancel2
tff(fact_1390_nat__mult__le__cancel__disj,axiom,
    ! [K2: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ) ).

% nat_mult_le_cancel_disj
tff(fact_1391_div__mult__self__is__m,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Nb)),Nb) = Ma ) ) ).

% div_mult_self_is_m
tff(fact_1392_div__mult__self1__is__m,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Ma)),Nb) = Ma ) ) ).

% div_mult_self1_is_m
tff(fact_1393_dbl__simps_I5_J,axiom,
    ! [K2: num] : aa_complex_complex(neg_nu7009210354673126013omplex,numera6690914467698888265omplex(K2)) = numera6690914467698888265omplex(aa_num_num(bit0,K2)) ).

% dbl_simps(5)
tff(fact_1394_dbl__simps_I5_J,axiom,
    ! [K2: num] : aa_real_real(neg_numeral_dbl_real,aa_num_real(numeral_numeral_real,K2)) = aa_num_real(numeral_numeral_real,aa_num_num(bit0,K2)) ).

% dbl_simps(5)
tff(fact_1395_dbl__simps_I5_J,axiom,
    ! [K2: num] : aa_rat_rat(neg_numeral_dbl_rat,aa_num_rat(numeral_numeral_rat,K2)) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,K2)) ).

% dbl_simps(5)
tff(fact_1396_dbl__simps_I5_J,axiom,
    ! [K2: num] : aa_int_int(neg_numeral_dbl_int,aa_num_int(numeral_numeral_int,K2)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,K2)) ).

% dbl_simps(5)
tff(fact_1397_both__member__options__from__complete__tree__to__child,axiom,
    ! [Degb: nat,Mi: nat,Ma: nat,TreeListb: list_VEBT_VEBT,Summaryb: vEBT_VEBT,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Degb)
     => ( aa_nat_o(vEBT_V8194947554948674370ptions(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),Degb,TreeListb),Summaryb)),X)
       => ( aa_nat_o(vEBT_V8194947554948674370ptions(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeListb),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Degb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Degb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
          | ( X = Mi )
          | ( X = Ma ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
tff(fact_1398_times__enat__simps_I4_J,axiom,
    ! [Ma: nat] :
      aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(extended_enat2,Ma)),extend5688581933313929465d_enat) = $ite(Ma = zero_zero_nat,zero_z5237406670263579293d_enat,extend5688581933313929465d_enat) ).

% times_enat_simps(4)
tff(fact_1399_times__enat__simps_I3_J,axiom,
    ! [Nb: nat] :
      aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(extend5688581933313929465d_enat),aa_nat_Extended_enat(extended_enat2,Nb)) = $ite(Nb = zero_zero_nat,zero_z5237406670263579293d_enat,extend5688581933313929465d_enat) ).

% times_enat_simps(3)
tff(fact_1400_divide__le__0__1__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(one_one_real),A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% divide_le_0_1_iff
tff(fact_1401_divide__le__0__1__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(one_one_rat),A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% divide_le_0_1_iff
tff(fact_1402_zero__le__divide__1__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(one_one_real),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% zero_le_divide_1_iff
tff(fact_1403_zero__le__divide__1__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(one_one_rat),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% zero_le_divide_1_iff
tff(fact_1404_divide__less__0__1__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(one_one_real),A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% divide_less_0_1_iff
tff(fact_1405_divide__less__0__1__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(one_one_rat),A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% divide_less_0_1_iff
tff(fact_1406_divide__less__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(B),A)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% divide_less_eq_1_neg
tff(fact_1407_divide__less__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B),A)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% divide_less_eq_1_neg
tff(fact_1408_divide__less__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(B),A)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ).

% divide_less_eq_1_pos
tff(fact_1409_divide__less__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B),A)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ).

% divide_less_eq_1_pos
tff(fact_1410_less__divide__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(divide_divide_real(B),A))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ).

% less_divide_eq_1_neg
tff(fact_1411_less__divide__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),A))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ).

% less_divide_eq_1_neg
tff(fact_1412_less__divide__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(divide_divide_real(B),A))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% less_divide_eq_1_pos
tff(fact_1413_less__divide__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),A))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% less_divide_eq_1_pos
tff(fact_1414_zero__less__divide__1__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(divide_divide_real(one_one_real),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% zero_less_divide_1_iff
tff(fact_1415_zero__less__divide__1__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(one_one_rat),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% zero_less_divide_1_iff
tff(fact_1416_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: complex,W: num,A: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),numera6690914467698888265omplex(W)) = A )
    <=> $ite(numera6690914467698888265omplex(W) != zero_zero_complex,B = aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(W)),A = zero_zero_complex) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_1417_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: real,W: num,A: real] :
      ( ( aa_real_real(divide_divide_real(B),aa_num_real(numeral_numeral_real,W)) = A )
    <=> $ite(aa_num_real(numeral_numeral_real,W) != zero_zero_real,B = aa_real_real(times_times_real(A),aa_num_real(numeral_numeral_real,W)),A = zero_zero_real) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_1418_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W)) = A )
    <=> $ite(aa_num_rat(numeral_numeral_rat,W) != zero_zero_rat,B = aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,W)),A = zero_zero_rat) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_1419_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: complex,B: complex,W: num] :
      ( ( A = aa_complex_complex(divide1717551699836669952omplex(B),numera6690914467698888265omplex(W)) )
    <=> $ite(numera6690914467698888265omplex(W) != zero_zero_complex,aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(W)) = B,A = zero_zero_complex) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_1420_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W: num] :
      ( ( A = aa_real_real(divide_divide_real(B),aa_num_real(numeral_numeral_real,W)) )
    <=> $ite(aa_num_real(numeral_numeral_real,W) != zero_zero_real,aa_real_real(times_times_real(A),aa_num_real(numeral_numeral_real,W)) = B,A = zero_zero_real) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_1421_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( A = aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W)) )
    <=> $ite(aa_num_rat(numeral_numeral_rat,W) != zero_zero_rat,aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,W)) = B,A = zero_zero_rat) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_1422_nonzero__divide__mult__cancel__left,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(times_times_complex(A),B)) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),B) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_1423_nonzero__divide__mult__cancel__left,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(divide_divide_real(A),aa_real_real(times_times_real(A),B)) = aa_real_real(divide_divide_real(one_one_real),B) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_1424_nonzero__divide__mult__cancel__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(A),aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(divide_divide_rat(one_one_rat),B) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_1425_nonzero__divide__mult__cancel__right,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(B),aa_complex_complex(times_times_complex(A),B)) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),A) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_1426_nonzero__divide__mult__cancel__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(divide_divide_real(B),aa_real_real(times_times_real(A),B)) = aa_real_real(divide_divide_real(one_one_real),A) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_1427_nonzero__divide__mult__cancel__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(B),aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(divide_divide_rat(one_one_rat),A) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_1428_div__mult__self1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self1
tff(fact_1429_div__mult__self1,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_mult_self1
tff(fact_1430_div__mult__self2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self2
tff(fact_1431_div__mult__self2,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_mult_self2
tff(fact_1432_div__mult__self3,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self3
tff(fact_1433_div__mult__self3,axiom,
    ! [B: int,C: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),B)),A)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_mult_self3
tff(fact_1434_div__mult__self4,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self4
tff(fact_1435_div__mult__self4,axiom,
    ! [B: int,C: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),C)),A)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_mult_self4
tff(fact_1436_power__mono__iff,axiom,
    ! [A: real,B: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb))
          <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_1437_power__mono__iff,axiom,
    ! [A: rat,B: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb))
          <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_1438_power__mono__iff,axiom,
    ! [A: nat,B: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb))
          <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_1439_power__mono__iff,axiom,
    ! [A: int,B: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Nb))
          <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_1440_half__negative__int__iff,axiom,
    ! [K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int) ) ).

% half_negative_int_iff
tff(fact_1441_half__nonnegative__int__iff,axiom,
    ! [K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2) ) ).

% half_nonnegative_int_iff
tff(fact_1442_Suc__numeral,axiom,
    ! [Nb: num] : aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(Nb),one)) ).

% Suc_numeral
tff(fact_1443_Suc__mod__mult__self1,axiom,
    ! [Ma: nat,K2: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Nb)))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Ma)),Nb) ).

% Suc_mod_mult_self1
tff(fact_1444_Suc__mod__mult__self2,axiom,
    ! [Ma: nat,Nb: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K2)))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Ma)),Nb) ).

% Suc_mod_mult_self2
tff(fact_1445_Suc__mod__mult__self3,axiom,
    ! [K2: nat,Nb: nat,Ma: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Nb)),Ma))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Ma)),Nb) ).

% Suc_mod_mult_self3
tff(fact_1446_Suc__mod__mult__self4,axiom,
    ! [Nb: nat,K2: nat,Ma: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K2)),Ma))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Ma)),Nb) ).

% Suc_mod_mult_self4
tff(fact_1447_divide__le__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(B),A)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% divide_le_eq_1_neg
tff(fact_1448_divide__le__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),A)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% divide_le_eq_1_neg
tff(fact_1449_divide__le__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(B),A)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ).

% divide_le_eq_1_pos
tff(fact_1450_divide__le__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),A)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).

% divide_le_eq_1_pos
tff(fact_1451_le__divide__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(divide_divide_real(B),A))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ).

% le_divide_eq_1_neg
tff(fact_1452_le__divide__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),A))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).

% le_divide_eq_1_neg
tff(fact_1453_le__divide__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(divide_divide_real(B),A))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% le_divide_eq_1_pos
tff(fact_1454_le__divide__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),A))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% le_divide_eq_1_pos
tff(fact_1455_power__strict__decreasing__iff,axiom,
    ! [B: real,Ma: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Ma)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Ma) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_1456_power__strict__decreasing__iff,axiom,
    ! [B: rat,Ma: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),one_one_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(B),Ma)),aa_nat_rat(power_power_rat(B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Ma) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_1457_power__strict__decreasing__iff,axiom,
    ! [B: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),one_one_nat)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Ma) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_1458_power__strict__decreasing__iff,axiom,
    ! [B: int,Ma: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Ma)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Ma) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_1459_zero__eq__power2,axiom,
    ! [A: rat] :
      ( ( aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% zero_eq_power2
tff(fact_1460_zero__eq__power2,axiom,
    ! [A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat )
    <=> ( A = zero_zero_nat ) ) ).

% zero_eq_power2
tff(fact_1461_zero__eq__power2,axiom,
    ! [A: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% zero_eq_power2
tff(fact_1462_zero__eq__power2,axiom,
    ! [A: int] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% zero_eq_power2
tff(fact_1463_zero__eq__power2,axiom,
    ! [A: complex] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_complex )
    <=> ( A = zero_zero_complex ) ) ).

% zero_eq_power2
tff(fact_1464_add__2__eq__Suc,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) = aa_nat_nat(suc,aa_nat_nat(suc,Nb)) ).

% add_2_eq_Suc
tff(fact_1465_add__2__eq__Suc_H,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(suc,Nb)) ).

% add_2_eq_Suc'
tff(fact_1466_Suc__1,axiom,
    aa_nat_nat(suc,one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% Suc_1
tff(fact_1467_div2__Suc__Suc,axiom,
    ! [Ma: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Ma))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% div2_Suc_Suc
tff(fact_1468_mod2__Suc__Suc,axiom,
    ! [Ma: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,Ma))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% mod2_Suc_Suc
tff(fact_1469_add__self__mod__2,axiom,
    ! [Ma: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Ma)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% add_self_mod_2
tff(fact_1470_Suc__times__numeral__mod__eq,axiom,
    ! [K2: num,Nb: nat] :
      ( ( aa_num_nat(numeral_numeral_nat,K2) != one_one_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,K2)),Nb))),aa_num_nat(numeral_numeral_nat,K2)) = one_one_nat ) ) ).

% Suc_times_numeral_mod_eq
tff(fact_1471_bits__1__div__2,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% bits_1_div_2
tff(fact_1472_bits__1__div__2,axiom,
    aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ).

% bits_1_div_2
tff(fact_1473_one__div__two__eq__zero,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% one_div_two_eq_zero
tff(fact_1474_one__div__two__eq__zero,axiom,
    aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ).

% one_div_two_eq_zero
tff(fact_1475_power__decreasing__iff,axiom,
    ! [B: real,Ma: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Ma)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma) ) ) ) ).

% power_decreasing_iff
tff(fact_1476_power__decreasing__iff,axiom,
    ! [B: rat,Ma: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),one_one_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(B),Ma)),aa_nat_rat(power_power_rat(B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma) ) ) ) ).

% power_decreasing_iff
tff(fact_1477_power__decreasing__iff,axiom,
    ! [B: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),one_one_nat)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma) ) ) ) ).

% power_decreasing_iff
tff(fact_1478_power__decreasing__iff,axiom,
    ! [B: int,Ma: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Ma)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma) ) ) ) ).

% power_decreasing_iff
tff(fact_1479_power2__less__eq__zero__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_real)
    <=> ( A = zero_zero_real ) ) ).

% power2_less_eq_zero_iff
tff(fact_1480_power2__less__eq__zero__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_rat)
    <=> ( A = zero_zero_rat ) ) ).

% power2_less_eq_zero_iff
tff(fact_1481_power2__less__eq__zero__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_int)
    <=> ( A = zero_zero_int ) ) ).

% power2_less_eq_zero_iff
tff(fact_1482_power2__eq__iff__nonneg,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y2)
       => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( X = Y2 ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_1483_power2__eq__iff__nonneg,axiom,
    ! [X: rat,Y2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y2)
       => ( ( aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( X = Y2 ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_1484_power2__eq__iff__nonneg,axiom,
    ! [X: nat,Y2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),X)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y2)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( X = Y2 ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_1485_power2__eq__iff__nonneg,axiom,
    ! [X: int,Y2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y2)
       => ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( X = Y2 ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_1486_zero__less__power2,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( A != zero_zero_real ) ) ).

% zero_less_power2
tff(fact_1487_zero__less__power2,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( A != zero_zero_rat ) ) ).

% zero_less_power2
tff(fact_1488_zero__less__power2,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( A != zero_zero_int ) ) ).

% zero_less_power2
tff(fact_1489_sum__power2__eq__zero__iff,axiom,
    ! [X: rat,Y2: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_rat )
    <=> ( ( X = zero_zero_rat )
        & ( Y2 = zero_zero_rat ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_1490_sum__power2__eq__zero__iff,axiom,
    ! [X: real,Y2: real] :
      ( ( aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_real )
    <=> ( ( X = zero_zero_real )
        & ( Y2 = zero_zero_real ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_1491_sum__power2__eq__zero__iff,axiom,
    ! [X: int,Y2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_int )
    <=> ( ( X = zero_zero_int )
        & ( Y2 = zero_zero_int ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_1492_mod2__gr__0,axiom,
    ! [Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ).

% mod2_gr_0
tff(fact_1493_mod__pos__neg__trivial,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),L)),zero_zero_int)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),L) ) ) ) ).

% mod_pos_neg_trivial
tff(fact_1494_zmod__zmult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ) ) ).

% zmod_zmult2_eq
tff(fact_1495_zdiv__zmult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ).

% zdiv_zmult2_eq
tff(fact_1496_unique__quotient__lemma__neg,axiom,
    ! [B: int,Q4: int,R2: int,Q: int,R: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q4)),R2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q)),R))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),R)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),R2)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q),Q4) ) ) ) ) ).

% unique_quotient_lemma_neg
tff(fact_1497_Euclidean__Division_Opos__mod__bound,axiom,
    ! [L: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),L)),L) ) ).

% Euclidean_Division.pos_mod_bound
tff(fact_1498_neg__mod__bound,axiom,
    ! [L: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,L),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),L)) ) ).

% neg_mod_bound
tff(fact_1499_nonneg1__imp__zdiv__pos__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))
      <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ) ) ).

% nonneg1_imp_zdiv_pos_iff
tff(fact_1500_Euclidean__Division_Opos__mod__sign,axiom,
    ! [L: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),L)) ) ).

% Euclidean_Division.pos_mod_sign
tff(fact_1501_neg__mod__sign,axiom,
    ! [L: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),L)),zero_zero_int) ) ).

% neg_mod_sign
tff(fact_1502_pos__imp__zdiv__nonneg__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ).

% pos_imp_zdiv_nonneg_iff
tff(fact_1503_neg__imp__zdiv__nonneg__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ) ).

% neg_imp_zdiv_nonneg_iff
tff(fact_1504_unique__quotient__lemma,axiom,
    ! [B: int,Q4: int,R2: int,Q: int,R: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q4)),R2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q)),R))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R2),B)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R),B)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q4),Q) ) ) ) ) ).

% unique_quotient_lemma
tff(fact_1505_zdiv__mono2__neg__lemma,axiom,
    ! [B: int,Q: int,R: int,B4: int,Q4: int,R2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q)),R) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B4),Q4)),R2) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B4),Q4)),R2)),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R),B)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R2)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B4)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B4),B)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q4),Q) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
tff(fact_1506_pos__imp__zdiv__pos__iff,axiom,
    ! [K2: int,I2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,I2),K2))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),I2) ) ) ).

% pos_imp_zdiv_pos_iff
tff(fact_1507_div__nonpos__pos__le0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),zero_zero_int) ) ) ).

% div_nonpos_pos_le0
tff(fact_1508_div__nonneg__neg__le0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),zero_zero_int) ) ) ).

% div_nonneg_neg_le0
tff(fact_1509_verit__le__mono__div__int,axiom,
    ! [A2: int,B3: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,
              aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),Nb)),
                $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B3),Nb) = zero_zero_int,one_one_int,zero_zero_int))),aa_int_int(aa_int_fun_int_int(divide_divide_int,B3),Nb)) ) ) ).

% verit_le_mono_div_int
tff(fact_1510_int__div__less__self,axiom,
    ! [X: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),K2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),K2)),X) ) ) ).

% int_div_less_self
tff(fact_1511_zmod__trivial__iff,axiom,
    ! [I2: int,K2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,I2),K2) = I2 )
    <=> ( ( K2 = zero_zero_int )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),I2)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,I2),K2) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,K2),I2) ) ) ) ).

% zmod_trivial_iff
tff(fact_1512_zdiv__mono2__lemma,axiom,
    ! [B: int,Q: int,R: int,B4: int,Q4: int,R2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q)),R) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B4),Q4)),R2) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B4),Q4)),R2))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R2),B4)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B4)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B4),B)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q),Q4) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
tff(fact_1513_div__positive__int,axiom,
    ! [L: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),K2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),L)) ) ) ).

% div_positive_int
tff(fact_1514_split__pos__lemma,axiom,
    ! [K2: int,P: fun_int_fun_int_o,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K2)
     => ( aa_int_o(aa_int_fun_int_o(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,Nb),K2)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Nb),K2))
      <=> ! [I: int,J3: int] :
            ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),J3)
              & aa_int_o(aa_int_fun_int_o(ord_less_int,J3),K2)
              & ( Nb = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K2),I)),J3) ) )
           => aa_int_o(aa_int_fun_int_o(P,I),J3) ) ) ) ).

% split_pos_lemma
tff(fact_1515_split__neg__lemma,axiom,
    ! [K2: int,P: fun_int_fun_int_o,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,Nb),K2)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Nb),K2))
      <=> ! [I: int,J3: int] :
            ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),J3)
              & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J3),zero_zero_int)
              & ( Nb = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K2),I)),J3) ) )
           => aa_int_o(aa_int_fun_int_o(P,I),J3) ) ) ) ).

% split_neg_lemma
tff(fact_1516_div__int__pos__iff,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),L))
    <=> ( ( K2 = zero_zero_int )
        | ( L = zero_zero_int )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ) ) ).

% div_int_pos_iff
tff(fact_1517_div__mod__decomp__int,axiom,
    ! [A2: int,Nb: int] : A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),Nb)),Nb)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),Nb)) ).

% div_mod_decomp_int
tff(fact_1518_zdiv__mono2__neg,axiom,
    ! [A: int,B4: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B4)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B4),B)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B4)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ) ).

% zdiv_mono2_neg
tff(fact_1519_zdiv__mono1__neg,axiom,
    ! [A: int,A4: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),A4)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A4),B)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% zdiv_mono1_neg
tff(fact_1520_int__mod__pos__eq,axiom,
    ! [A: int,B: int,Q: int,R: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q)),R) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R),B)
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = R ) ) ) ) ).

% int_mod_pos_eq
tff(fact_1521_int__mod__neg__eq,axiom,
    ! [A: int,B: int,Q: int,R: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q)),R) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),R)
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = R ) ) ) ) ).

% int_mod_neg_eq
tff(fact_1522_int__div__pos__eq,axiom,
    ! [A: int,B: int,Q: int,R: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q)),R) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R),B)
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = Q ) ) ) ) ).

% int_div_pos_eq
tff(fact_1523_int__div__neg__eq,axiom,
    ! [A: int,B: int,Q: int,R: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q)),R) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),R)
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = Q ) ) ) ) ).

% int_div_neg_eq
tff(fact_1524_zdiv__eq__0__iff,axiom,
    ! [I2: int,K2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,I2),K2) = zero_zero_int )
    <=> ( ( K2 = zero_zero_int )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),I2)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,I2),K2) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,K2),I2) ) ) ) ).

% zdiv_eq_0_iff
tff(fact_1525_zdiv__mono__strict,axiom,
    ! [A2: int,B3: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Nb)
       => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),Nb) = zero_zero_int )
         => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B3),Nb) = zero_zero_int )
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),Nb)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B3),Nb)) ) ) ) ) ).

% zdiv_mono_strict
tff(fact_1526_pos__mod__conj,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))
        & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),B) ) ) ).

% pos_mod_conj
tff(fact_1527_neg__mod__conj,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),zero_zero_int)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ) ) ).

% neg_mod_conj
tff(fact_1528_q__pos__lemma,axiom,
    ! [B4: int,Q4: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B4),Q4)),R2))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R2),B4)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B4)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Q4) ) ) ) ).

% q_pos_lemma
tff(fact_1529_zdiv__mono2,axiom,
    ! [A: int,B4: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B4)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B4),B)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B4)) ) ) ) ).

% zdiv_mono2
tff(fact_1530_zdiv__mono1,axiom,
    ! [A: int,A4: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),A4)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A4),B)) ) ) ).

% zdiv_mono1
tff(fact_1531_split__zmod,axiom,
    ! [P: fun_int_o,Nb: int,K2: int] :
      ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Nb),K2))
    <=> ( ( ( K2 = zero_zero_int )
         => aa_int_o(P,Nb) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K2)
         => ! [I: int,J3: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),J3)
                & aa_int_o(aa_int_fun_int_o(ord_less_int,J3),K2)
                & ( Nb = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K2),I)),J3) ) )
             => aa_int_o(P,J3) ) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int)
         => ! [I: int,J3: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),J3)
                & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J3),zero_zero_int)
                & ( Nb = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K2),I)),J3) ) )
             => aa_int_o(P,J3) ) ) ) ) ).

% split_zmod
tff(fact_1532_split__zdiv,axiom,
    ! [P: fun_int_o,Nb: int,K2: int] :
      ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,Nb),K2))
    <=> ( ( ( K2 = zero_zero_int )
         => aa_int_o(P,zero_zero_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K2)
         => ! [I: int,J3: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),J3)
                & aa_int_o(aa_int_fun_int_o(ord_less_int,J3),K2)
                & ( Nb = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K2),I)),J3) ) )
             => aa_int_o(P,I) ) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int)
         => ! [I: int,J3: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),J3)
                & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J3),zero_zero_int)
                & ( Nb = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K2),I)),J3) ) )
             => aa_int_o(P,I) ) ) ) ) ).

% split_zdiv
tff(fact_1533_zmod__eq__0__iff,axiom,
    ! [Ma: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Ma),D) = zero_zero_int )
    <=> ? [Q5: int] : Ma = aa_int_int(aa_int_fun_int_int(times_times_int,D),Q5) ) ).

% zmod_eq_0_iff
tff(fact_1534_zmod__eq__0D,axiom,
    ! [Ma: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Ma),D) = zero_zero_int )
     => ? [Q3: int] : Ma = aa_int_int(aa_int_fun_int_int(times_times_int,D),Q3) ) ).

% zmod_eq_0D
tff(fact_1535_zmod__le__nonneg__dividend,axiom,
    ! [Ma: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ma)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Ma),K2)),Ma) ) ).

% zmod_le_nonneg_dividend
tff(fact_1536_zero__one__enat__neq_I1_J,axiom,
    zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).

% zero_one_enat_neq(1)
tff(fact_1537_zero__reorient,axiom,
    ! [X: complex] :
      ( ( zero_zero_complex = X )
    <=> ( X = zero_zero_complex ) ) ).

% zero_reorient
tff(fact_1538_zero__reorient,axiom,
    ! [X: real] :
      ( ( zero_zero_real = X )
    <=> ( X = zero_zero_real ) ) ).

% zero_reorient
tff(fact_1539_zero__reorient,axiom,
    ! [X: rat] :
      ( ( zero_zero_rat = X )
    <=> ( X = zero_zero_rat ) ) ).

% zero_reorient
tff(fact_1540_zero__reorient,axiom,
    ! [X: nat] :
      ( ( zero_zero_nat = X )
    <=> ( X = zero_zero_nat ) ) ).

% zero_reorient
tff(fact_1541_zero__reorient,axiom,
    ! [X: int] :
      ( ( zero_zero_int = X )
    <=> ( X = zero_zero_int ) ) ).

% zero_reorient
tff(fact_1542_vebt__buildup_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero_nat )
     => ( ( X != aa_nat_nat(suc,zero_zero_nat) )
       => ~ ! [Va: nat] : X != aa_nat_nat(suc,aa_nat_nat(suc,Va)) ) ) ).

% vebt_buildup.cases
tff(fact_1543_not0__implies__Suc,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero_nat )
     => ? [M2: nat] : Nb = aa_nat_nat(suc,M2) ) ).

% not0_implies_Suc
tff(fact_1544_Zero__not__Suc,axiom,
    ! [Ma: nat] : zero_zero_nat != aa_nat_nat(suc,Ma) ).

% Zero_not_Suc
tff(fact_1545_Zero__neq__Suc,axiom,
    ! [Ma: nat] : zero_zero_nat != aa_nat_nat(suc,Ma) ).

% Zero_neq_Suc
tff(fact_1546_Suc__neq__Zero,axiom,
    ! [Ma: nat] : aa_nat_nat(suc,Ma) != zero_zero_nat ).

% Suc_neq_Zero
tff(fact_1547_zero__induct,axiom,
    ! [P: fun_nat_o,K2: nat] :
      ( aa_nat_o(P,K2)
     => ( ! [N: nat] :
            ( aa_nat_o(P,aa_nat_nat(suc,N))
           => aa_nat_o(P,N) )
       => aa_nat_o(P,zero_zero_nat) ) ) ).

% zero_induct
tff(fact_1548_n__not__Suc__n,axiom,
    ! [Nb: nat] : Nb != aa_nat_nat(suc,Nb) ).

% n_not_Suc_n
tff(fact_1549_diff__induct,axiom,
    ! [P: fun_nat_fun_nat_o,Ma: nat,Nb: nat] :
      ( ! [X4: nat] : aa_nat_o(aa_nat_fun_nat_o(P,X4),zero_zero_nat)
     => ( ! [Y4: nat] : aa_nat_o(aa_nat_fun_nat_o(P,zero_zero_nat),aa_nat_nat(suc,Y4))
       => ( ! [X4: nat,Y4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(P,X4),Y4)
             => aa_nat_o(aa_nat_fun_nat_o(P,aa_nat_nat(suc,X4)),aa_nat_nat(suc,Y4)) )
         => aa_nat_o(aa_nat_fun_nat_o(P,Ma),Nb) ) ) ) ).

% diff_induct
tff(fact_1550_nat__induct,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ! [N: nat] :
            ( aa_nat_o(P,N)
           => aa_nat_o(P,aa_nat_nat(suc,N)) )
       => aa_nat_o(P,Nb) ) ) ).

% nat_induct
tff(fact_1551_Suc__inject,axiom,
    ! [X: nat,Y2: nat] :
      ( ( aa_nat_nat(suc,X) = aa_nat_nat(suc,Y2) )
     => ( X = Y2 ) ) ).

% Suc_inject
tff(fact_1552_old_Onat_Oexhaust,axiom,
    ! [Y2: nat] :
      ( ( Y2 != zero_zero_nat )
     => ~ ! [Nat2: nat] : Y2 != aa_nat_nat(suc,Nat2) ) ).

% old.nat.exhaust
tff(fact_1553_nat_OdiscI,axiom,
    ! [Nat: nat,X23: nat] :
      ( ( Nat = aa_nat_nat(suc,X23) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
tff(fact_1554_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat3: nat] : zero_zero_nat != aa_nat_nat(suc,Nat3) ).

% old.nat.distinct(1)
tff(fact_1555_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat3: nat] : aa_nat_nat(suc,Nat3) != zero_zero_nat ).

% old.nat.distinct(2)
tff(fact_1556_nat_Odistinct_I1_J,axiom,
    ! [X23: nat] : zero_zero_nat != aa_nat_nat(suc,X23) ).

% nat.distinct(1)
tff(fact_1557_less__Suc__eq__0__disj,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(suc,Nb))
    <=> ( ( Ma = zero_zero_nat )
        | ? [J3: nat] :
            ( ( Ma = aa_nat_nat(suc,J3) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),Nb) ) ) ) ).

% less_Suc_eq_0_disj
tff(fact_1558_gr0__implies__Suc,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ? [M2: nat] : Nb = aa_nat_nat(suc,M2) ) ).

% gr0_implies_Suc
tff(fact_1559_All__less__Suc2,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ! [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(suc,Nb))
         => aa_nat_o(P,I) )
    <=> ( aa_nat_o(P,zero_zero_nat)
        & ! [I: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Nb)
           => aa_nat_o(P,aa_nat_nat(suc,I)) ) ) ) ).

% All_less_Suc2
tff(fact_1560_gr0__conv__Suc,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
    <=> ? [M3: nat] : Nb = aa_nat_nat(suc,M3) ) ).

% gr0_conv_Suc
tff(fact_1561_Ex__less__Suc2,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ? [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(suc,Nb))
          & aa_nat_o(P,I) )
    <=> ( aa_nat_o(P,zero_zero_nat)
        | ? [I: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Nb)
            & aa_nat_o(P,aa_nat_nat(suc,I)) ) ) ) ).

% Ex_less_Suc2
tff(fact_1562_one__is__add,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb) )
    <=> ( ( ( Ma = aa_nat_nat(suc,zero_zero_nat) )
          & ( Nb = zero_zero_nat ) )
        | ( ( Ma = zero_zero_nat )
          & ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% one_is_add
tff(fact_1563_add__is__1,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( ( Ma = aa_nat_nat(suc,zero_zero_nat) )
          & ( Nb = zero_zero_nat ) )
        | ( ( Ma = zero_zero_nat )
          & ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% add_is_1
tff(fact_1564_realpow__pos__nth2,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ? [R3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R3)
          & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,R3),aa_nat_nat(suc,Nb)) = A ) ) ) ).

% realpow_pos_nth2
tff(fact_1565_One__nat__def,axiom,
    one_one_nat = aa_nat_nat(suc,zero_zero_nat) ).

% One_nat_def
tff(fact_1566_mod__Suc,axiom,
    ! [Ma: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Ma)),Nb) = $ite(aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),Nb)) = Nb,zero_zero_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),Nb))) ).

% mod_Suc
tff(fact_1567_enat__0__iff_I2_J,axiom,
    ! [X: nat] :
      ( ( zero_z5237406670263579293d_enat = aa_nat_Extended_enat(extended_enat2,X) )
    <=> ( X = zero_zero_nat ) ) ).

% enat_0_iff(2)
tff(fact_1568_enat__0__iff_I1_J,axiom,
    ! [X: nat] :
      ( ( aa_nat_Extended_enat(extended_enat2,X) = zero_z5237406670263579293d_enat )
    <=> ( X = zero_zero_nat ) ) ).

% enat_0_iff(1)
tff(fact_1569_zero__enat__def,axiom,
    zero_z5237406670263579293d_enat = aa_nat_Extended_enat(extended_enat2,zero_zero_nat) ).

% zero_enat_def
tff(fact_1570_power__inject__base,axiom,
    ! [A: real,Nb: nat,B: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_nat_nat(suc,Nb)) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_1571_power__inject__base,axiom,
    ! [A: rat,Nb: nat,B: rat] :
      ( ( aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Nb)) = aa_nat_rat(power_power_rat(B),aa_nat_nat(suc,Nb)) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_1572_power__inject__base,axiom,
    ! [A: nat,Nb: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(suc,Nb)) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_1573_power__inject__base,axiom,
    ! [A: int,Nb: nat,B: int] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(suc,Nb)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,B),aa_nat_nat(suc,Nb)) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
         => ( A = B ) ) ) ) ).

% power_inject_base
tff(fact_1574_power__le__imp__le__base,axiom,
    ! [A: real,Nb: nat,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_nat_nat(suc,Nb)))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% power_le_imp_le_base
tff(fact_1575_power__le__imp__le__base,axiom,
    ! [A: rat,Nb: nat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Nb))),aa_nat_rat(power_power_rat(B),aa_nat_nat(suc,Nb)))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% power_le_imp_le_base
tff(fact_1576_power__le__imp__le__base,axiom,
    ! [A: nat,Nb: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(suc,Nb)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).

% power_le_imp_le_base
tff(fact_1577_power__le__imp__le__base,axiom,
    ! [A: int,Nb: nat,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(suc,Nb))),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),aa_nat_nat(suc,Nb)))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% power_le_imp_le_base
tff(fact_1578_numeral__1__eq__Suc__0,axiom,
    aa_num_nat(numeral_numeral_nat,one) = aa_nat_nat(suc,zero_zero_nat) ).

% numeral_1_eq_Suc_0
tff(fact_1579_num_Osize_I5_J,axiom,
    ! [X23: num] : aa_num_nat(size_size_num,aa_num_num(bit0,X23)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_size_num,X23)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size(5)
tff(fact_1580_ex__least__nat__less,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,Nb)
     => ( ~ aa_nat_o(P,zero_zero_nat)
       => ? [K: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),Nb)
            & ! [I4: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I4),K)
               => ~ aa_nat_o(P,I4) )
            & aa_nat_o(P,aa_nat_nat(suc,K)) ) ) ) ).

% ex_least_nat_less
tff(fact_1581_one__less__mult,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Ma)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Nb)) ) ) ).

% one_less_mult
tff(fact_1582_n__less__m__mult__n,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Ma)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Nb)) ) ) ).

% n_less_m_mult_n
tff(fact_1583_n__less__n__mult__m,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Ma)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Ma)) ) ) ).

% n_less_n_mult_m
tff(fact_1584_power__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_rat(power_power_rat(zero_zero_rat),Nb) = $ite(Nb = zero_zero_nat,one_one_rat,zero_zero_rat) ).

% power_0_left
tff(fact_1585_power__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),Nb) = $ite(Nb = zero_zero_nat,one_one_nat,zero_zero_nat) ).

% power_0_left
tff(fact_1586_power__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Nb) = $ite(Nb = zero_zero_nat,one_one_real,zero_zero_real) ).

% power_0_left
tff(fact_1587_power__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_int(aa_int_fun_nat_int(power_power_int,zero_zero_int),Nb) = $ite(Nb = zero_zero_nat,one_one_int,zero_zero_int) ).

% power_0_left
tff(fact_1588_power__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,zero_zero_complex),Nb) = $ite(Nb = zero_zero_nat,one_one_complex,zero_zero_complex) ).

% power_0_left
tff(fact_1589_nat__induct__non__zero,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(P,one_one_nat)
       => ( ! [N: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
             => ( aa_nat_o(P,N)
               => aa_nat_o(P,aa_nat_nat(suc,N)) ) )
         => aa_nat_o(P,Nb) ) ) ) ).

% nat_induct_non_zero
tff(fact_1590_power__gt__expt,axiom,
    ! [Nb: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),K2)) ) ).

% power_gt_expt
tff(fact_1591_zero__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_rat(power_power_rat(zero_zero_rat),Nb) = zero_zero_rat ) ) ).

% zero_power
tff(fact_1592_zero__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),Nb) = zero_zero_nat ) ) ).

% zero_power
tff(fact_1593_zero__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Nb) = zero_zero_real ) ) ).

% zero_power
tff(fact_1594_zero__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,zero_zero_int),Nb) = zero_zero_int ) ) ).

% zero_power
tff(fact_1595_zero__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,zero_zero_complex),Nb) = zero_zero_complex ) ) ).

% zero_power
tff(fact_1596_nat__one__le__power,axiom,
    ! [I2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),I2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I2),Nb)) ) ).

% nat_one_le_power
tff(fact_1597_realpow__pos__nth__unique,axiom,
    ! [Nb: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => ? [X4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X4)
            & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,X4),Nb) = A )
            & ! [Y5: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y5)
                  & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y5),Nb) = A ) )
               => ( Y5 = X4 ) ) ) ) ) ).

% realpow_pos_nth_unique
tff(fact_1598_realpow__pos__nth,axiom,
    ! [Nb: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => ? [R3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R3)
            & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,R3),Nb) = A ) ) ) ) ).

% realpow_pos_nth
tff(fact_1599_Nat_OlessE,axiom,
    ! [I2: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),K2)
     => ( ( K2 != aa_nat_nat(suc,I2) )
       => ~ ! [J2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J2)
             => ( K2 != aa_nat_nat(suc,J2) ) ) ) ) ).

% Nat.lessE
tff(fact_1600_Suc__lessD,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Ma)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb) ) ).

% Suc_lessD
tff(fact_1601_Suc__lessE,axiom,
    ! [I2: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I2)),K2)
     => ~ ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J2)
           => ( K2 != aa_nat_nat(suc,J2) ) ) ) ).

% Suc_lessE
tff(fact_1602_Suc__lessI,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
     => ( ( aa_nat_nat(suc,Ma) != Nb )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Ma)),Nb) ) ) ).

% Suc_lessI
tff(fact_1603_less__SucE,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(suc,Nb))
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
       => ( Ma = Nb ) ) ) ).

% less_SucE
tff(fact_1604_less__SucI,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(suc,Nb)) ) ).

% less_SucI
tff(fact_1605_Ex__less__Suc,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ? [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(suc,Nb))
          & aa_nat_o(P,I) )
    <=> ( aa_nat_o(P,Nb)
        | ? [I: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Nb)
            & aa_nat_o(P,I) ) ) ) ).

% Ex_less_Suc
tff(fact_1606_less__Suc__eq,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(suc,Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
        | ( Ma = Nb ) ) ) ).

% less_Suc_eq
tff(fact_1607_not__less__eq,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Ma)) ) ).

% not_less_eq
tff(fact_1608_All__less__Suc,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ! [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(suc,Nb))
         => aa_nat_o(P,I) )
    <=> ( aa_nat_o(P,Nb)
        & ! [I: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Nb)
           => aa_nat_o(P,I) ) ) ) ).

% All_less_Suc
tff(fact_1609_Suc__less__eq2,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Ma)
    <=> ? [M4: nat] :
          ( ( Ma = aa_nat_nat(suc,M4) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M4) ) ) ).

% Suc_less_eq2
tff(fact_1610_less__antisym,axiom,
    ! [Nb: nat,Ma: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Ma)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Ma))
       => ( Ma = Nb ) ) ) ).

% less_antisym
tff(fact_1611_Suc__less__SucD,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Ma)),aa_nat_nat(suc,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb) ) ).

% Suc_less_SucD
tff(fact_1612_less__trans__Suc,axiom,
    ! [I2: nat,J: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),K2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I2)),K2) ) ) ).

% less_trans_Suc
tff(fact_1613_less__Suc__induct,axiom,
    ! [I2: nat,J: nat,P: fun_nat_fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
     => ( ! [I3: nat] : aa_nat_o(aa_nat_fun_nat_o(P,I3),aa_nat_nat(suc,I3))
       => ( ! [I3: nat,J2: nat,K: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),J2)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),K)
               => ( aa_nat_o(aa_nat_fun_nat_o(P,I3),J2)
                 => ( aa_nat_o(aa_nat_fun_nat_o(P,J2),K)
                   => aa_nat_o(aa_nat_fun_nat_o(P,I3),K) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(P,I2),J) ) ) ) ).

% less_Suc_induct
tff(fact_1614_strict__inc__induct,axiom,
    ! [I2: nat,J: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
     => ( ! [I3: nat] :
            ( ( J = aa_nat_nat(suc,I3) )
           => aa_nat_o(P,I3) )
       => ( ! [I3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),J)
             => ( aa_nat_o(P,aa_nat_nat(suc,I3))
               => aa_nat_o(P,I3) ) )
         => aa_nat_o(P,I2) ) ) ) ).

% strict_inc_induct
tff(fact_1615_not__less__less__Suc__eq,axiom,
    ! [Nb: nat,Ma: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Ma)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Ma))
      <=> ( Nb = Ma ) ) ) ).

% not_less_less_Suc_eq
tff(fact_1616_invar__vebt_Ointros_I1_J,axiom,
    ! [A: $o,B: $o] : aa_nat_o(vEBT_invar_vebt(aa_o_VEBT_VEBT(vEBT_Leaf((A)),(B))),aa_nat_nat(suc,zero_zero_nat)) ).

% invar_vebt.intros(1)
tff(fact_1617_transitive__stepwise__le,axiom,
    ! [Ma: nat,Nb: nat,R4: fun_nat_fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => ( ! [X4: nat] : aa_nat_o(aa_nat_fun_nat_o(R4,X4),X4)
       => ( ! [X4: nat,Y4: nat,Z: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(R4,X4),Y4)
             => ( aa_nat_o(aa_nat_fun_nat_o(R4,Y4),Z)
               => aa_nat_o(aa_nat_fun_nat_o(R4,X4),Z) ) )
         => ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(R4,N),aa_nat_nat(suc,N))
           => aa_nat_o(aa_nat_fun_nat_o(R4,Ma),Nb) ) ) ) ) ).

% transitive_stepwise_le
tff(fact_1618_nat__induct__at__least,axiom,
    ! [Ma: nat,Nb: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => ( aa_nat_o(P,Ma)
       => ( ! [N: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),N)
             => ( aa_nat_o(P,N)
               => aa_nat_o(P,aa_nat_nat(suc,N)) ) )
         => aa_nat_o(P,Nb) ) ) ) ).

% nat_induct_at_least
tff(fact_1619_full__nat__induct,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( ! [N: nat] :
          ( ! [M: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M)),N)
             => aa_nat_o(P,M) )
         => aa_nat_o(P,N) )
     => aa_nat_o(P,Nb) ) ).

% full_nat_induct
tff(fact_1620_not__less__eq__eq,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),Ma) ) ).

% not_less_eq_eq
tff(fact_1621_Suc__n__not__le__n,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),Nb) ).

% Suc_n_not_le_n
tff(fact_1622_le__Suc__eq,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),aa_nat_nat(suc,Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
        | ( Ma = aa_nat_nat(suc,Nb) ) ) ) ).

% le_Suc_eq
tff(fact_1623_Suc__le__D,axiom,
    ! [Nb: nat,M5: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),M5)
     => ? [M2: nat] : M5 = aa_nat_nat(suc,M2) ) ).

% Suc_le_D
tff(fact_1624_le__SucI,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),aa_nat_nat(suc,Nb)) ) ).

% le_SucI
tff(fact_1625_le__SucE,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),aa_nat_nat(suc,Nb))
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
       => ( Ma = aa_nat_nat(suc,Nb) ) ) ) ).

% le_SucE
tff(fact_1626_Suc__leD,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Ma)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ).

% Suc_leD
tff(fact_1627_nat__arith_Osuc1,axiom,
    ! [A2: nat,K2: nat,A: nat] :
      ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),A) )
     => ( aa_nat_nat(suc,A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),aa_nat_nat(suc,A)) ) ) ).

% nat_arith.suc1
tff(fact_1628_add__Suc,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,Ma)),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb)) ).

% add_Suc
tff(fact_1629_add__Suc__shift,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,Ma)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),aa_nat_nat(suc,Nb)) ).

% add_Suc_shift
tff(fact_1630_Suc__mult__cancel1,axiom,
    ! [K2: nat,Ma: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K2)),Ma) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K2)),Nb) )
    <=> ( Ma = Nb ) ) ).

% Suc_mult_cancel1
tff(fact_1631_le__numeral__extra_I3_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),zero_zero_real) ).

% le_numeral_extra(3)
tff(fact_1632_le__numeral__extra_I3_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),zero_zero_rat) ).

% le_numeral_extra(3)
tff(fact_1633_le__numeral__extra_I3_J,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),zero_zero_nat) ).

% le_numeral_extra(3)
tff(fact_1634_le__numeral__extra_I3_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),zero_zero_int) ).

% le_numeral_extra(3)
tff(fact_1635_zero__le,axiom,
    ! [X: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),X) ).

% zero_le
tff(fact_1636_field__lbound__gt__zero,axiom,
    ! [D1: real,D22: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D1)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D22)
       => ? [E2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,E2),D1)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,E2),D22) ) ) ) ).

% field_lbound_gt_zero
tff(fact_1637_field__lbound__gt__zero,axiom,
    ! [D1: rat,D22: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),D1)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),D22)
       => ? [E2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E2)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,E2),D1)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,E2),D22) ) ) ) ).

% field_lbound_gt_zero
tff(fact_1638_less__numeral__extra_I3_J,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),zero_zero_real) ).

% less_numeral_extra(3)
tff(fact_1639_less__numeral__extra_I3_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),zero_zero_rat) ).

% less_numeral_extra(3)
tff(fact_1640_less__numeral__extra_I3_J,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),zero_zero_nat) ).

% less_numeral_extra(3)
tff(fact_1641_less__numeral__extra_I3_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),zero_zero_int) ).

% less_numeral_extra(3)
tff(fact_1642_gr__zeroI,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% gr_zeroI
tff(fact_1643_not__less__zero,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),zero_zero_nat) ).

% not_less_zero
tff(fact_1644_gr__implies__not__zero,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
     => ( Nb != zero_zero_nat ) ) ).

% gr_implies_not_zero
tff(fact_1645_zero__less__iff__neq__zero,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
    <=> ( Nb != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
tff(fact_1646_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_zero_complex != numera6690914467698888265omplex(Nb) ).

% zero_neq_numeral
tff(fact_1647_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_zero_real != aa_num_real(numeral_numeral_real,Nb) ).

% zero_neq_numeral
tff(fact_1648_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_zero_rat != aa_num_rat(numeral_numeral_rat,Nb) ).

% zero_neq_numeral
tff(fact_1649_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_zero_nat != aa_num_nat(numeral_numeral_nat,Nb) ).

% zero_neq_numeral
tff(fact_1650_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_zero_int != aa_num_int(numeral_numeral_int,Nb) ).

% zero_neq_numeral
tff(fact_1651_mult__not__zero,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(A),B) != zero_zero_complex )
     => ( ( A != zero_zero_complex )
        & ( B != zero_zero_complex ) ) ) ).

% mult_not_zero
tff(fact_1652_mult__not__zero,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(times_times_real(A),B) != zero_zero_real )
     => ( ( A != zero_zero_real )
        & ( B != zero_zero_real ) ) ) ).

% mult_not_zero
tff(fact_1653_mult__not__zero,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(A),B) != zero_zero_rat )
     => ( ( A != zero_zero_rat )
        & ( B != zero_zero_rat ) ) ) ).

% mult_not_zero
tff(fact_1654_mult__not__zero,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) != zero_zero_nat )
     => ( ( A != zero_zero_nat )
        & ( B != zero_zero_nat ) ) ) ).

% mult_not_zero
tff(fact_1655_mult__not__zero,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) != zero_zero_int )
     => ( ( A != zero_zero_int )
        & ( B != zero_zero_int ) ) ) ).

% mult_not_zero
tff(fact_1656_divisors__zero,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(A),B) = zero_zero_complex )
     => ( ( A = zero_zero_complex )
        | ( B = zero_zero_complex ) ) ) ).

% divisors_zero
tff(fact_1657_divisors__zero,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(times_times_real(A),B) = zero_zero_real )
     => ( ( A = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% divisors_zero
tff(fact_1658_divisors__zero,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(A),B) = zero_zero_rat )
     => ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% divisors_zero
tff(fact_1659_divisors__zero,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = zero_zero_nat )
     => ( ( A = zero_zero_nat )
        | ( B = zero_zero_nat ) ) ) ).

% divisors_zero
tff(fact_1660_divisors__zero,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) = zero_zero_int )
     => ( ( A = zero_zero_int )
        | ( B = zero_zero_int ) ) ) ).

% divisors_zero
tff(fact_1661_no__zero__divisors,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B != zero_zero_complex )
       => ( aa_complex_complex(times_times_complex(A),B) != zero_zero_complex ) ) ) ).

% no_zero_divisors
tff(fact_1662_no__zero__divisors,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( aa_real_real(times_times_real(A),B) != zero_zero_real ) ) ) ).

% no_zero_divisors
tff(fact_1663_no__zero__divisors,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( aa_rat_rat(times_times_rat(A),B) != zero_zero_rat ) ) ) ).

% no_zero_divisors
tff(fact_1664_no__zero__divisors,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( B != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) != zero_zero_nat ) ) ) ).

% no_zero_divisors
tff(fact_1665_no__zero__divisors,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( B != zero_zero_int )
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) != zero_zero_int ) ) ) ).

% no_zero_divisors
tff(fact_1666_mult__left__cancel,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( aa_complex_complex(times_times_complex(C),A) = aa_complex_complex(times_times_complex(C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_1667_mult__left__cancel,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(times_times_real(C),A) = aa_real_real(times_times_real(C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_1668_mult__left__cancel,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(times_times_rat(C),A) = aa_rat_rat(times_times_rat(C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_1669_mult__left__cancel,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_1670_mult__left__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C),A) = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_1671_mult__right__cancel,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( aa_complex_complex(times_times_complex(A),C) = aa_complex_complex(times_times_complex(B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_1672_mult__right__cancel,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(times_times_real(A),C) = aa_real_real(times_times_real(B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_1673_mult__right__cancel,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(times_times_rat(A),C) = aa_rat_rat(times_times_rat(B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_1674_mult__right__cancel,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_1675_mult__right__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),C) = aa_int_int(aa_int_fun_int_int(times_times_int,B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_1676_zero__neq__one,axiom,
    zero_zero_complex != one_one_complex ).

% zero_neq_one
tff(fact_1677_zero__neq__one,axiom,
    zero_zero_real != one_one_real ).

% zero_neq_one
tff(fact_1678_zero__neq__one,axiom,
    zero_zero_rat != one_one_rat ).

% zero_neq_one
tff(fact_1679_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
tff(fact_1680_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
tff(fact_1681_verit__sum__simplify,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(A),zero_zero_complex) = A ).

% verit_sum_simplify
tff(fact_1682_verit__sum__simplify,axiom,
    ! [A: real] : aa_real_real(plus_plus_real(A),zero_zero_real) = A ).

% verit_sum_simplify
tff(fact_1683_verit__sum__simplify,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(A),zero_zero_rat) = A ).

% verit_sum_simplify
tff(fact_1684_verit__sum__simplify,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),zero_zero_nat) = A ).

% verit_sum_simplify
tff(fact_1685_verit__sum__simplify,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),zero_zero_int) = A ).

% verit_sum_simplify
tff(fact_1686_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(zero_zero_complex),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_1687_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: real] : aa_real_real(plus_plus_real(zero_zero_real),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_1688_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(zero_zero_rat),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_1689_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_1690_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_1691_add_Ocomm__neutral,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(A),zero_zero_complex) = A ).

% add.comm_neutral
tff(fact_1692_add_Ocomm__neutral,axiom,
    ! [A: real] : aa_real_real(plus_plus_real(A),zero_zero_real) = A ).

% add.comm_neutral
tff(fact_1693_add_Ocomm__neutral,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(A),zero_zero_rat) = A ).

% add.comm_neutral
tff(fact_1694_add_Ocomm__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),zero_zero_nat) = A ).

% add.comm_neutral
tff(fact_1695_add_Ocomm__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),zero_zero_int) = A ).

% add.comm_neutral
tff(fact_1696_add_Ogroup__left__neutral,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(zero_zero_complex),A) = A ).

% add.group_left_neutral
tff(fact_1697_add_Ogroup__left__neutral,axiom,
    ! [A: real] : aa_real_real(plus_plus_real(zero_zero_real),A) = A ).

% add.group_left_neutral
tff(fact_1698_add_Ogroup__left__neutral,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(zero_zero_rat),A) = A ).

% add.group_left_neutral
tff(fact_1699_add_Ogroup__left__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),A) = A ).

% add.group_left_neutral
tff(fact_1700_power__not__zero,axiom,
    ! [A: rat,Nb: nat] :
      ( ( A != zero_zero_rat )
     => ( aa_nat_rat(power_power_rat(A),Nb) != zero_zero_rat ) ) ).

% power_not_zero
tff(fact_1701_power__not__zero,axiom,
    ! [A: nat,Nb: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb) != zero_zero_nat ) ) ).

% power_not_zero
tff(fact_1702_power__not__zero,axiom,
    ! [A: real,Nb: nat] :
      ( ( A != zero_zero_real )
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb) != zero_zero_real ) ) ).

% power_not_zero
tff(fact_1703_power__not__zero,axiom,
    ! [A: int,Nb: nat] :
      ( ( A != zero_zero_int )
     => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb) != zero_zero_int ) ) ).

% power_not_zero
tff(fact_1704_power__not__zero,axiom,
    ! [A: complex,Nb: nat] :
      ( ( A != zero_zero_complex )
     => ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),Nb) != zero_zero_complex ) ) ).

% power_not_zero
tff(fact_1705_num_Osize_I4_J,axiom,
    aa_num_nat(size_size_num,one) = zero_zero_nat ).

% num.size(4)
tff(fact_1706_mod__Suc__Suc__eq,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),Nb)))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,Ma))),Nb) ).

% mod_Suc_Suc_eq
tff(fact_1707_mod__Suc__eq,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),Nb))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Ma)),Nb) ).

% mod_Suc_eq
tff(fact_1708_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat) ).

% bot_nat_0.extremum_strict
tff(fact_1709_gr0I,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% gr0I
tff(fact_1710_not__gr0,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
    <=> ( Nb = zero_zero_nat ) ) ).

% not_gr0
tff(fact_1711_not__less0,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),zero_zero_nat) ).

% not_less0
tff(fact_1712_less__zeroE,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),zero_zero_nat) ).

% less_zeroE
tff(fact_1713_gr__implies__not0,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
     => ( Nb != zero_zero_nat ) ) ).

% gr_implies_not0
tff(fact_1714_infinite__descent0,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ! [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
           => ( ~ aa_nat_o(P,N)
             => ? [M: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
                  & ~ aa_nat_o(P,M) ) ) )
       => aa_nat_o(P,Nb) ) ) ).

% infinite_descent0
tff(fact_1715_le__0__eq,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),zero_zero_nat)
    <=> ( Nb = zero_zero_nat ) ) ).

% le_0_eq
tff(fact_1716_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_uniqueI
tff(fact_1717_bot__nat__0_Oextremum__unique,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
    <=> ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_unique
tff(fact_1718_less__eq__nat_Osimps_I1_J,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Nb) ).

% less_eq_nat.simps(1)
tff(fact_1719_plus__nat_Oadd__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),Nb) = Nb ).

% plus_nat.add_0
tff(fact_1720_add__eq__self__zero,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb) = Ma )
     => ( Nb = zero_zero_nat ) ) ).

% add_eq_self_zero
tff(fact_1721_nat__mult__eq__cancel__disj,axiom,
    ! [K2: nat,Ma: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Ma) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Nb) )
    <=> ( ( K2 = zero_zero_nat )
        | ( Ma = Nb ) ) ) ).

% nat_mult_eq_cancel_disj
tff(fact_1722_mult__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat),Nb) = zero_zero_nat ).

% mult_0
tff(fact_1723_real__arch__pow__inv,axiom,
    ! [Y2: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),one_one_real)
       => ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),N)),Y2) ) ) ).

% real_arch_pow_inv
tff(fact_1724_infinity__ne__i0,axiom,
    extend5688581933313929465d_enat != zero_z5237406670263579293d_enat ).

% infinity_ne_i0
tff(fact_1725_VEBT_Osize_I4_J,axiom,
    ! [X21: $o,X22: $o] : aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,aa_o_VEBT_VEBT(vEBT_Leaf((X21)),(X22))) = zero_zero_nat ).

% VEBT.size(4)
tff(fact_1726_power__Suc__le__self,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Nb))),A) ) ) ).

% power_Suc_le_self
tff(fact_1727_power__Suc__le__self,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Nb))),A) ) ) ).

% power_Suc_le_self
tff(fact_1728_power__Suc__le__self,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb))),A) ) ) ).

% power_Suc_le_self
tff(fact_1729_power__Suc__le__self,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(suc,Nb))),A) ) ) ).

% power_Suc_le_self
tff(fact_1730_power__Suc__less__one,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Nb))),one_one_real) ) ) ).

% power_Suc_less_one
tff(fact_1731_power__Suc__less__one,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Nb))),one_one_rat) ) ) ).

% power_Suc_less_one
tff(fact_1732_power__Suc__less__one,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb))),one_one_nat) ) ) ).

% power_Suc_less_one
tff(fact_1733_power__Suc__less__one,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(suc,Nb))),one_one_int) ) ) ).

% power_Suc_less_one
tff(fact_1734_not__iless0,axiom,
    ! [Nb: extended_enat] : ~ aa_Extended_enat_o(ord_le72135733267957522d_enat(Nb),zero_z5237406670263579293d_enat) ).

% not_iless0
tff(fact_1735_numeral__2__eq__2,axiom,
    aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) = aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat)) ).

% numeral_2_eq_2
tff(fact_1736_iadd__is__0,axiom,
    ! [Ma: extended_enat,Nb: extended_enat] :
      ( ( plus_p3455044024723400733d_enat(Ma,Nb) = zero_z5237406670263579293d_enat )
    <=> ( ( Ma = zero_z5237406670263579293d_enat )
        & ( Nb = zero_z5237406670263579293d_enat ) ) ) ).

% iadd_is_0
tff(fact_1737_ile0__eq,axiom,
    ! [Nb: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(Nb),zero_z5237406670263579293d_enat)
    <=> ( Nb = zero_z5237406670263579293d_enat ) ) ).

% ile0_eq
tff(fact_1738_i0__lb,axiom,
    ! [Nb: extended_enat] : aa_Extended_enat_o(ord_le2932123472753598470d_enat(zero_z5237406670263579293d_enat),Nb) ).

% i0_lb
tff(fact_1739_power__eq__iff__eq__base,axiom,
    ! [Nb: nat,A: real,B: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
         => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_1740_power__eq__iff__eq__base,axiom,
    ! [Nb: nat,A: rat,B: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
         => ( ( aa_nat_rat(power_power_rat(A),Nb) = aa_nat_rat(power_power_rat(B),Nb) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_1741_power__eq__iff__eq__base,axiom,
    ! [Nb: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_1742_power__eq__iff__eq__base,axiom,
    ! [Nb: nat,A: int,B: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
         => ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb) = aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Nb) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_1743_power__eq__imp__eq__base,axiom,
    ! [A: real,Nb: nat,B: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_1744_power__eq__imp__eq__base,axiom,
    ! [A: rat,Nb: nat,B: rat] :
      ( ( aa_nat_rat(power_power_rat(A),Nb) = aa_nat_rat(power_power_rat(B),Nb) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_1745_power__eq__imp__eq__base,axiom,
    ! [A: nat,Nb: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_1746_power__eq__imp__eq__base,axiom,
    ! [A: int,Nb: nat,B: int] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb) = aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Nb) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_1747_imult__is__0,axiom,
    ! [Ma: extended_enat,Nb: extended_enat] :
      ( ( aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(Ma),Nb) = zero_z5237406670263579293d_enat )
    <=> ( ( Ma = zero_z5237406670263579293d_enat )
        | ( Nb = zero_z5237406670263579293d_enat ) ) ) ).

% imult_is_0
tff(fact_1748_lambda__zero,axiom,
    aTP_Lamm_as = times_times_complex(zero_zero_complex) ).

% lambda_zero
tff(fact_1749_lambda__zero,axiom,
    aTP_Lamm_at = times_times_real(zero_zero_real) ).

% lambda_zero
tff(fact_1750_lambda__zero,axiom,
    aTP_Lamm_au = times_times_rat(zero_zero_rat) ).

% lambda_zero
tff(fact_1751_lambda__zero,axiom,
    aTP_Lamm_av = aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat) ).

% lambda_zero
tff(fact_1752_lambda__zero,axiom,
    aTP_Lamm_aw = aa_int_fun_int_int(times_times_int,zero_zero_int) ).

% lambda_zero
tff(fact_1753_less__2__cases,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))
     => ( ( Nb = zero_zero_nat )
        | ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases
tff(fact_1754_less__2__cases__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))
    <=> ( ( Nb = zero_zero_nat )
        | ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases_iff
tff(fact_1755_power__strict__mono,axiom,
    ! [A: real,B: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)) ) ) ) ).

% power_strict_mono
tff(fact_1756_power__strict__mono,axiom,
    ! [A: rat,B: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb)) ) ) ) ).

% power_strict_mono
tff(fact_1757_power__strict__mono,axiom,
    ! [A: nat,B: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)) ) ) ) ).

% power_strict_mono
tff(fact_1758_power__strict__mono,axiom,
    ! [A: int,B: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Nb)) ) ) ) ).

% power_strict_mono
tff(fact_1759_split__div_H,axiom,
    ! [P: fun_nat_o,Ma: nat,Nb: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),Nb))
    <=> ( ( ( Nb = zero_zero_nat )
          & aa_nat_o(P,zero_zero_nat) )
        | ? [Q5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q5)),Ma)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(suc,Q5)))
            & aa_nat_o(P,Q5) ) ) ) ).

% split_div'
tff(fact_1760_power__Suc2,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),aa_nat_nat(suc,Nb)) = aa_complex_complex(times_times_complex(aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),Nb)),A) ).

% power_Suc2
tff(fact_1761_power__Suc2,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Nb)) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),A) ).

% power_Suc2
tff(fact_1762_power__Suc2,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Nb)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),Nb)),A) ).

% power_Suc2
tff(fact_1763_power__Suc2,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),A) ).

% power_Suc2
tff(fact_1764_power__Suc2,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)),A) ).

% power_Suc2
tff(fact_1765_power__Suc,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),aa_nat_nat(suc,Nb)) = aa_complex_complex(times_times_complex(A),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),Nb)) ).

% power_Suc
tff(fact_1766_power__Suc,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Nb)) = aa_real_real(times_times_real(A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ).

% power_Suc
tff(fact_1767_power__Suc,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Nb)) = aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),Nb)) ).

% power_Suc
tff(fact_1768_power__Suc,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ).

% power_Suc
tff(fact_1769_power__Suc,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)) ).

% power_Suc
tff(fact_1770_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_real,Nb: nat,N3: nat] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,N)),aa_nat_real(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N3)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,Nb)),aa_nat_real(F,N3)) ) ) ).

% lift_Suc_mono_less
tff(fact_1771_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_rat,Nb: nat,N3: nat] :
      ( ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,N)),aa_nat_rat(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N3)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,Nb)),aa_nat_rat(F,N3)) ) ) ).

% lift_Suc_mono_less
tff(fact_1772_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_num,Nb: nat,N3: nat] :
      ( ! [N: nat] : aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(F,N)),aa_nat_num(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N3)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(F,Nb)),aa_nat_num(F,N3)) ) ) ).

% lift_Suc_mono_less
tff(fact_1773_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_nat,Nb: nat,N3: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,N)),aa_nat_nat(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N3)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,Nb)),aa_nat_nat(F,N3)) ) ) ).

% lift_Suc_mono_less
tff(fact_1774_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_int,Nb: nat,N3: nat] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,N)),aa_nat_int(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N3)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,Nb)),aa_nat_int(F,N3)) ) ) ).

% lift_Suc_mono_less
tff(fact_1775_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_real,Nb: nat,Ma: nat] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,N)),aa_nat_real(F,aa_nat_nat(suc,N)))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,Nb)),aa_nat_real(F,Ma))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Ma) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_1776_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_rat,Nb: nat,Ma: nat] :
      ( ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,N)),aa_nat_rat(F,aa_nat_nat(suc,N)))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,Nb)),aa_nat_rat(F,Ma))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Ma) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_1777_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_num,Nb: nat,Ma: nat] :
      ( ! [N: nat] : aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(F,N)),aa_nat_num(F,aa_nat_nat(suc,N)))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(F,Nb)),aa_nat_num(F,Ma))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Ma) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_1778_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_nat,Nb: nat,Ma: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,N)),aa_nat_nat(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,Nb)),aa_nat_nat(F,Ma))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Ma) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_1779_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_int,Nb: nat,Ma: nat] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,N)),aa_nat_int(F,aa_nat_nat(suc,N)))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,Nb)),aa_nat_int(F,Ma))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Ma) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_1780_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_set_int,Nb: nat,N3: nat] :
      ( ! [N: nat] : ord_less_eq_set_int(aa_nat_set_int(F,N),aa_nat_set_int(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N3)
       => ord_less_eq_set_int(aa_nat_set_int(F,Nb),aa_nat_set_int(F,N3)) ) ) ).

% lift_Suc_mono_le
tff(fact_1781_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_rat,Nb: nat,N3: nat] :
      ( ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,N)),aa_nat_rat(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N3)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,Nb)),aa_nat_rat(F,N3)) ) ) ).

% lift_Suc_mono_le
tff(fact_1782_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_num,Nb: nat,N3: nat] :
      ( ! [N: nat] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,N)),aa_nat_num(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N3)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,Nb)),aa_nat_num(F,N3)) ) ) ).

% lift_Suc_mono_le
tff(fact_1783_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_nat,Nb: nat,N3: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,N)),aa_nat_nat(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N3)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,Nb)),aa_nat_nat(F,N3)) ) ) ).

% lift_Suc_mono_le
tff(fact_1784_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_int,Nb: nat,N3: nat] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,N)),aa_nat_int(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N3)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,Nb)),aa_nat_int(F,N3)) ) ) ).

% lift_Suc_mono_le
tff(fact_1785_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_set_int,Nb: nat,N3: nat] :
      ( ! [N: nat] : ord_less_eq_set_int(aa_nat_set_int(F,aa_nat_nat(suc,N)),aa_nat_set_int(F,N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N3)
       => ord_less_eq_set_int(aa_nat_set_int(F,N3),aa_nat_set_int(F,Nb)) ) ) ).

% lift_Suc_antimono_le
tff(fact_1786_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_rat,Nb: nat,N3: nat] :
      ( ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,aa_nat_nat(suc,N))),aa_nat_rat(F,N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N3)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,N3)),aa_nat_rat(F,Nb)) ) ) ).

% lift_Suc_antimono_le
tff(fact_1787_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_num,Nb: nat,N3: nat] :
      ( ! [N: nat] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,aa_nat_nat(suc,N))),aa_nat_num(F,N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N3)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,N3)),aa_nat_num(F,Nb)) ) ) ).

% lift_Suc_antimono_le
tff(fact_1788_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_nat,Nb: nat,N3: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,aa_nat_nat(suc,N))),aa_nat_nat(F,N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N3)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,N3)),aa_nat_nat(F,Nb)) ) ) ).

% lift_Suc_antimono_le
tff(fact_1789_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_int,Nb: nat,N3: nat] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,aa_nat_nat(suc,N))),aa_nat_int(F,N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N3)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,N3)),aa_nat_int(F,Nb)) ) ) ).

% lift_Suc_antimono_le
tff(fact_1790_Suc__times__mod__eq,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Ma)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Nb))),Ma) = one_one_nat ) ) ).

% Suc_times_mod_eq
tff(fact_1791_le__imp__less__Suc,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(suc,Nb)) ) ).

% le_imp_less_Suc
tff(fact_1792_less__eq__Suc__le,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Ma)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),Ma) ) ).

% less_eq_Suc_le
tff(fact_1793_less__Suc__eq__le,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ).

% less_Suc_eq_le
tff(fact_1794_le__less__Suc__eq,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Ma))
      <=> ( Nb = Ma ) ) ) ).

% le_less_Suc_eq
tff(fact_1795_Suc__le__lessD,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Ma)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb) ) ).

% Suc_le_lessD
tff(fact_1796_inc__induct,axiom,
    ! [I2: nat,J: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => ( aa_nat_o(P,J)
       => ( ! [N: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),N)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),J)
               => ( aa_nat_o(P,aa_nat_nat(suc,N))
                 => aa_nat_o(P,N) ) ) )
         => aa_nat_o(P,I2) ) ) ) ).

% inc_induct
tff(fact_1797_dec__induct,axiom,
    ! [I2: nat,J: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => ( aa_nat_o(P,I2)
       => ( ! [N: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),N)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),J)
               => ( aa_nat_o(P,N)
                 => aa_nat_o(P,aa_nat_nat(suc,N)) ) ) )
         => aa_nat_o(P,J) ) ) ) ).

% dec_induct
tff(fact_1798_Suc__le__eq,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Ma)),Nb)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb) ) ).

% Suc_le_eq
tff(fact_1799_Suc__leI,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Ma)),Nb) ) ).

% Suc_leI
tff(fact_1800_less__natE,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
     => ~ ! [Q3: nat] : Nb != aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Q3)) ) ).

% less_natE
tff(fact_1801_less__add__Suc1,axiom,
    ! [I2: nat,Ma: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),Ma))) ).

% less_add_Suc1
tff(fact_1802_less__add__Suc2,axiom,
    ! [I2: nat,Ma: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),I2))) ).

% less_add_Suc2
tff(fact_1803_less__iff__Suc__add,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
    <=> ? [K3: nat] : Nb = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),K3)) ) ).

% less_iff_Suc_add
tff(fact_1804_less__imp__Suc__add,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
     => ? [K: nat] : Nb = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),K)) ) ).

% less_imp_Suc_add
tff(fact_1805_Suc__mult__less__cancel1,axiom,
    ! [K2: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K2)),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K2)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb) ) ).

% Suc_mult_less_cancel1
tff(fact_1806_Suc__mult__le__cancel1,axiom,
    ! [K2: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K2)),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K2)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ).

% Suc_mult_le_cancel1
tff(fact_1807_dbl__def,axiom,
    ! [X: real] : aa_real_real(neg_numeral_dbl_real,X) = aa_real_real(plus_plus_real(X),X) ).

% dbl_def
tff(fact_1808_dbl__def,axiom,
    ! [X: rat] : aa_rat_rat(neg_numeral_dbl_rat,X) = aa_rat_rat(plus_plus_rat(X),X) ).

% dbl_def
tff(fact_1809_dbl__def,axiom,
    ! [X: int] : aa_int_int(neg_numeral_dbl_int,X) = aa_int_int(aa_int_fun_int_int(plus_plus_int,X),X) ).

% dbl_def
tff(fact_1810_mult__Suc,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Ma)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Nb)) ).

% mult_Suc
tff(fact_1811_Suc__div__le__mono,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,Ma)),Nb)) ).

% Suc_div_le_mono
tff(fact_1812_Suc__eq__plus1__left,axiom,
    ! [Nb: nat] : aa_nat_nat(suc,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),Nb) ).

% Suc_eq_plus1_left
tff(fact_1813_plus__1__eq__Suc,axiom,
    aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat) = suc ).

% plus_1_eq_Suc
tff(fact_1814_Suc__eq__plus1,axiom,
    ! [Nb: nat] : aa_nat_nat(suc,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat) ).

% Suc_eq_plus1
tff(fact_1815_zero__le__numeral,axiom,
    ! [Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_num_real(numeral_numeral_real,Nb)) ).

% zero_le_numeral
tff(fact_1816_zero__le__numeral,axiom,
    ! [Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,Nb)) ).

% zero_le_numeral
tff(fact_1817_zero__le__numeral,axiom,
    ! [Nb: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,Nb)) ).

% zero_le_numeral
tff(fact_1818_zero__le__numeral,axiom,
    ! [Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_num_int(numeral_numeral_int,Nb)) ).

% zero_le_numeral
tff(fact_1819_not__numeral__le__zero,axiom,
    ! [Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Nb)),zero_zero_real) ).

% not_numeral_le_zero
tff(fact_1820_not__numeral__le__zero,axiom,
    ! [Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Nb)),zero_zero_rat) ).

% not_numeral_le_zero
tff(fact_1821_not__numeral__le__zero,axiom,
    ! [Nb: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Nb)),zero_zero_nat) ).

% not_numeral_le_zero
tff(fact_1822_not__numeral__le__zero,axiom,
    ! [Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Nb)),zero_zero_int) ).

% not_numeral_le_zero
tff(fact_1823_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_1824_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_1825_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_1826_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_1827_zero__le__mult__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(times_times_real(A),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) ) ) ) ).

% zero_le_mult_iff
tff(fact_1828_zero__le__mult__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) ) ) ) ).

% zero_le_mult_iff
tff(fact_1829_zero__le__mult__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) ) ) ) ).

% zero_le_mult_iff
tff(fact_1830_mult__nonneg__nonpos2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(B),A)),zero_zero_real) ) ) ).

% mult_nonneg_nonpos2
tff(fact_1831_mult__nonneg__nonpos2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(B),A)),zero_zero_rat) ) ) ).

% mult_nonneg_nonpos2
tff(fact_1832_mult__nonneg__nonpos2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),zero_zero_nat) ) ) ).

% mult_nonneg_nonpos2
tff(fact_1833_mult__nonneg__nonpos2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),zero_zero_int) ) ) ).

% mult_nonneg_nonpos2
tff(fact_1834_mult__nonpos__nonneg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),B)),zero_zero_real) ) ) ).

% mult_nonpos_nonneg
tff(fact_1835_mult__nonpos__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat) ) ) ).

% mult_nonpos_nonneg
tff(fact_1836_mult__nonpos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ) ).

% mult_nonpos_nonneg
tff(fact_1837_mult__nonpos__nonneg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int) ) ) ).

% mult_nonpos_nonneg
tff(fact_1838_mult__nonneg__nonpos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),B)),zero_zero_real) ) ) ).

% mult_nonneg_nonpos
tff(fact_1839_mult__nonneg__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat) ) ) ).

% mult_nonneg_nonpos
tff(fact_1840_mult__nonneg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ) ).

% mult_nonneg_nonpos
tff(fact_1841_mult__nonneg__nonpos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int) ) ) ).

% mult_nonneg_nonpos
tff(fact_1842_mult__nonneg__nonneg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(times_times_real(A),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_1843_mult__nonneg__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_1844_mult__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_1845_mult__nonneg__nonneg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_1846_split__mult__neg__le,axiom,
    ! [A: real,B: real] :
      ( ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),B)),zero_zero_real) ) ).

% split_mult_neg_le
tff(fact_1847_split__mult__neg__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat) ) ).

% split_mult_neg_le
tff(fact_1848_split__mult__neg__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat) )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B) ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ).

% split_mult_neg_le
tff(fact_1849_split__mult__neg__le,axiom,
    ! [A: int,B: int] :
      ( ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int) ) ).

% split_mult_neg_le
tff(fact_1850_mult__le__0__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),B)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) ) ) ).

% mult_le_0_iff
tff(fact_1851_mult__le__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) ) ) ).

% mult_le_0_iff
tff(fact_1852_mult__le__0__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) ) ) ) ).

% mult_le_0_iff
tff(fact_1853_mult__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),C)) ) ) ).

% mult_right_mono
tff(fact_1854_mult__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) ) ) ).

% mult_right_mono
tff(fact_1855_mult__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ) ).

% mult_right_mono
tff(fact_1856_mult__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% mult_right_mono
tff(fact_1857_mult__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),C)) ) ) ).

% mult_right_mono_neg
tff(fact_1858_mult__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) ) ) ).

% mult_right_mono_neg
tff(fact_1859_mult__right__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% mult_right_mono_neg
tff(fact_1860_mult__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B)) ) ) ).

% mult_left_mono
tff(fact_1861_mult__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)) ) ) ).

% mult_left_mono
tff(fact_1862_mult__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ).

% mult_left_mono
tff(fact_1863_mult__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% mult_left_mono
tff(fact_1864_mult__nonpos__nonpos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(times_times_real(A),B)) ) ) ).

% mult_nonpos_nonpos
tff(fact_1865_mult__nonpos__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)) ) ) ).

% mult_nonpos_nonpos
tff(fact_1866_mult__nonpos__nonpos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ) ).

% mult_nonpos_nonpos
tff(fact_1867_mult__left__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B)) ) ) ).

% mult_left_mono_neg
tff(fact_1868_mult__left__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)) ) ) ).

% mult_left_mono_neg
tff(fact_1869_mult__left__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% mult_left_mono_neg
tff(fact_1870_split__mult__pos__le,axiom,
    ! [A: real,B: real] :
      ( ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(times_times_real(A),B)) ) ).

% split_mult_pos_le
tff(fact_1871_split__mult__pos__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)) ) ).

% split_mult_pos_le
tff(fact_1872_split__mult__pos__le,axiom,
    ! [A: int,B: int] :
      ( ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ).

% split_mult_pos_le
tff(fact_1873_zero__le__square,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(times_times_real(A),A)) ).

% zero_le_square
tff(fact_1874_zero__le__square,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),A)) ).

% zero_le_square
tff(fact_1875_zero__le__square,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),A)) ).

% zero_le_square
tff(fact_1876_mult__mono_H,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),D)) ) ) ) ) ).

% mult_mono'
tff(fact_1877_mult__mono_H,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D)) ) ) ) ) ).

% mult_mono'
tff(fact_1878_mult__mono_H,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_mono'
tff(fact_1879_mult__mono_H,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ) ).

% mult_mono'
tff(fact_1880_mult__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),D)) ) ) ) ) ).

% mult_mono
tff(fact_1881_mult__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D)) ) ) ) ) ).

% mult_mono
tff(fact_1882_mult__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_mono
tff(fact_1883_mult__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ) ).

% mult_mono
tff(fact_1884_zero__less__numeral,axiom,
    ! [Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_num_real(numeral_numeral_real,Nb)) ).

% zero_less_numeral
tff(fact_1885_zero__less__numeral,axiom,
    ! [Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,Nb)) ).

% zero_less_numeral
tff(fact_1886_zero__less__numeral,axiom,
    ! [Nb: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,Nb)) ).

% zero_less_numeral
tff(fact_1887_zero__less__numeral,axiom,
    ! [Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_num_int(numeral_numeral_int,Nb)) ).

% zero_less_numeral
tff(fact_1888_not__numeral__less__zero,axiom,
    ! [Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,Nb)),zero_zero_real) ).

% not_numeral_less_zero
tff(fact_1889_not__numeral__less__zero,axiom,
    ! [Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,Nb)),zero_zero_rat) ).

% not_numeral_less_zero
tff(fact_1890_not__numeral__less__zero,axiom,
    ! [Nb: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Nb)),zero_zero_nat) ).

% not_numeral_less_zero
tff(fact_1891_not__numeral__less__zero,axiom,
    ! [Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Nb)),zero_zero_int) ).

% not_numeral_less_zero
tff(fact_1892_zero__less__one__class_Ozero__le__one,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),one_one_real) ).

% zero_less_one_class.zero_le_one
tff(fact_1893_zero__less__one__class_Ozero__le__one,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),one_one_rat) ).

% zero_less_one_class.zero_le_one
tff(fact_1894_zero__less__one__class_Ozero__le__one,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),one_one_nat) ).

% zero_less_one_class.zero_le_one
tff(fact_1895_zero__less__one__class_Ozero__le__one,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),one_one_int) ).

% zero_less_one_class.zero_le_one
tff(fact_1896_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),one_one_real) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_1897_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),one_one_rat) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_1898_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),one_one_nat) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_1899_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),one_one_int) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_1900_not__one__le__zero,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),zero_zero_real) ).

% not_one_le_zero
tff(fact_1901_not__one__le__zero,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),zero_zero_rat) ).

% not_one_le_zero
tff(fact_1902_not__one__le__zero,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),zero_zero_nat) ).

% not_one_le_zero
tff(fact_1903_not__one__le__zero,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),zero_zero_int) ).

% not_one_le_zero
tff(fact_1904_add__decreasing,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(A),C)),B) ) ) ).

% add_decreasing
tff(fact_1905_add__decreasing,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),C)),B) ) ) ).

% add_decreasing
tff(fact_1906_add__decreasing,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),B) ) ) ).

% add_decreasing
tff(fact_1907_add__decreasing,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B) ) ) ).

% add_decreasing
tff(fact_1908_add__increasing,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(plus_plus_real(A),C)) ) ) ).

% add_increasing
tff(fact_1909_add__increasing,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(plus_plus_rat(A),C)) ) ) ).

% add_increasing
tff(fact_1910_add__increasing,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% add_increasing
tff(fact_1911_add__increasing,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).

% add_increasing
tff(fact_1912_add__decreasing2,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(A),C)),B) ) ) ).

% add_decreasing2
tff(fact_1913_add__decreasing2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),C)),B) ) ) ).

% add_decreasing2
tff(fact_1914_add__decreasing2,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),B) ) ) ).

% add_decreasing2
tff(fact_1915_add__decreasing2,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B) ) ) ).

% add_decreasing2
tff(fact_1916_add__increasing2,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(plus_plus_real(A),C)) ) ) ).

% add_increasing2
tff(fact_1917_add__increasing2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(plus_plus_rat(A),C)) ) ) ).

% add_increasing2
tff(fact_1918_add__increasing2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% add_increasing2
tff(fact_1919_add__increasing2,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).

% add_increasing2
tff(fact_1920_add__nonneg__nonneg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(plus_plus_real(A),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_1921_add__nonneg__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_1922_add__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_1923_add__nonneg__nonneg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_1924_add__nonpos__nonpos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(A),B)),zero_zero_real) ) ) ).

% add_nonpos_nonpos
tff(fact_1925_add__nonpos__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),B)),zero_zero_rat) ) ) ).

% add_nonpos_nonpos
tff(fact_1926_add__nonpos__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat) ) ) ).

% add_nonpos_nonpos
tff(fact_1927_add__nonpos__nonpos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int) ) ) ).

% add_nonpos_nonpos
tff(fact_1928_add__nonneg__eq__0__iff,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y2)
       => ( ( aa_real_real(plus_plus_real(X),Y2) = zero_zero_real )
        <=> ( ( X = zero_zero_real )
            & ( Y2 = zero_zero_real ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_1929_add__nonneg__eq__0__iff,axiom,
    ! [X: rat,Y2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y2)
       => ( ( aa_rat_rat(plus_plus_rat(X),Y2) = zero_zero_rat )
        <=> ( ( X = zero_zero_rat )
            & ( Y2 = zero_zero_rat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_1930_add__nonneg__eq__0__iff,axiom,
    ! [X: nat,Y2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),X)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y2)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y2) = zero_zero_nat )
        <=> ( ( X = zero_zero_nat )
            & ( Y2 = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_1931_add__nonneg__eq__0__iff,axiom,
    ! [X: int,Y2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y2)
       => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y2) = zero_zero_int )
        <=> ( ( X = zero_zero_int )
            & ( Y2 = zero_zero_int ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_1932_add__nonpos__eq__0__iff,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),zero_zero_real)
       => ( ( aa_real_real(plus_plus_real(X),Y2) = zero_zero_real )
        <=> ( ( X = zero_zero_real )
            & ( Y2 = zero_zero_real ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_1933_add__nonpos__eq__0__iff,axiom,
    ! [X: rat,Y2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y2),zero_zero_rat)
       => ( ( aa_rat_rat(plus_plus_rat(X),Y2) = zero_zero_rat )
        <=> ( ( X = zero_zero_rat )
            & ( Y2 = zero_zero_rat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_1934_add__nonpos__eq__0__iff,axiom,
    ! [X: nat,Y2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y2),zero_zero_nat)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y2) = zero_zero_nat )
        <=> ( ( X = zero_zero_nat )
            & ( Y2 = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_1935_add__nonpos__eq__0__iff,axiom,
    ! [X: int,Y2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y2),zero_zero_int)
       => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y2) = zero_zero_int )
        <=> ( ( X = zero_zero_int )
            & ( Y2 = zero_zero_int ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_1936_mult__neg__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(times_times_real(A),B)) ) ) ).

% mult_neg_neg
tff(fact_1937_mult__neg__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)) ) ) ).

% mult_neg_neg
tff(fact_1938_mult__neg__neg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ) ).

% mult_neg_neg
tff(fact_1939_not__square__less__zero,axiom,
    ! [A: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),A)),zero_zero_real) ).

% not_square_less_zero
tff(fact_1940_not__square__less__zero,axiom,
    ! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),A)),zero_zero_rat) ).

% not_square_less_zero
tff(fact_1941_not__square__less__zero,axiom,
    ! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),A)),zero_zero_int) ).

% not_square_less_zero
tff(fact_1942_mult__less__0__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),B)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ) ) ).

% mult_less_0_iff
tff(fact_1943_mult__less__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ) ) ).

% mult_less_0_iff
tff(fact_1944_mult__less__0__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ) ) ).

% mult_less_0_iff
tff(fact_1945_mult__neg__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),B)),zero_zero_real) ) ) ).

% mult_neg_pos
tff(fact_1946_mult__neg__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat) ) ) ).

% mult_neg_pos
tff(fact_1947_mult__neg__pos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ) ).

% mult_neg_pos
tff(fact_1948_mult__neg__pos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int) ) ) ).

% mult_neg_pos
tff(fact_1949_mult__pos__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),B)),zero_zero_real) ) ) ).

% mult_pos_neg
tff(fact_1950_mult__pos__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat) ) ) ).

% mult_pos_neg
tff(fact_1951_mult__pos__neg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ) ).

% mult_pos_neg
tff(fact_1952_mult__pos__neg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int) ) ) ).

% mult_pos_neg
tff(fact_1953_mult__pos__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(times_times_real(A),B)) ) ) ).

% mult_pos_pos
tff(fact_1954_mult__pos__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B)) ) ) ).

% mult_pos_pos
tff(fact_1955_mult__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) ) ) ).

% mult_pos_pos
tff(fact_1956_mult__pos__pos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ) ) ).

% mult_pos_pos
tff(fact_1957_mult__pos__neg2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(B),A)),zero_zero_real) ) ) ).

% mult_pos_neg2
tff(fact_1958_mult__pos__neg2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(B),A)),zero_zero_rat) ) ) ).

% mult_pos_neg2
tff(fact_1959_mult__pos__neg2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),zero_zero_nat) ) ) ).

% mult_pos_neg2
tff(fact_1960_mult__pos__neg2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),zero_zero_int) ) ) ).

% mult_pos_neg2
tff(fact_1961_zero__less__mult__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(times_times_real(A),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real) ) ) ) ).

% zero_less_mult_iff
tff(fact_1962_zero__less__mult__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat) ) ) ) ).

% zero_less_mult_iff
tff(fact_1963_zero__less__mult__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int) ) ) ) ).

% zero_less_mult_iff
tff(fact_1964_zero__less__mult__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(times_times_real(A),B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ) ).

% zero_less_mult_pos
tff(fact_1965_zero__less__mult__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ) ).

% zero_less_mult_pos
tff(fact_1966_zero__less__mult__pos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ) ).

% zero_less_mult_pos
tff(fact_1967_zero__less__mult__pos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ) ).

% zero_less_mult_pos
tff(fact_1968_zero__less__mult__pos2,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(times_times_real(B),A))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ) ).

% zero_less_mult_pos2
tff(fact_1969_zero__less__mult__pos2,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(B),A))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ) ).

% zero_less_mult_pos2
tff(fact_1970_zero__less__mult__pos2,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ) ).

% zero_less_mult_pos2
tff(fact_1971_zero__less__mult__pos2,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,B),A))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ) ).

% zero_less_mult_pos2
tff(fact_1972_mult__less__cancel__left__neg,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ).

% mult_less_cancel_left_neg
tff(fact_1973_mult__less__cancel__left__neg,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ).

% mult_less_cancel_left_neg
tff(fact_1974_mult__less__cancel__left__neg,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ).

% mult_less_cancel_left_neg
tff(fact_1975_mult__less__cancel__left__pos,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% mult_less_cancel_left_pos
tff(fact_1976_mult__less__cancel__left__pos,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% mult_less_cancel_left_pos
tff(fact_1977_mult__less__cancel__left__pos,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% mult_less_cancel_left_pos
tff(fact_1978_mult__strict__left__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_1979_mult__strict__left__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_1980_mult__strict__left__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_1981_mult__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B)) ) ) ).

% mult_strict_left_mono
tff(fact_1982_mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)) ) ) ).

% mult_strict_left_mono
tff(fact_1983_mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ).

% mult_strict_left_mono
tff(fact_1984_mult__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% mult_strict_left_mono
tff(fact_1985_mult__less__cancel__left__disj,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_1986_mult__less__cancel__left__disj,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_1987_mult__less__cancel__left__disj,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_1988_mult__strict__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),C)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_1989_mult__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_1990_mult__strict__right__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_1991_mult__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),C)) ) ) ).

% mult_strict_right_mono
tff(fact_1992_mult__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) ) ) ).

% mult_strict_right_mono
tff(fact_1993_mult__strict__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ) ).

% mult_strict_right_mono
tff(fact_1994_mult__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% mult_strict_right_mono
tff(fact_1995_mult__less__cancel__right__disj,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),C))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_1996_mult__less__cancel__right__disj,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_1997_mult__less__cancel__right__disj,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_1998_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_1999_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2000_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2001_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2002_less__numeral__extra_I1_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),one_one_real) ).

% less_numeral_extra(1)
tff(fact_2003_less__numeral__extra_I1_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),one_one_rat) ).

% less_numeral_extra(1)
tff(fact_2004_less__numeral__extra_I1_J,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),one_one_nat) ).

% less_numeral_extra(1)
tff(fact_2005_less__numeral__extra_I1_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),one_one_int) ).

% less_numeral_extra(1)
tff(fact_2006_not__one__less__zero,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),zero_zero_real) ).

% not_one_less_zero
tff(fact_2007_not__one__less__zero,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),zero_zero_rat) ).

% not_one_less_zero
tff(fact_2008_not__one__less__zero,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),zero_zero_nat) ).

% not_one_less_zero
tff(fact_2009_not__one__less__zero,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),zero_zero_int) ).

% not_one_less_zero
tff(fact_2010_zero__less__one,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),one_one_real) ).

% zero_less_one
tff(fact_2011_zero__less__one,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),one_one_rat) ).

% zero_less_one
tff(fact_2012_zero__less__one,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),one_one_nat) ).

% zero_less_one
tff(fact_2013_zero__less__one,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),one_one_int) ).

% zero_less_one
tff(fact_2014_add__neg__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(A),B)),zero_zero_real) ) ) ).

% add_neg_neg
tff(fact_2015_add__neg__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),B)),zero_zero_rat) ) ) ).

% add_neg_neg
tff(fact_2016_add__neg__neg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat) ) ) ).

% add_neg_neg
tff(fact_2017_add__neg__neg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int) ) ) ).

% add_neg_neg
tff(fact_2018_add__pos__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(plus_plus_real(A),B)) ) ) ).

% add_pos_pos
tff(fact_2019_add__pos__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A),B)) ) ) ).

% add_pos_pos
tff(fact_2020_add__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% add_pos_pos
tff(fact_2021_add__pos__pos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% add_pos_pos
tff(fact_2022_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ~ ! [C3: nat] :
            ( ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C3) )
           => ( C3 = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_2023_pos__add__strict,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(plus_plus_real(A),C)) ) ) ).

% pos_add_strict
tff(fact_2024_pos__add__strict,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(plus_plus_rat(A),C)) ) ) ).

% pos_add_strict
tff(fact_2025_pos__add__strict,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% pos_add_strict
tff(fact_2026_pos__add__strict,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).

% pos_add_strict
tff(fact_2027_add__less__zeroD,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(X),Y2)),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),zero_zero_real)
        | aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),zero_zero_real) ) ) ).

% add_less_zeroD
tff(fact_2028_add__less__zeroD,axiom,
    ! [X: rat,Y2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(X),Y2)),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),zero_zero_rat)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y2),zero_zero_rat) ) ) ).

% add_less_zeroD
tff(fact_2029_add__less__zeroD,axiom,
    ! [X: int,Y2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y2)),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),zero_zero_int)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,Y2),zero_zero_int) ) ) ).

% add_less_zeroD
tff(fact_2030_divide__le__0__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(A),B)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) ) ) ) ).

% divide_le_0_iff
tff(fact_2031_divide__le__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(A),B)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) ) ) ) ).

% divide_le_0_iff
tff(fact_2032_divide__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(A),C)),aa_real_real(divide_divide_real(B),C)) ) ) ).

% divide_right_mono
tff(fact_2033_divide__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C)) ) ) ).

% divide_right_mono
tff(fact_2034_zero__le__divide__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(A),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) ) ) ) ).

% zero_le_divide_iff
tff(fact_2035_zero__le__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(A),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) ) ) ) ).

% zero_le_divide_iff
tff(fact_2036_divide__nonneg__nonneg,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(X),Y2)) ) ) ).

% divide_nonneg_nonneg
tff(fact_2037_divide__nonneg__nonneg,axiom,
    ! [X: rat,Y2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(X),Y2)) ) ) ).

% divide_nonneg_nonneg
tff(fact_2038_divide__nonneg__nonpos,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(X),Y2)),zero_zero_real) ) ) ).

% divide_nonneg_nonpos
tff(fact_2039_divide__nonneg__nonpos,axiom,
    ! [X: rat,Y2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(X),Y2)),zero_zero_rat) ) ) ).

% divide_nonneg_nonpos
tff(fact_2040_divide__nonpos__nonneg,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(X),Y2)),zero_zero_real) ) ) ).

% divide_nonpos_nonneg
tff(fact_2041_divide__nonpos__nonneg,axiom,
    ! [X: rat,Y2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(X),Y2)),zero_zero_rat) ) ) ).

% divide_nonpos_nonneg
tff(fact_2042_divide__nonpos__nonpos,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(X),Y2)) ) ) ).

% divide_nonpos_nonpos
tff(fact_2043_divide__nonpos__nonpos,axiom,
    ! [X: rat,Y2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(X),Y2)) ) ) ).

% divide_nonpos_nonpos
tff(fact_2044_divide__right__mono__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(B),C)),aa_real_real(divide_divide_real(A),C)) ) ) ).

% divide_right_mono_neg
tff(fact_2045_divide__right__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C)),aa_rat_rat(divide_divide_rat(A),C)) ) ) ).

% divide_right_mono_neg
tff(fact_2046_mod__induct,axiom,
    ! [P: fun_nat_o,Nb: nat,P2: nat,Ma: nat] :
      ( aa_nat_o(P,Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),P2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),P2)
         => ( ! [N: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),P2)
               => ( aa_nat_o(P,N)
                 => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,N)),P2)) ) )
           => aa_nat_o(P,Ma) ) ) ) ) ).

% mod_induct
tff(fact_2047_power__mono,axiom,
    ! [A: real,B: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)) ) ) ).

% power_mono
tff(fact_2048_power__mono,axiom,
    ! [A: rat,B: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb)) ) ) ).

% power_mono
tff(fact_2049_power__mono,axiom,
    ! [A: nat,B: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)) ) ) ).

% power_mono
tff(fact_2050_power__mono,axiom,
    ! [A: int,B: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Nb)) ) ) ).

% power_mono
tff(fact_2051_zero__le__power,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ).

% zero_le_power
tff(fact_2052_zero__le__power,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Nb)) ) ).

% zero_le_power
tff(fact_2053_zero__le__power,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ).

% zero_le_power
tff(fact_2054_zero__le__power,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)) ) ).

% zero_le_power
tff(fact_2055_divide__neg__neg,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(divide_divide_real(X),Y2)) ) ) ).

% divide_neg_neg
tff(fact_2056_divide__neg__neg,axiom,
    ! [X: rat,Y2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(X),Y2)) ) ) ).

% divide_neg_neg
tff(fact_2057_divide__neg__pos,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(X),Y2)),zero_zero_real) ) ) ).

% divide_neg_pos
tff(fact_2058_divide__neg__pos,axiom,
    ! [X: rat,Y2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(X),Y2)),zero_zero_rat) ) ) ).

% divide_neg_pos
tff(fact_2059_divide__pos__neg,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(X),Y2)),zero_zero_real) ) ) ).

% divide_pos_neg
tff(fact_2060_divide__pos__neg,axiom,
    ! [X: rat,Y2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(X),Y2)),zero_zero_rat) ) ) ).

% divide_pos_neg
tff(fact_2061_divide__pos__pos,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(divide_divide_real(X),Y2)) ) ) ).

% divide_pos_pos
tff(fact_2062_divide__pos__pos,axiom,
    ! [X: rat,Y2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(X),Y2)) ) ) ).

% divide_pos_pos
tff(fact_2063_divide__less__0__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(A),B)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) ) ) ) ).

% divide_less_0_iff
tff(fact_2064_divide__less__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(A),B)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) ) ) ) ).

% divide_less_0_iff
tff(fact_2065_divide__less__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(A),C)),aa_real_real(divide_divide_real(B),C))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) )
        & ( C != zero_zero_real ) ) ) ).

% divide_less_cancel
tff(fact_2066_divide__less__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) )
        & ( C != zero_zero_rat ) ) ) ).

% divide_less_cancel
tff(fact_2067_zero__less__divide__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(divide_divide_real(A),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real) ) ) ) ).

% zero_less_divide_iff
tff(fact_2068_zero__less__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(A),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat) ) ) ) ).

% zero_less_divide_iff
tff(fact_2069_divide__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(A),C)),aa_real_real(divide_divide_real(B),C)) ) ) ).

% divide_strict_right_mono
tff(fact_2070_divide__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C)) ) ) ).

% divide_strict_right_mono
tff(fact_2071_divide__strict__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(A),C)),aa_real_real(divide_divide_real(B),C)) ) ) ).

% divide_strict_right_mono_neg
tff(fact_2072_divide__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C)) ) ) ).

% divide_strict_right_mono_neg
tff(fact_2073_mod__Suc__le__divisor,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),aa_nat_nat(suc,Nb))),Nb) ).

% mod_Suc_le_divisor
tff(fact_2074_zero__less__power,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ).

% zero_less_power
tff(fact_2075_zero__less__power,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Nb)) ) ).

% zero_less_power
tff(fact_2076_zero__less__power,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ).

% zero_less_power
tff(fact_2077_zero__less__power,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)) ) ).

% zero_less_power
tff(fact_2078_frac__eq__eq,axiom,
    ! [Y2: complex,Z2: complex,X: complex,W: complex] :
      ( ( Y2 != zero_zero_complex )
     => ( ( Z2 != zero_zero_complex )
       => ( ( aa_complex_complex(divide1717551699836669952omplex(X),Y2) = aa_complex_complex(divide1717551699836669952omplex(W),Z2) )
        <=> ( aa_complex_complex(times_times_complex(X),Z2) = aa_complex_complex(times_times_complex(W),Y2) ) ) ) ) ).

% frac_eq_eq
tff(fact_2079_frac__eq__eq,axiom,
    ! [Y2: real,Z2: real,X: real,W: real] :
      ( ( Y2 != zero_zero_real )
     => ( ( Z2 != zero_zero_real )
       => ( ( aa_real_real(divide_divide_real(X),Y2) = aa_real_real(divide_divide_real(W),Z2) )
        <=> ( aa_real_real(times_times_real(X),Z2) = aa_real_real(times_times_real(W),Y2) ) ) ) ) ).

% frac_eq_eq
tff(fact_2080_frac__eq__eq,axiom,
    ! [Y2: rat,Z2: rat,X: rat,W: rat] :
      ( ( Y2 != zero_zero_rat )
     => ( ( Z2 != zero_zero_rat )
       => ( ( aa_rat_rat(divide_divide_rat(X),Y2) = aa_rat_rat(divide_divide_rat(W),Z2) )
        <=> ( aa_rat_rat(times_times_rat(X),Z2) = aa_rat_rat(times_times_rat(W),Y2) ) ) ) ) ).

% frac_eq_eq
tff(fact_2081_divide__eq__eq,axiom,
    ! [B: complex,C: complex,A: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),C) = A )
    <=> $ite(C != zero_zero_complex,B = aa_complex_complex(times_times_complex(A),C),A = zero_zero_complex) ) ).

% divide_eq_eq
tff(fact_2082_divide__eq__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( aa_real_real(divide_divide_real(B),C) = A )
    <=> $ite(C != zero_zero_real,B = aa_real_real(times_times_real(A),C),A = zero_zero_real) ) ).

% divide_eq_eq
tff(fact_2083_divide__eq__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( aa_rat_rat(divide_divide_rat(B),C) = A )
    <=> $ite(C != zero_zero_rat,B = aa_rat_rat(times_times_rat(A),C),A = zero_zero_rat) ) ).

% divide_eq_eq
tff(fact_2084_eq__divide__eq,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( A = aa_complex_complex(divide1717551699836669952omplex(B),C) )
    <=> $ite(C != zero_zero_complex,aa_complex_complex(times_times_complex(A),C) = B,A = zero_zero_complex) ) ).

% eq_divide_eq
tff(fact_2085_eq__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A = aa_real_real(divide_divide_real(B),C) )
    <=> $ite(C != zero_zero_real,aa_real_real(times_times_real(A),C) = B,A = zero_zero_real) ) ).

% eq_divide_eq
tff(fact_2086_eq__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = aa_rat_rat(divide_divide_rat(B),C) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(times_times_rat(A),C) = B,A = zero_zero_rat) ) ).

% eq_divide_eq
tff(fact_2087_divide__eq__imp,axiom,
    ! [C: complex,B: complex,A: complex] :
      ( ( C != zero_zero_complex )
     => ( ( B = aa_complex_complex(times_times_complex(A),C) )
       => ( aa_complex_complex(divide1717551699836669952omplex(B),C) = A ) ) ) ).

% divide_eq_imp
tff(fact_2088_divide__eq__imp,axiom,
    ! [C: real,B: real,A: real] :
      ( ( C != zero_zero_real )
     => ( ( B = aa_real_real(times_times_real(A),C) )
       => ( aa_real_real(divide_divide_real(B),C) = A ) ) ) ).

% divide_eq_imp
tff(fact_2089_divide__eq__imp,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( C != zero_zero_rat )
     => ( ( B = aa_rat_rat(times_times_rat(A),C) )
       => ( aa_rat_rat(divide_divide_rat(B),C) = A ) ) ) ).

% divide_eq_imp
tff(fact_2090_eq__divide__imp,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( aa_complex_complex(times_times_complex(A),C) = B )
       => ( A = aa_complex_complex(divide1717551699836669952omplex(B),C) ) ) ) ).

% eq_divide_imp
tff(fact_2091_eq__divide__imp,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(times_times_real(A),C) = B )
       => ( A = aa_real_real(divide_divide_real(B),C) ) ) ) ).

% eq_divide_imp
tff(fact_2092_eq__divide__imp,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(times_times_rat(A),C) = B )
       => ( A = aa_rat_rat(divide_divide_rat(B),C) ) ) ) ).

% eq_divide_imp
tff(fact_2093_nonzero__divide__eq__eq,axiom,
    ! [C: complex,B: complex,A: complex] :
      ( ( C != zero_zero_complex )
     => ( ( aa_complex_complex(divide1717551699836669952omplex(B),C) = A )
      <=> ( B = aa_complex_complex(times_times_complex(A),C) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_2094_nonzero__divide__eq__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(divide_divide_real(B),C) = A )
      <=> ( B = aa_real_real(times_times_real(A),C) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_2095_nonzero__divide__eq__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(divide_divide_rat(B),C) = A )
      <=> ( B = aa_rat_rat(times_times_rat(A),C) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_2096_nonzero__eq__divide__eq,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( A = aa_complex_complex(divide1717551699836669952omplex(B),C) )
      <=> ( aa_complex_complex(times_times_complex(A),C) = B ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_2097_nonzero__eq__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( A = aa_real_real(divide_divide_real(B),C) )
      <=> ( aa_real_real(times_times_real(A),C) = B ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_2098_nonzero__eq__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( A = aa_rat_rat(divide_divide_rat(B),C) )
      <=> ( aa_rat_rat(times_times_rat(A),C) = B ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_2099_right__inverse__eq,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( aa_complex_complex(divide1717551699836669952omplex(A),B) = one_one_complex )
      <=> ( A = B ) ) ) ).

% right_inverse_eq
tff(fact_2100_right__inverse__eq,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( aa_real_real(divide_divide_real(A),B) = one_one_real )
      <=> ( A = B ) ) ) ).

% right_inverse_eq
tff(fact_2101_right__inverse__eq,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( aa_rat_rat(divide_divide_rat(A),B) = one_one_rat )
      <=> ( A = B ) ) ) ).

% right_inverse_eq
tff(fact_2102_not__exp__less__eq__0__int,axiom,
    ! [Nb: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),zero_zero_int) ).

% not_exp_less_eq_0_int
tff(fact_2103_neg__zdiv__mult__2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),one_one_int)),A) ) ) ).

% neg_zdiv_mult_2
tff(fact_2104_pos__zdiv__mult__2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A) ) ) ).

% pos_zdiv_mult_2
tff(fact_2105_pos__zmod__mult__2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),A))) ) ) ).

% pos_zmod_mult_2
tff(fact_2106_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,modulo364778990260209775nteger(A,B)),A) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_2107_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),A) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_2108_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),A) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_2109_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),B) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_2110_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),B) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_2111_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,modulo364778990260209775nteger(A,B)),B) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_2112_power__0,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(A),zero_zero_nat) = one_one_rat ).

% power_0
tff(fact_2113_power__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),zero_zero_nat) = one_one_nat ).

% power_0
tff(fact_2114_power__0,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),zero_zero_nat) = one_one_real ).

% power_0
tff(fact_2115_power__0,axiom,
    ! [A: int] : aa_nat_int(aa_int_fun_nat_int(power_power_int,A),zero_zero_nat) = one_one_int ).

% power_0
tff(fact_2116_power__0,axiom,
    ! [A: complex] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),zero_zero_nat) = one_one_complex ).

% power_0
tff(fact_2117_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = A )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = zero_zero_nat ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_2118_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = A )
    <=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = zero_zero_int ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_2119_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger(A,B) = A )
    <=> ( divide6298287555418463151nteger(A,B) = zero_z3403309356797280102nteger ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_2120_ex__least__nat__le,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,Nb)
     => ( ~ aa_nat_o(P,zero_zero_nat)
       => ? [K: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
            & ! [I4: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),K)
               => ~ aa_nat_o(P,I4) )
            & aa_nat_o(P,K) ) ) ) ).

% ex_least_nat_le
tff(fact_2121_less__imp__add__positive,axiom,
    ! [I2: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
     => ? [K: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
          & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K) = J ) ) ) ).

% less_imp_add_positive
tff(fact_2122_nat__mult__less__cancel1,axiom,
    ! [K2: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb) ) ) ).

% nat_mult_less_cancel1
tff(fact_2123_nat__mult__eq__cancel1,axiom,
    ! [K2: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Ma) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Nb) )
      <=> ( Ma = Nb ) ) ) ).

% nat_mult_eq_cancel1
tff(fact_2124_mult__less__mono2,axiom,
    ! [I2: nat,J: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),I2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),J)) ) ) ).

% mult_less_mono2
tff(fact_2125_mult__less__mono1,axiom,
    ! [I2: nat,J: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),K2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),K2)) ) ) ).

% mult_less_mono1
tff(fact_2126_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),Nb) = zero_zero_nat )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
        | ( Nb = zero_zero_nat ) ) ) ).

% Euclidean_Division.div_eq_0_iff
tff(fact_2127_nat__power__less__imp__less,axiom,
    ! [I2: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),I2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I2),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I2),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb) ) ) ).

% nat_power_less_imp_less
tff(fact_2128_mult__eq__self__implies__10,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( Ma = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Nb) )
     => ( ( Nb = one_one_nat )
        | ( Ma = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
tff(fact_2129_mod__less__divisor,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),Nb)),Nb) ) ).

% mod_less_divisor
tff(fact_2130_mod__eq__0D,axiom,
    ! [Ma: nat,D: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),D) = zero_zero_nat )
     => ? [Q3: nat] : Ma = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,D),Q3) ) ).

% mod_eq_0D
tff(fact_2131_nat__bit__induct,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ! [N: nat] :
            ( aa_nat_o(P,N)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
             => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ) )
       => ( ! [N: nat] :
              ( aa_nat_o(P,N)
             => aa_nat_o(P,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) )
         => aa_nat_o(P,Nb) ) ) ) ).

% nat_bit_induct
tff(fact_2132_div__2__gt__zero,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% div_2_gt_zero
tff(fact_2133_Suc__n__div__2__gt__zero,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% Suc_n_div_2_gt_zero
tff(fact_2134_imult__is__infinity,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(A),B) = extend5688581933313929465d_enat )
    <=> ( ( ( A = extend5688581933313929465d_enat )
          & ( B != zero_z5237406670263579293d_enat ) )
        | ( ( B = extend5688581933313929465d_enat )
          & ( A != zero_z5237406670263579293d_enat ) ) ) ) ).

% imult_is_infinity
tff(fact_2135_enat__0__less__mult__iff,axiom,
    ! [Ma: extended_enat,Nb: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(Ma),Nb))
    <=> ( aa_Extended_enat_o(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),Ma)
        & aa_Extended_enat_o(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),Nb) ) ) ).

% enat_0_less_mult_iff
tff(fact_2136_odd__0__le__power__imp__0__le,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% odd_0_le_power_imp_0_le
tff(fact_2137_odd__0__le__power__imp__0__le,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% odd_0_le_power_imp_0_le
tff(fact_2138_odd__0__le__power__imp__0__le,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).

% odd_0_le_power_imp_0_le
tff(fact_2139_odd__power__less__zero,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),zero_zero_real) ) ).

% odd_power_less_zero
tff(fact_2140_odd__power__less__zero,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),zero_zero_rat) ) ).

% odd_power_less_zero
tff(fact_2141_odd__power__less__zero,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),zero_zero_int) ) ).

% odd_power_less_zero
tff(fact_2142_power__gt1,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Nb))) ) ).

% power_gt1
tff(fact_2143_power__gt1,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Nb))) ) ).

% power_gt1
tff(fact_2144_power__gt1,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb))) ) ).

% power_gt1
tff(fact_2145_power__gt1,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(suc,Nb))) ) ).

% power_gt1
tff(fact_2146_mult__le__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).

% mult_le_cancel_left
tff(fact_2147_mult__le__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).

% mult_le_cancel_left
tff(fact_2148_mult__le__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ) ).

% mult_le_cancel_left
tff(fact_2149_mult__le__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),C))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).

% mult_le_cancel_right
tff(fact_2150_mult__le__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).

% mult_le_cancel_right
tff(fact_2151_mult__le__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ) ).

% mult_le_cancel_right
tff(fact_2152_mult__left__less__imp__less,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_2153_mult__left__less__imp__less,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_2154_mult__left__less__imp__less,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_2155_mult__left__less__imp__less,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_2156_mult__strict__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_2157_mult__strict__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_2158_mult__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_2159_mult__strict__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_2160_mult__less__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).

% mult_less_cancel_left
tff(fact_2161_mult__less__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% mult_less_cancel_left
tff(fact_2162_mult__less__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).

% mult_less_cancel_left
tff(fact_2163_mult__right__less__imp__less,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),C))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_2164_mult__right__less__imp__less,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_2165_mult__right__less__imp__less,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_2166_mult__right__less__imp__less,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_2167_mult__strict__mono_H,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2168_mult__strict__mono_H,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2169_mult__strict__mono_H,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2170_mult__strict__mono_H,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2171_mult__less__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),C))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).

% mult_less_cancel_right
tff(fact_2172_mult__less__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% mult_less_cancel_right
tff(fact_2173_mult__less__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ) ).

% mult_less_cancel_right
tff(fact_2174_mult__le__cancel__left__neg,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ).

% mult_le_cancel_left_neg
tff(fact_2175_mult__le__cancel__left__neg,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).

% mult_le_cancel_left_neg
tff(fact_2176_mult__le__cancel__left__neg,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ).

% mult_le_cancel_left_neg
tff(fact_2177_mult__le__cancel__left__pos,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% mult_le_cancel_left_pos
tff(fact_2178_mult__le__cancel__left__pos,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% mult_le_cancel_left_pos
tff(fact_2179_mult__le__cancel__left__pos,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% mult_le_cancel_left_pos
tff(fact_2180_mult__left__le__imp__le,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_2181_mult__left__le__imp__le,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_2182_mult__left__le__imp__le,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_2183_mult__left__le__imp__le,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_2184_mult__right__le__imp__le,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),C))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_2185_mult__right__le__imp__le,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_2186_mult__right__le__imp__le,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_2187_mult__right__le__imp__le,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_2188_mult__le__less__imp__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2189_mult__le__less__imp__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2190_mult__le__less__imp__less,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2191_mult__le__less__imp__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2192_mult__less__le__imp__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2193_mult__less__le__imp__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2194_mult__less__le__imp__less,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2195_mult__less__le__imp__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2196_field__le__epsilon,axiom,
    ! [X: real,Y2: real] :
      ( ! [E2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(plus_plus_real(Y2),E2)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y2) ) ).

% field_le_epsilon
tff(fact_2197_field__le__epsilon,axiom,
    ! [X: rat,Y2: rat] :
      ( ! [E2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_rat_rat(plus_plus_rat(Y2),E2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y2) ) ).

% field_le_epsilon
tff(fact_2198_add__neg__nonpos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(A),B)),zero_zero_real) ) ) ).

% add_neg_nonpos
tff(fact_2199_add__neg__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),B)),zero_zero_rat) ) ) ).

% add_neg_nonpos
tff(fact_2200_add__neg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat) ) ) ).

% add_neg_nonpos
tff(fact_2201_add__neg__nonpos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int) ) ) ).

% add_neg_nonpos
tff(fact_2202_add__nonneg__pos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(plus_plus_real(A),B)) ) ) ).

% add_nonneg_pos
tff(fact_2203_add__nonneg__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A),B)) ) ) ).

% add_nonneg_pos
tff(fact_2204_add__nonneg__pos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% add_nonneg_pos
tff(fact_2205_add__nonneg__pos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% add_nonneg_pos
tff(fact_2206_add__nonpos__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(A),B)),zero_zero_real) ) ) ).

% add_nonpos_neg
tff(fact_2207_add__nonpos__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),B)),zero_zero_rat) ) ) ).

% add_nonpos_neg
tff(fact_2208_add__nonpos__neg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),zero_zero_nat) ) ) ).

% add_nonpos_neg
tff(fact_2209_add__nonpos__neg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),zero_zero_int) ) ) ).

% add_nonpos_neg
tff(fact_2210_add__pos__nonneg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(plus_plus_real(A),B)) ) ) ).

% add_pos_nonneg
tff(fact_2211_add__pos__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A),B)) ) ) ).

% add_pos_nonneg
tff(fact_2212_add__pos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% add_pos_nonneg
tff(fact_2213_add__pos__nonneg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% add_pos_nonneg
tff(fact_2214_add__strict__increasing,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(plus_plus_real(A),C)) ) ) ).

% add_strict_increasing
tff(fact_2215_add__strict__increasing,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(plus_plus_rat(A),C)) ) ) ).

% add_strict_increasing
tff(fact_2216_add__strict__increasing,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% add_strict_increasing
tff(fact_2217_add__strict__increasing,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).

% add_strict_increasing
tff(fact_2218_add__strict__increasing2,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(plus_plus_real(A),C)) ) ) ).

% add_strict_increasing2
tff(fact_2219_add__strict__increasing2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(plus_plus_rat(A),C)) ) ) ).

% add_strict_increasing2
tff(fact_2220_add__strict__increasing2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% add_strict_increasing2
tff(fact_2221_add__strict__increasing2,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).

% add_strict_increasing2
tff(fact_2222_mult__left__le,axiom,
    ! [C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),C)),A) ) ) ).

% mult_left_le
tff(fact_2223_mult__left__le,axiom,
    ! [C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),one_one_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),A) ) ) ).

% mult_left_le
tff(fact_2224_mult__left__le,axiom,
    ! [C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),A) ) ) ).

% mult_left_le
tff(fact_2225_mult__left__le,axiom,
    ! [C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),A) ) ) ).

% mult_left_le
tff(fact_2226_mult__le__one,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),B)),one_one_real) ) ) ) ).

% mult_le_one
tff(fact_2227_mult__le__one,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),one_one_rat) ) ) ) ).

% mult_le_one
tff(fact_2228_mult__le__one,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),one_one_nat)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),one_one_nat) ) ) ) ).

% mult_le_one
tff(fact_2229_mult__le__one,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),one_one_int) ) ) ) ).

% mult_le_one
tff(fact_2230_mult__right__le__one__le,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(X),Y2)),X) ) ) ) ).

% mult_right_le_one_le
tff(fact_2231_mult__right__le__one__le,axiom,
    ! [X: rat,Y2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y2)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y2),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(X),Y2)),X) ) ) ) ).

% mult_right_le_one_le
tff(fact_2232_mult__right__le__one__le,axiom,
    ! [X: int,Y2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y2),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),Y2)),X) ) ) ) ).

% mult_right_le_one_le
tff(fact_2233_mult__left__le__one__le,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(Y2),X)),X) ) ) ) ).

% mult_left_le_one_le
tff(fact_2234_mult__left__le__one__le,axiom,
    ! [X: rat,Y2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y2)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y2),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Y2),X)),X) ) ) ) ).

% mult_left_le_one_le
tff(fact_2235_mult__left__le__one__le,axiom,
    ! [X: int,Y2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y2),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Y2),X)),X) ) ) ) ).

% mult_left_le_one_le
tff(fact_2236_sum__squares__le__zero__iff,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_real_real(times_times_real(X),X)),aa_real_real(times_times_real(Y2),Y2))),zero_zero_real)
    <=> ( ( X = zero_zero_real )
        & ( Y2 = zero_zero_real ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_2237_sum__squares__le__zero__iff,axiom,
    ! [X: rat,Y2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(X),X)),aa_rat_rat(times_times_rat(Y2),Y2))),zero_zero_rat)
    <=> ( ( X = zero_zero_rat )
        & ( Y2 = zero_zero_rat ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_2238_sum__squares__le__zero__iff,axiom,
    ! [X: int,Y2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y2),Y2))),zero_zero_int)
    <=> ( ( X = zero_zero_int )
        & ( Y2 = zero_zero_int ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_2239_sum__squares__ge__zero,axiom,
    ! [X: real,Y2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(plus_plus_real(aa_real_real(times_times_real(X),X)),aa_real_real(times_times_real(Y2),Y2))) ).

% sum_squares_ge_zero
tff(fact_2240_sum__squares__ge__zero,axiom,
    ! [X: rat,Y2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(X),X)),aa_rat_rat(times_times_rat(Y2),Y2))) ).

% sum_squares_ge_zero
tff(fact_2241_sum__squares__ge__zero,axiom,
    ! [X: int,Y2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y2),Y2))) ).

% sum_squares_ge_zero
tff(fact_2242_frac__le,axiom,
    ! [Y2: real,X: real,W: real,Z2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),W)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,W),Z2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(X),Z2)),aa_real_real(divide_divide_real(Y2),W)) ) ) ) ) ).

% frac_le
tff(fact_2243_frac__le,axiom,
    ! [Y2: rat,X: rat,W: rat,Z2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y2)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),W)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,W),Z2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(X),Z2)),aa_rat_rat(divide_divide_rat(Y2),W)) ) ) ) ) ).

% frac_le
tff(fact_2244_frac__less,axiom,
    ! [X: real,Y2: real,W: real,Z2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),W)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,W),Z2)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(X),Z2)),aa_real_real(divide_divide_real(Y2),W)) ) ) ) ) ).

% frac_less
tff(fact_2245_frac__less,axiom,
    ! [X: rat,Y2: rat,W: rat,Z2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y2)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),W)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,W),Z2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(X),Z2)),aa_rat_rat(divide_divide_rat(Y2),W)) ) ) ) ) ).

% frac_less
tff(fact_2246_frac__less2,axiom,
    ! [X: real,Y2: real,W: real,Z2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),W)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,W),Z2)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(X),Z2)),aa_real_real(divide_divide_real(Y2),W)) ) ) ) ) ).

% frac_less2
tff(fact_2247_frac__less2,axiom,
    ! [X: rat,Y2: rat,W: rat,Z2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y2)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),W)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,W),Z2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(X),Z2)),aa_rat_rat(divide_divide_rat(Y2),W)) ) ) ) ) ).

% frac_less2
tff(fact_2248_divide__le__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(A),C)),aa_real_real(divide_divide_real(B),C))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).

% divide_le_cancel
tff(fact_2249_divide__le__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).

% divide_le_cancel
tff(fact_2250_divide__nonneg__neg,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(X),Y2)),zero_zero_real) ) ) ).

% divide_nonneg_neg
tff(fact_2251_divide__nonneg__neg,axiom,
    ! [X: rat,Y2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(X),Y2)),zero_zero_rat) ) ) ).

% divide_nonneg_neg
tff(fact_2252_divide__nonneg__pos,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(X),Y2)) ) ) ).

% divide_nonneg_pos
tff(fact_2253_divide__nonneg__pos,axiom,
    ! [X: rat,Y2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(X),Y2)) ) ) ).

% divide_nonneg_pos
tff(fact_2254_divide__nonpos__neg,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(X),Y2)) ) ) ).

% divide_nonpos_neg
tff(fact_2255_divide__nonpos__neg,axiom,
    ! [X: rat,Y2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(X),Y2)) ) ) ).

% divide_nonpos_neg
tff(fact_2256_divide__nonpos__pos,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(X),Y2)),zero_zero_real) ) ) ).

% divide_nonpos_pos
tff(fact_2257_divide__nonpos__pos,axiom,
    ! [X: rat,Y2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(X),Y2)),zero_zero_rat) ) ) ).

% divide_nonpos_pos
tff(fact_2258_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = zero_zero_nat ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_2259_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = zero_zero_int ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_2260_div__positive,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_positive
tff(fact_2261_div__positive,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_positive
tff(fact_2262_sum__squares__gt__zero__iff,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(plus_plus_real(aa_real_real(times_times_real(X),X)),aa_real_real(times_times_real(Y2),Y2)))
    <=> ( ( X != zero_zero_real )
        | ( Y2 != zero_zero_real ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_2263_sum__squares__gt__zero__iff,axiom,
    ! [X: rat,Y2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(X),X)),aa_rat_rat(times_times_rat(Y2),Y2)))
    <=> ( ( X != zero_zero_rat )
        | ( Y2 != zero_zero_rat ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_2264_sum__squares__gt__zero__iff,axiom,
    ! [X: int,Y2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y2),Y2)))
    <=> ( ( X != zero_zero_int )
        | ( Y2 != zero_zero_int ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_2265_not__sum__squares__lt__zero,axiom,
    ! [X: real,Y2: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(aa_real_real(times_times_real(X),X)),aa_real_real(times_times_real(Y2),Y2))),zero_zero_real) ).

% not_sum_squares_lt_zero
tff(fact_2266_not__sum__squares__lt__zero,axiom,
    ! [X: rat,Y2: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(X),X)),aa_rat_rat(times_times_rat(Y2),Y2))),zero_zero_rat) ).

% not_sum_squares_lt_zero
tff(fact_2267_not__sum__squares__lt__zero,axiom,
    ! [X: int,Y2: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y2),Y2))),zero_zero_int) ).

% not_sum_squares_lt_zero
tff(fact_2268_power__less__imp__less__base,axiom,
    ! [A: real,Nb: nat,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% power_less_imp_less_base
tff(fact_2269_power__less__imp__less__base,axiom,
    ! [A: rat,Nb: nat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% power_less_imp_less_base
tff(fact_2270_power__less__imp__less__base,axiom,
    ! [A: nat,Nb: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% power_less_imp_less_base
tff(fact_2271_power__less__imp__less__base,axiom,
    ! [A: int,Nb: nat,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Nb))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% power_less_imp_less_base
tff(fact_2272_zero__less__two,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(plus_plus_real(one_one_real),one_one_real)) ).

% zero_less_two
tff(fact_2273_zero__less__two,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(one_one_rat),one_one_rat)) ).

% zero_less_two
tff(fact_2274_zero__less__two,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat)) ).

% zero_less_two
tff(fact_2275_zero__less__two,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),one_one_int)) ).

% zero_less_two
tff(fact_2276_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_2277_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_2278_divide__less__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(B),C)),A)
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(times_times_real(A),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)) ) ) ).

% divide_less_eq
tff(fact_2279_divide__less__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B),C)),A)
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(A),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)) ) ) ).

% divide_less_eq
tff(fact_2280_less__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(divide_divide_real(B),C))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),C)),B),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(times_times_real(A),C)),aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)) ) ) ).

% less_divide_eq
tff(fact_2281_less__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(divide_divide_rat(B),C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(A),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)) ) ) ).

% less_divide_eq
tff(fact_2282_neg__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(B),C)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),C)),B) ) ) ).

% neg_divide_less_eq
tff(fact_2283_neg__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B),C)),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),B) ) ) ).

% neg_divide_less_eq
tff(fact_2284_neg__less__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(divide_divide_real(B),C))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(times_times_real(A),C)) ) ) ).

% neg_less_divide_eq
tff(fact_2285_neg__less__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(divide_divide_rat(B),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(A),C)) ) ) ).

% neg_less_divide_eq
tff(fact_2286_pos__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(B),C)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(times_times_real(A),C)) ) ) ).

% pos_divide_less_eq
tff(fact_2287_pos__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B),C)),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(A),C)) ) ) ).

% pos_divide_less_eq
tff(fact_2288_pos__less__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(divide_divide_real(B),C))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),C)),B) ) ) ).

% pos_less_divide_eq
tff(fact_2289_pos__less__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(divide_divide_rat(B),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),B) ) ) ).

% pos_less_divide_eq
tff(fact_2290_mult__imp__div__pos__less,axiom,
    ! [Y2: real,X: real,Z2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(times_times_real(Z2),Y2))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(X),Y2)),Z2) ) ) ).

% mult_imp_div_pos_less
tff(fact_2291_mult__imp__div__pos__less,axiom,
    ! [Y2: rat,X: rat,Z2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_rat_rat(times_times_rat(Z2),Y2))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(X),Y2)),Z2) ) ) ).

% mult_imp_div_pos_less
tff(fact_2292_mult__imp__less__div__pos,axiom,
    ! [Y2: real,Z2: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(Z2),Y2)),X)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),aa_real_real(divide_divide_real(X),Y2)) ) ) ).

% mult_imp_less_div_pos
tff(fact_2293_mult__imp__less__div__pos,axiom,
    ! [Y2: rat,Z2: rat,X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(Z2),Y2)),X)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),aa_rat_rat(divide_divide_rat(X),Y2)) ) ) ).

% mult_imp_less_div_pos
tff(fact_2294_divide__strict__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(times_times_real(A),B))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(C),A)),aa_real_real(divide_divide_real(C),B)) ) ) ) ).

% divide_strict_left_mono
tff(fact_2295_divide__strict__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(C),A)),aa_rat_rat(divide_divide_rat(C),B)) ) ) ) ).

% divide_strict_left_mono
tff(fact_2296_divide__strict__left__mono__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(times_times_real(A),B))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(C),A)),aa_real_real(divide_divide_real(C),B)) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_2297_divide__strict__left__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(C),A)),aa_rat_rat(divide_divide_rat(C),B)) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_2298_power__le__one,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),one_one_real) ) ) ).

% power_le_one
tff(fact_2299_power__le__one,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Nb)),one_one_rat) ) ) ).

% power_le_one
tff(fact_2300_power__le__one,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),one_one_nat) ) ) ).

% power_le_one
tff(fact_2301_power__le__one,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)),one_one_int) ) ) ).

% power_le_one
tff(fact_2302_divide__less__eq__1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(B),A)),one_one_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        | ( A = zero_zero_real ) ) ) ).

% divide_less_eq_1
tff(fact_2303_divide__less__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B),A)),one_one_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        | ( A = zero_zero_rat ) ) ) ).

% divide_less_eq_1
tff(fact_2304_less__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(divide_divide_real(B),A))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).

% less_divide_eq_1
tff(fact_2305_less__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),A))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% less_divide_eq_1
tff(fact_2306_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: complex,C: complex,W: num] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),C) = numera6690914467698888265omplex(W) )
    <=> $ite(C != zero_zero_complex,B = aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),C),numera6690914467698888265omplex(W) = zero_zero_complex) ) ).

% divide_eq_eq_numeral(1)
tff(fact_2307_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W: num] :
      ( ( aa_real_real(divide_divide_real(B),C) = aa_num_real(numeral_numeral_real,W) )
    <=> $ite(C != zero_zero_real,B = aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,W)),C),aa_num_real(numeral_numeral_real,W) = zero_zero_real) ) ).

% divide_eq_eq_numeral(1)
tff(fact_2308_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( aa_rat_rat(divide_divide_rat(B),C) = aa_num_rat(numeral_numeral_rat,W) )
    <=> $ite(C != zero_zero_rat,B = aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C),aa_num_rat(numeral_numeral_rat,W) = zero_zero_rat) ) ).

% divide_eq_eq_numeral(1)
tff(fact_2309_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: complex,C: complex] :
      ( ( numera6690914467698888265omplex(W) = aa_complex_complex(divide1717551699836669952omplex(B),C) )
    <=> $ite(C != zero_zero_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),C) = B,numera6690914467698888265omplex(W) = zero_zero_complex) ) ).

% eq_divide_eq_numeral(1)
tff(fact_2310_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: real,C: real] :
      ( ( aa_num_real(numeral_numeral_real,W) = aa_real_real(divide_divide_real(B),C) )
    <=> $ite(C != zero_zero_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,W)),C) = B,aa_num_real(numeral_numeral_real,W) = zero_zero_real) ) ).

% eq_divide_eq_numeral(1)
tff(fact_2311_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( aa_num_rat(numeral_numeral_rat,W) = aa_rat_rat(divide_divide_rat(B),C) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C) = B,aa_num_rat(numeral_numeral_rat,W) = zero_zero_rat) ) ).

% eq_divide_eq_numeral(1)
tff(fact_2312_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: complex,Z2: complex,B: complex] :
      aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(A),Z2)),B) = $ite(Z2 = zero_zero_complex,B,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(A),aa_complex_complex(times_times_complex(B),Z2))),Z2)) ).

% add_divide_eq_if_simps(2)
tff(fact_2313_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: real,Z2: real,B: real] :
      aa_real_real(plus_plus_real(aa_real_real(divide_divide_real(A),Z2)),B) = $ite(Z2 = zero_zero_real,B,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(A),aa_real_real(times_times_real(B),Z2))),Z2)) ).

% add_divide_eq_if_simps(2)
tff(fact_2314_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: rat,Z2: rat,B: rat] :
      aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(A),Z2)),B) = $ite(Z2 = zero_zero_rat,B,aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(A),aa_rat_rat(times_times_rat(B),Z2))),Z2)) ).

% add_divide_eq_if_simps(2)
tff(fact_2315_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: complex,B: complex,Z2: complex] :
      aa_complex_complex(plus_plus_complex(A),aa_complex_complex(divide1717551699836669952omplex(B),Z2)) = $ite(Z2 = zero_zero_complex,A,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),Z2)),B)),Z2)) ).

% add_divide_eq_if_simps(1)
tff(fact_2316_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: real,B: real,Z2: real] :
      aa_real_real(plus_plus_real(A),aa_real_real(divide_divide_real(B),Z2)) = $ite(Z2 = zero_zero_real,A,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),Z2)),B)),Z2)) ).

% add_divide_eq_if_simps(1)
tff(fact_2317_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: rat,B: rat,Z2: rat] :
      aa_rat_rat(plus_plus_rat(A),aa_rat_rat(divide_divide_rat(B),Z2)) = $ite(Z2 = zero_zero_rat,A,aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),Z2)),B)),Z2)) ).

% add_divide_eq_if_simps(1)
tff(fact_2318_add__frac__eq,axiom,
    ! [Y2: complex,Z2: complex,X: complex,W: complex] :
      ( ( Y2 != zero_zero_complex )
     => ( ( Z2 != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(X),Y2)),aa_complex_complex(divide1717551699836669952omplex(W),Z2)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(X),Z2)),aa_complex_complex(times_times_complex(W),Y2))),aa_complex_complex(times_times_complex(Y2),Z2)) ) ) ) ).

% add_frac_eq
tff(fact_2319_add__frac__eq,axiom,
    ! [Y2: real,Z2: real,X: real,W: real] :
      ( ( Y2 != zero_zero_real )
     => ( ( Z2 != zero_zero_real )
       => ( aa_real_real(plus_plus_real(aa_real_real(divide_divide_real(X),Y2)),aa_real_real(divide_divide_real(W),Z2)) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(times_times_real(X),Z2)),aa_real_real(times_times_real(W),Y2))),aa_real_real(times_times_real(Y2),Z2)) ) ) ) ).

% add_frac_eq
tff(fact_2320_add__frac__eq,axiom,
    ! [Y2: rat,Z2: rat,X: rat,W: rat] :
      ( ( Y2 != zero_zero_rat )
     => ( ( Z2 != zero_zero_rat )
       => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(X),Y2)),aa_rat_rat(divide_divide_rat(W),Z2)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(X),Z2)),aa_rat_rat(times_times_rat(W),Y2))),aa_rat_rat(times_times_rat(Y2),Z2)) ) ) ) ).

% add_frac_eq
tff(fact_2321_add__frac__num,axiom,
    ! [Y2: complex,X: complex,Z2: complex] :
      ( ( Y2 != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(X),Y2)),Z2) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(X),aa_complex_complex(times_times_complex(Z2),Y2))),Y2) ) ) ).

% add_frac_num
tff(fact_2322_add__frac__num,axiom,
    ! [Y2: real,X: real,Z2: real] :
      ( ( Y2 != zero_zero_real )
     => ( aa_real_real(plus_plus_real(aa_real_real(divide_divide_real(X),Y2)),Z2) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(X),aa_real_real(times_times_real(Z2),Y2))),Y2) ) ) ).

% add_frac_num
tff(fact_2323_add__frac__num,axiom,
    ! [Y2: rat,X: rat,Z2: rat] :
      ( ( Y2 != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(X),Y2)),Z2) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(X),aa_rat_rat(times_times_rat(Z2),Y2))),Y2) ) ) ).

% add_frac_num
tff(fact_2324_add__num__frac,axiom,
    ! [Y2: complex,Z2: complex,X: complex] :
      ( ( Y2 != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(Z2),aa_complex_complex(divide1717551699836669952omplex(X),Y2)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(X),aa_complex_complex(times_times_complex(Z2),Y2))),Y2) ) ) ).

% add_num_frac
tff(fact_2325_add__num__frac,axiom,
    ! [Y2: real,Z2: real,X: real] :
      ( ( Y2 != zero_zero_real )
     => ( aa_real_real(plus_plus_real(Z2),aa_real_real(divide_divide_real(X),Y2)) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(X),aa_real_real(times_times_real(Z2),Y2))),Y2) ) ) ).

% add_num_frac
tff(fact_2326_add__num__frac,axiom,
    ! [Y2: rat,Z2: rat,X: rat] :
      ( ( Y2 != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(Z2),aa_rat_rat(divide_divide_rat(X),Y2)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(X),aa_rat_rat(times_times_rat(Z2),Y2))),Y2) ) ) ).

% add_num_frac
tff(fact_2327_add__divide__eq__iff,axiom,
    ! [Z2: complex,X: complex,Y2: complex] :
      ( ( Z2 != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(X),aa_complex_complex(divide1717551699836669952omplex(Y2),Z2)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(X),Z2)),Y2)),Z2) ) ) ).

% add_divide_eq_iff
tff(fact_2328_add__divide__eq__iff,axiom,
    ! [Z2: real,X: real,Y2: real] :
      ( ( Z2 != zero_zero_real )
     => ( aa_real_real(plus_plus_real(X),aa_real_real(divide_divide_real(Y2),Z2)) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(times_times_real(X),Z2)),Y2)),Z2) ) ) ).

% add_divide_eq_iff
tff(fact_2329_add__divide__eq__iff,axiom,
    ! [Z2: rat,X: rat,Y2: rat] :
      ( ( Z2 != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(X),aa_rat_rat(divide_divide_rat(Y2),Z2)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(X),Z2)),Y2)),Z2) ) ) ).

% add_divide_eq_iff
tff(fact_2330_divide__add__eq__iff,axiom,
    ! [Z2: complex,X: complex,Y2: complex] :
      ( ( Z2 != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(X),Z2)),Y2) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(X),aa_complex_complex(times_times_complex(Y2),Z2))),Z2) ) ) ).

% divide_add_eq_iff
tff(fact_2331_divide__add__eq__iff,axiom,
    ! [Z2: real,X: real,Y2: real] :
      ( ( Z2 != zero_zero_real )
     => ( aa_real_real(plus_plus_real(aa_real_real(divide_divide_real(X),Z2)),Y2) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(X),aa_real_real(times_times_real(Y2),Z2))),Z2) ) ) ).

% divide_add_eq_iff
tff(fact_2332_divide__add__eq__iff,axiom,
    ! [Z2: rat,X: rat,Y2: rat] :
      ( ( Z2 != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(X),Z2)),Y2) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(X),aa_rat_rat(times_times_rat(Y2),Z2))),Z2) ) ) ).

% divide_add_eq_iff
tff(fact_2333_div__add__self1,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),one_one_nat) ) ) ).

% div_add_self1
tff(fact_2334_div__add__self1,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),one_one_int) ) ) ).

% div_add_self1
tff(fact_2335_div__add__self2,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),one_one_nat) ) ) ).

% div_add_self2
tff(fact_2336_div__add__self2,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),one_one_int) ) ) ).

% div_add_self2
tff(fact_2337_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
       => ( modulo364778990260209775nteger(A,B) = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_2338_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_2339_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_2340_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),modulo364778990260209775nteger(A,B)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_2341_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_2342_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_2343_subset__iff__psubset__eq,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ord_less_eq_set_int(A2,B3)
    <=> ( ord_less_set_int(A2,B3)
        | ( A2 = B3 ) ) ) ).

% subset_iff_psubset_eq
tff(fact_2344_subset__psubset__trans,axiom,
    ! [A2: set_int,B3: set_int,C4: set_int] :
      ( ord_less_eq_set_int(A2,B3)
     => ( ord_less_set_int(B3,C4)
       => ord_less_set_int(A2,C4) ) ) ).

% subset_psubset_trans
tff(fact_2345_subset__not__subset__eq,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ord_less_set_int(A2,B3)
    <=> ( ord_less_eq_set_int(A2,B3)
        & ~ ord_less_eq_set_int(B3,A2) ) ) ).

% subset_not_subset_eq
tff(fact_2346_psubset__subset__trans,axiom,
    ! [A2: set_int,B3: set_int,C4: set_int] :
      ( ord_less_set_int(A2,B3)
     => ( ord_less_eq_set_int(B3,C4)
       => ord_less_set_int(A2,C4) ) ) ).

% psubset_subset_trans
tff(fact_2347_psubset__imp__subset,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ord_less_set_int(A2,B3)
     => ord_less_eq_set_int(A2,B3) ) ).

% psubset_imp_subset
tff(fact_2348_psubset__eq,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ord_less_set_int(A2,B3)
    <=> ( ord_less_eq_set_int(A2,B3)
        & ( A2 != B3 ) ) ) ).

% psubset_eq
tff(fact_2349_psubsetE,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ord_less_set_int(A2,B3)
     => ~ ( ord_less_eq_set_int(A2,B3)
         => ord_less_eq_set_int(B3,A2) ) ) ).

% psubsetE
tff(fact_2350_in__mono,axiom,
    ! [A2: set_complex,B3: set_complex,X: complex] :
      ( ord_le211207098394363844omplex(A2,B3)
     => ( member_complex(X,A2)
       => member_complex(X,B3) ) ) ).

% in_mono
tff(fact_2351_in__mono,axiom,
    ! [A2: set_real,B3: set_real,X: real] :
      ( ord_less_eq_set_real(A2,B3)
     => ( member_real(X,A2)
       => member_real(X,B3) ) ) ).

% in_mono
tff(fact_2352_in__mono,axiom,
    ! [A2: set_set_nat,B3: set_set_nat,X: set_nat] :
      ( ord_le6893508408891458716et_nat(A2,B3)
     => ( member_set_nat(X,A2)
       => member_set_nat(X,B3) ) ) ).

% in_mono
tff(fact_2353_in__mono,axiom,
    ! [A2: set_nat,B3: set_nat,X: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B3)
     => ( aa_set_nat_o(member_nat(X),A2)
       => aa_set_nat_o(member_nat(X),B3) ) ) ).

% in_mono
tff(fact_2354_in__mono,axiom,
    ! [A2: set_int,B3: set_int,X: int] :
      ( ord_less_eq_set_int(A2,B3)
     => ( member_int(X,A2)
       => member_int(X,B3) ) ) ).

% in_mono
tff(fact_2355_subsetD,axiom,
    ! [A2: set_complex,B3: set_complex,C: complex] :
      ( ord_le211207098394363844omplex(A2,B3)
     => ( member_complex(C,A2)
       => member_complex(C,B3) ) ) ).

% subsetD
tff(fact_2356_subsetD,axiom,
    ! [A2: set_real,B3: set_real,C: real] :
      ( ord_less_eq_set_real(A2,B3)
     => ( member_real(C,A2)
       => member_real(C,B3) ) ) ).

% subsetD
tff(fact_2357_subsetD,axiom,
    ! [A2: set_set_nat,B3: set_set_nat,C: set_nat] :
      ( ord_le6893508408891458716et_nat(A2,B3)
     => ( member_set_nat(C,A2)
       => member_set_nat(C,B3) ) ) ).

% subsetD
tff(fact_2358_subsetD,axiom,
    ! [A2: set_nat,B3: set_nat,C: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B3)
     => ( aa_set_nat_o(member_nat(C),A2)
       => aa_set_nat_o(member_nat(C),B3) ) ) ).

% subsetD
tff(fact_2359_subsetD,axiom,
    ! [A2: set_int,B3: set_int,C: int] :
      ( ord_less_eq_set_int(A2,B3)
     => ( member_int(C,A2)
       => member_int(C,B3) ) ) ).

% subsetD
tff(fact_2360_equalityE,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( A2 = B3 )
     => ~ ( ord_less_eq_set_int(A2,B3)
         => ~ ord_less_eq_set_int(B3,A2) ) ) ).

% equalityE
tff(fact_2361_subset__eq,axiom,
    ! [A2: set_complex,B3: set_complex] :
      ( ord_le211207098394363844omplex(A2,B3)
    <=> ! [X3: complex] :
          ( member_complex(X3,A2)
         => member_complex(X3,B3) ) ) ).

% subset_eq
tff(fact_2362_subset__eq,axiom,
    ! [A2: set_real,B3: set_real] :
      ( ord_less_eq_set_real(A2,B3)
    <=> ! [X3: real] :
          ( member_real(X3,A2)
         => member_real(X3,B3) ) ) ).

% subset_eq
tff(fact_2363_subset__eq,axiom,
    ! [A2: set_set_nat,B3: set_set_nat] :
      ( ord_le6893508408891458716et_nat(A2,B3)
    <=> ! [X3: set_nat] :
          ( member_set_nat(X3,A2)
         => member_set_nat(X3,B3) ) ) ).

% subset_eq
tff(fact_2364_subset__eq,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B3)
    <=> ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A2)
         => aa_set_nat_o(member_nat(X3),B3) ) ) ).

% subset_eq
tff(fact_2365_subset__eq,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ord_less_eq_set_int(A2,B3)
    <=> ! [X3: int] :
          ( member_int(X3,A2)
         => member_int(X3,B3) ) ) ).

% subset_eq
tff(fact_2366_equalityD1,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( A2 = B3 )
     => ord_less_eq_set_int(A2,B3) ) ).

% equalityD1
tff(fact_2367_equalityD2,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( A2 = B3 )
     => ord_less_eq_set_int(B3,A2) ) ).

% equalityD2
tff(fact_2368_subset__iff,axiom,
    ! [A2: set_complex,B3: set_complex] :
      ( ord_le211207098394363844omplex(A2,B3)
    <=> ! [T2: complex] :
          ( member_complex(T2,A2)
         => member_complex(T2,B3) ) ) ).

% subset_iff
tff(fact_2369_subset__iff,axiom,
    ! [A2: set_real,B3: set_real] :
      ( ord_less_eq_set_real(A2,B3)
    <=> ! [T2: real] :
          ( member_real(T2,A2)
         => member_real(T2,B3) ) ) ).

% subset_iff
tff(fact_2370_subset__iff,axiom,
    ! [A2: set_set_nat,B3: set_set_nat] :
      ( ord_le6893508408891458716et_nat(A2,B3)
    <=> ! [T2: set_nat] :
          ( member_set_nat(T2,A2)
         => member_set_nat(T2,B3) ) ) ).

% subset_iff
tff(fact_2371_subset__iff,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B3)
    <=> ! [T2: nat] :
          ( aa_set_nat_o(member_nat(T2),A2)
         => aa_set_nat_o(member_nat(T2),B3) ) ) ).

% subset_iff
tff(fact_2372_subset__iff,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ord_less_eq_set_int(A2,B3)
    <=> ! [T2: int] :
          ( member_int(T2,A2)
         => member_int(T2,B3) ) ) ).

% subset_iff
tff(fact_2373_subset__refl,axiom,
    ! [A2: set_int] : ord_less_eq_set_int(A2,A2) ).

% subset_refl
tff(fact_2374_Collect__mono,axiom,
    ! [P: fun_complex_o,Q2: fun_complex_o] :
      ( ! [X4: complex] :
          ( aa_complex_o(P,X4)
         => aa_complex_o(Q2,X4) )
     => ord_le211207098394363844omplex(collect_complex(P),collect_complex(Q2)) ) ).

% Collect_mono
tff(fact_2375_Collect__mono,axiom,
    ! [P: fun_list_nat_o,Q2: fun_list_nat_o] :
      ( ! [X4: list_nat] :
          ( aa_list_nat_o(P,X4)
         => aa_list_nat_o(Q2,X4) )
     => ord_le6045566169113846134st_nat(collect_list_nat(P),collect_list_nat(Q2)) ) ).

% Collect_mono
tff(fact_2376_Collect__mono,axiom,
    ! [P: fun_set_nat_o,Q2: fun_set_nat_o] :
      ( ! [X4: set_nat] :
          ( aa_set_nat_o(P,X4)
         => aa_set_nat_o(Q2,X4) )
     => ord_le6893508408891458716et_nat(collect_set_nat(P),collect_set_nat(Q2)) ) ).

% Collect_mono
tff(fact_2377_Collect__mono,axiom,
    ! [P: fun_nat_o,Q2: fun_nat_o] :
      ( ! [X4: nat] :
          ( aa_nat_o(P,X4)
         => aa_nat_o(Q2,X4) )
     => aa_set_nat_o(ord_less_eq_set_nat(collect_nat(P)),collect_nat(Q2)) ) ).

% Collect_mono
tff(fact_2378_Collect__mono,axiom,
    ! [P: fun_int_o,Q2: fun_int_o] :
      ( ! [X4: int] :
          ( aa_int_o(P,X4)
         => aa_int_o(Q2,X4) )
     => ord_less_eq_set_int(collect_int(P),collect_int(Q2)) ) ).

% Collect_mono
tff(fact_2379_subset__trans,axiom,
    ! [A2: set_int,B3: set_int,C4: set_int] :
      ( ord_less_eq_set_int(A2,B3)
     => ( ord_less_eq_set_int(B3,C4)
       => ord_less_eq_set_int(A2,C4) ) ) ).

% subset_trans
tff(fact_2380_set__eq__subset,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( A2 = B3 )
    <=> ( ord_less_eq_set_int(A2,B3)
        & ord_less_eq_set_int(B3,A2) ) ) ).

% set_eq_subset
tff(fact_2381_Collect__mono__iff,axiom,
    ! [P: fun_complex_o,Q2: fun_complex_o] :
      ( ord_le211207098394363844omplex(collect_complex(P),collect_complex(Q2))
    <=> ! [X3: complex] :
          ( aa_complex_o(P,X3)
         => aa_complex_o(Q2,X3) ) ) ).

% Collect_mono_iff
tff(fact_2382_Collect__mono__iff,axiom,
    ! [P: fun_list_nat_o,Q2: fun_list_nat_o] :
      ( ord_le6045566169113846134st_nat(collect_list_nat(P),collect_list_nat(Q2))
    <=> ! [X3: list_nat] :
          ( aa_list_nat_o(P,X3)
         => aa_list_nat_o(Q2,X3) ) ) ).

% Collect_mono_iff
tff(fact_2383_Collect__mono__iff,axiom,
    ! [P: fun_set_nat_o,Q2: fun_set_nat_o] :
      ( ord_le6893508408891458716et_nat(collect_set_nat(P),collect_set_nat(Q2))
    <=> ! [X3: set_nat] :
          ( aa_set_nat_o(P,X3)
         => aa_set_nat_o(Q2,X3) ) ) ).

% Collect_mono_iff
tff(fact_2384_Collect__mono__iff,axiom,
    ! [P: fun_nat_o,Q2: fun_nat_o] :
      ( aa_set_nat_o(ord_less_eq_set_nat(collect_nat(P)),collect_nat(Q2))
    <=> ! [X3: nat] :
          ( aa_nat_o(P,X3)
         => aa_nat_o(Q2,X3) ) ) ).

% Collect_mono_iff
tff(fact_2385_Collect__mono__iff,axiom,
    ! [P: fun_int_o,Q2: fun_int_o] :
      ( ord_less_eq_set_int(collect_int(P),collect_int(Q2))
    <=> ! [X3: int] :
          ( aa_int_o(P,X3)
         => aa_int_o(Q2,X3) ) ) ).

% Collect_mono_iff
tff(fact_2386_cong__exp__iff__simps_I2_J,axiom,
    ! [Nb: num,Q: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q))) = zero_zero_nat )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,Q)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(2)
tff(fact_2387_cong__exp__iff__simps_I2_J,axiom,
    ! [Nb: num,Q: num] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q))) = zero_zero_int )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,Q)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(2)
tff(fact_2388_cong__exp__iff__simps_I2_J,axiom,
    ! [Nb: num,Q: num] :
      ( ( modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,Nb)),numera6620942414471956472nteger(aa_num_num(bit0,Q))) = zero_z3403309356797280102nteger )
    <=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(Nb),numera6620942414471956472nteger(Q)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(2)
tff(fact_2389_cong__exp__iff__simps_I1_J,axiom,
    ! [Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,one)) = zero_zero_nat ).

% cong_exp_iff_simps(1)
tff(fact_2390_cong__exp__iff__simps_I1_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,one)) = zero_zero_int ).

% cong_exp_iff_simps(1)
tff(fact_2391_cong__exp__iff__simps_I1_J,axiom,
    ! [Nb: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(Nb),numera6620942414471956472nteger(one)) = zero_z3403309356797280102nteger ).

% cong_exp_iff_simps(1)
tff(fact_2392_bounded__Max__nat,axiom,
    ! [P: fun_nat_o,X: nat,M6: nat] :
      ( aa_nat_o(P,X)
     => ( ! [X4: nat] :
            ( aa_nat_o(P,X4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),M6) )
       => ~ ! [M2: nat] :
              ( aa_nat_o(P,M2)
             => ~ ! [X2: nat] :
                    ( aa_nat_o(P,X2)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),M2) ) ) ) ) ).

% bounded_Max_nat
tff(fact_2393_length__pos__if__in__set,axiom,
    ! [X: complex,Xs: list_complex] :
      ( member_complex(X,set_complex2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_s3451745648224563538omplex(Xs)) ) ).

% length_pos_if_in_set
tff(fact_2394_length__pos__if__in__set,axiom,
    ! [X: real,Xs: list_real] :
      ( member_real(X,set_real2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_size_list_real(Xs)) ) ).

% length_pos_if_in_set
tff(fact_2395_length__pos__if__in__set,axiom,
    ! [X: set_nat,Xs: list_set_nat] :
      ( member_set_nat(X,set_set_nat2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_s3254054031482475050et_nat(Xs)) ) ).

% length_pos_if_in_set
tff(fact_2396_length__pos__if__in__set,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( member_VEBT_VEBT(X,set_VEBT_VEBT2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_s6755466524823107622T_VEBT(Xs)) ) ).

% length_pos_if_in_set
tff(fact_2397_length__pos__if__in__set,axiom,
    ! [X: $o,Xs: list_o] :
      ( member_o((X),set_o2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_size_list_o(Xs)) ) ).

% length_pos_if_in_set
tff(fact_2398_length__pos__if__in__set,axiom,
    ! [X: nat,Xs: list_nat] :
      ( aa_set_nat_o(member_nat(X),set_nat2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_size_list_nat(Xs)) ) ).

% length_pos_if_in_set
tff(fact_2399_length__pos__if__in__set,axiom,
    ! [X: int,Xs: list_int] :
      ( member_int(X,set_int2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_size_list_int(Xs)) ) ).

% length_pos_if_in_set
tff(fact_2400_times__enat__def,axiom,
    ! [Ma: extended_enat,Nb: extended_enat] :
      aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(Ma),Nb) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(aa_fun2186929853097002313d_enat(extend3600170679010898289d_enat,aTP_Lamm_ay(Nb)),
          $ite(Nb = zero_z5237406670263579293d_enat,zero_z5237406670263579293d_enat,extend5688581933313929465d_enat)),Ma) ).

% times_enat_def
tff(fact_2401_nat__mult__le__cancel1,axiom,
    ! [K2: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ) ).

% nat_mult_le_cancel1
tff(fact_2402_div__le__mono2,axiom,
    ! [Ma: nat,Nb: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ma)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K2),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K2),Ma)) ) ) ).

% div_le_mono2
tff(fact_2403_div__greater__zero__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% div_greater_zero_iff
tff(fact_2404_nat__mult__div__cancel1,axiom,
    ! [K2: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),Nb) ) ) ).

% nat_mult_div_cancel1
tff(fact_2405_div__less__iff__less__mult,axiom,
    ! [Q: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Q)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),Q)),Nb)
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q)) ) ) ).

% div_less_iff_less_mult
tff(fact_2406_mod__le__divisor,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),Nb)),Nb) ) ).

% mod_le_divisor
tff(fact_2407_div__less__dividend,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ma)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),Nb)),Ma) ) ) ).

% div_less_dividend
tff(fact_2408_div__eq__dividend__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ma)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),Nb) = Ma )
      <=> ( Nb = one_one_nat ) ) ) ).

% div_eq_dividend_iff
tff(fact_2409_div__less__mono,axiom,
    ! [A2: nat,B3: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),Nb) = zero_zero_nat )
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B3),Nb) = zero_zero_nat )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B3),Nb)) ) ) ) ) ).

% div_less_mono
tff(fact_2410_Suc__ile__eq,axiom,
    ! [Ma: nat,Nb: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(aa_nat_Extended_enat(extended_enat2,aa_nat_nat(suc,Ma))),Nb)
    <=> aa_Extended_enat_o(ord_le72135733267957522d_enat(aa_nat_Extended_enat(extended_enat2,Ma)),Nb) ) ).

% Suc_ile_eq
tff(fact_2411_imult__infinity,axiom,
    ! [Nb: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),Nb)
     => ( aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(extend5688581933313929465d_enat),Nb) = extend5688581933313929465d_enat ) ) ).

% imult_infinity
tff(fact_2412_imult__infinity__right,axiom,
    ! [Nb: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),Nb)
     => ( aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(Nb),extend5688581933313929465d_enat) = extend5688581933313929465d_enat ) ) ).

% imult_infinity_right
tff(fact_2413_div__nat__eqI,axiom,
    ! [Nb: nat,Q: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q)),Ma)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(suc,Q)))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),Nb) = Q ) ) ) ).

% div_nat_eqI
tff(fact_2414_field__le__mult__one__interval,axiom,
    ! [X: real,Y2: real] :
      ( ! [Z: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),one_one_real)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(Z),X)),Y2) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y2) ) ).

% field_le_mult_one_interval
tff(fact_2415_field__le__mult__one__interval,axiom,
    ! [X: rat,Y2: rat] :
      ( ! [Z: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),one_one_rat)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Z),X)),Y2) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y2) ) ).

% field_le_mult_one_interval
tff(fact_2416_mult__le__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),aa_real_real(times_times_real(C),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),one_one_real) ) ) ) ).

% mult_le_cancel_left1
tff(fact_2417_mult__le__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),aa_rat_rat(times_times_rat(C),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),one_one_rat) ) ) ) ).

% mult_le_cancel_left1
tff(fact_2418_mult__le__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),one_one_int) ) ) ) ).

% mult_le_cancel_left1
tff(fact_2419_mult__le__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(C),A)),C)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A) ) ) ) ).

% mult_le_cancel_left2
tff(fact_2420_mult__le__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),A)),C)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A) ) ) ) ).

% mult_le_cancel_left2
tff(fact_2421_mult__le__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),C)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A) ) ) ) ).

% mult_le_cancel_left2
tff(fact_2422_mult__le__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),aa_real_real(times_times_real(B),C))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),one_one_real) ) ) ) ).

% mult_le_cancel_right1
tff(fact_2423_mult__le__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),aa_rat_rat(times_times_rat(B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),one_one_rat) ) ) ) ).

% mult_le_cancel_right1
tff(fact_2424_mult__le__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),one_one_int) ) ) ) ).

% mult_le_cancel_right1
tff(fact_2425_mult__le__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),C)),C)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A) ) ) ) ).

% mult_le_cancel_right2
tff(fact_2426_mult__le__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),C)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A) ) ) ) ).

% mult_le_cancel_right2
tff(fact_2427_mult__le__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),C)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A) ) ) ) ).

% mult_le_cancel_right2
tff(fact_2428_mult__less__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(times_times_real(C),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B),one_one_real) ) ) ) ).

% mult_less_cancel_left1
tff(fact_2429_mult__less__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(times_times_rat(C),B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),one_one_rat) ) ) ) ).

% mult_less_cancel_left1
tff(fact_2430_mult__less__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int) ) ) ) ).

% mult_less_cancel_left1
tff(fact_2431_mult__less__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(C),A)),C)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A) ) ) ) ).

% mult_less_cancel_left2
tff(fact_2432_mult__less__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),A)),C)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A) ) ) ) ).

% mult_less_cancel_left2
tff(fact_2433_mult__less__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),C)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A) ) ) ) ).

% mult_less_cancel_left2
tff(fact_2434_mult__less__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(times_times_real(B),C))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B),one_one_real) ) ) ) ).

% mult_less_cancel_right1
tff(fact_2435_mult__less__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(times_times_rat(B),C))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),one_one_rat) ) ) ) ).

% mult_less_cancel_right1
tff(fact_2436_mult__less__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int) ) ) ) ).

% mult_less_cancel_right1
tff(fact_2437_mult__less__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),C)),C)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A) ) ) ) ).

% mult_less_cancel_right2
tff(fact_2438_mult__less__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),C)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A) ) ) ) ).

% mult_less_cancel_right2
tff(fact_2439_mult__less__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),C)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A) ) ) ) ).

% mult_less_cancel_right2
tff(fact_2440_divide__le__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(B),C)),A)
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(times_times_real(A),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)) ) ) ).

% divide_le_eq
tff(fact_2441_divide__le__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C)),A)
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)) ) ) ).

% divide_le_eq
tff(fact_2442_le__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(divide_divide_real(B),C))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),C)),B),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(times_times_real(A),C)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)) ) ) ).

% le_divide_eq
tff(fact_2443_le__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(divide_divide_rat(B),C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)) ) ) ).

% le_divide_eq
tff(fact_2444_divide__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(times_times_real(A),B))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(C),A)),aa_real_real(divide_divide_real(C),B)) ) ) ) ).

% divide_left_mono
tff(fact_2445_divide__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(C),A)),aa_rat_rat(divide_divide_rat(C),B)) ) ) ) ).

% divide_left_mono
tff(fact_2446_neg__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(B),C)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),C)),B) ) ) ).

% neg_divide_le_eq
tff(fact_2447_neg__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C)),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),B) ) ) ).

% neg_divide_le_eq
tff(fact_2448_neg__le__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(divide_divide_real(B),C))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(times_times_real(A),C)) ) ) ).

% neg_le_divide_eq
tff(fact_2449_neg__le__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(divide_divide_rat(B),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),C)) ) ) ).

% neg_le_divide_eq
tff(fact_2450_pos__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(B),C)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(times_times_real(A),C)) ) ) ).

% pos_divide_le_eq
tff(fact_2451_pos__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C)),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),C)) ) ) ).

% pos_divide_le_eq
tff(fact_2452_pos__le__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(divide_divide_real(B),C))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),C)),B) ) ) ).

% pos_le_divide_eq
tff(fact_2453_pos__le__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(divide_divide_rat(B),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),B) ) ) ).

% pos_le_divide_eq
tff(fact_2454_mult__imp__div__pos__le,axiom,
    ! [Y2: real,X: real,Z2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(times_times_real(Z2),Y2))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(X),Y2)),Z2) ) ) ).

% mult_imp_div_pos_le
tff(fact_2455_mult__imp__div__pos__le,axiom,
    ! [Y2: rat,X: rat,Z2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_rat_rat(times_times_rat(Z2),Y2))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(X),Y2)),Z2) ) ) ).

% mult_imp_div_pos_le
tff(fact_2456_mult__imp__le__div__pos,axiom,
    ! [Y2: real,Z2: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(Z2),Y2)),X)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z2),aa_real_real(divide_divide_real(X),Y2)) ) ) ).

% mult_imp_le_div_pos
tff(fact_2457_mult__imp__le__div__pos,axiom,
    ! [Y2: rat,Z2: rat,X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Z2),Y2)),X)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z2),aa_rat_rat(divide_divide_rat(X),Y2)) ) ) ).

% mult_imp_le_div_pos
tff(fact_2458_divide__left__mono__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(times_times_real(A),B))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(C),A)),aa_real_real(divide_divide_real(C),B)) ) ) ) ).

% divide_left_mono_neg
tff(fact_2459_divide__left__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(C),A)),aa_rat_rat(divide_divide_rat(C),B)) ) ) ) ).

% divide_left_mono_neg
tff(fact_2460_convex__bound__le,axiom,
    ! [X: real,A: real,Y2: real,U: real,V: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),U)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),V)
           => ( ( aa_real_real(plus_plus_real(U),V) = one_one_real )
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_real_real(times_times_real(U),X)),aa_real_real(times_times_real(V),Y2))),A) ) ) ) ) ) ).

% convex_bound_le
tff(fact_2461_convex__bound__le,axiom,
    ! [X: rat,A: rat,Y2: rat,U: rat,V: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y2),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),U)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),V)
           => ( ( aa_rat_rat(plus_plus_rat(U),V) = one_one_rat )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(U),X)),aa_rat_rat(times_times_rat(V),Y2))),A) ) ) ) ) ) ).

% convex_bound_le
tff(fact_2462_convex__bound__le,axiom,
    ! [X: int,A: int,Y2: int,U: int,V: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y2),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),U)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),V)
           => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,U),V) = one_one_int )
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,U),X)),aa_int_int(aa_int_fun_int_int(times_times_int,V),Y2))),A) ) ) ) ) ) ).

% convex_bound_le
tff(fact_2463_divide__le__eq__1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(B),A)),one_one_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) )
        | ( A = zero_zero_real ) ) ) ).

% divide_le_eq_1
tff(fact_2464_divide__le__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),A)),one_one_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) )
        | ( A = zero_zero_rat ) ) ) ).

% divide_le_eq_1
tff(fact_2465_le__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(divide_divide_real(B),A))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).

% le_divide_eq_1
tff(fact_2466_le__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B),A))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).

% le_divide_eq_1
tff(fact_2467_Suc__nat__number__of__add,axiom,
    ! [V: num,Nb: nat] : aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,V)),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(V),one))),Nb) ).

% Suc_nat_number_of_add
tff(fact_2468_divide__less__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(B),C)),aa_num_real(numeral_numeral_real,W))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,W)),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,W)),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_num_real(numeral_numeral_real,W))) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_2469_divide__less__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B),C)),aa_num_rat(numeral_numeral_rat,W))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,W))) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_2470_less__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,W)),aa_real_real(divide_divide_real(B),C))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,W)),C)),B),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,W)),C)),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,W)),zero_zero_real)) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_2471_less__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,W)),aa_rat_rat(divide_divide_rat(B),C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,W)),zero_zero_rat)) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_2472_power__Suc__less,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ) ).

% power_Suc_less
tff(fact_2473_power__Suc__less,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),Nb))),aa_nat_rat(power_power_rat(A),Nb)) ) ) ).

% power_Suc_less
tff(fact_2474_power__Suc__less,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ) ).

% power_Suc_less
tff(fact_2475_power__Suc__less,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb))),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)) ) ) ).

% power_Suc_less
tff(fact_2476_power__strict__decreasing,axiom,
    ! [Nb: nat,N2: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ) ) ).

% power_strict_decreasing
tff(fact_2477_power__strict__decreasing,axiom,
    ! [Nb: nat,N2: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),N2)),aa_nat_rat(power_power_rat(A),Nb)) ) ) ) ).

% power_strict_decreasing
tff(fact_2478_power__strict__decreasing,axiom,
    ! [Nb: nat,N2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),one_one_nat)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ) ) ).

% power_strict_decreasing
tff(fact_2479_power__strict__decreasing,axiom,
    ! [Nb: nat,N2: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N2)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)) ) ) ) ).

% power_strict_decreasing
tff(fact_2480_power__decreasing,axiom,
    ! [Nb: nat,N2: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ) ) ).

% power_decreasing
tff(fact_2481_power__decreasing,axiom,
    ! [Nb: nat,N2: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),N2)),aa_nat_rat(power_power_rat(A),Nb)) ) ) ) ).

% power_decreasing
tff(fact_2482_power__decreasing,axiom,
    ! [Nb: nat,N2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),one_one_nat)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ) ) ).

% power_decreasing
tff(fact_2483_power__decreasing,axiom,
    ! [Nb: nat,N2: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),N2)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)) ) ) ) ).

% power_decreasing
tff(fact_2484_zero__power2,axiom,
    aa_nat_rat(power_power_rat(zero_zero_rat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_rat ).

% zero_power2
tff(fact_2485_zero__power2,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% zero_power2
tff(fact_2486_zero__power2,axiom,
    aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_real ).

% zero_power2
tff(fact_2487_zero__power2,axiom,
    aa_nat_int(aa_int_fun_nat_int(power_power_int,zero_zero_int),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_int ).

% zero_power2
tff(fact_2488_zero__power2,axiom,
    aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,zero_zero_complex),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_complex ).

% zero_power2
tff(fact_2489_self__le__power,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ) ).

% self_le_power
tff(fact_2490_self__le__power,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_nat_rat(power_power_rat(A),Nb)) ) ) ).

% self_le_power
tff(fact_2491_self__le__power,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ) ).

% self_le_power
tff(fact_2492_self__le__power,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)) ) ) ).

% self_le_power
tff(fact_2493_one__less__power,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ) ).

% one_less_power
tff(fact_2494_one__less__power,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_nat_rat(power_power_rat(A),Nb)) ) ) ).

% one_less_power
tff(fact_2495_one__less__power,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ) ).

% one_less_power
tff(fact_2496_one__less__power,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)) ) ) ).

% one_less_power
tff(fact_2497_pos2,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% pos2
tff(fact_2498_invar__vebt_Osimps,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( aa_nat_o(vEBT_invar_vebt(A1),A22)
    <=> ( ( ? [A5: $o,B5: $o] : A1 = aa_o_VEBT_VEBT(vEBT_Leaf((A5)),(B5))
          & ( A22 = aa_nat_nat(suc,zero_zero_nat) ) )
        | ? [TreeList2: list_VEBT_VEBT,N4: nat,Summary2: vEBT_VEBT] :
            ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,A22,TreeList2),Summary2) )
            & ! [X3: vEBT_VEBT] :
                ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList2))
               => aa_nat_o(vEBT_invar_vebt(X3),N4) )
            & aa_nat_o(vEBT_invar_vebt(Summary2),N4)
            & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N4),N4) )
            & ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),X_12)
            & ! [X3: vEBT_VEBT] :
                ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList2))
               => ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X_12) ) )
        | ? [TreeList2: list_VEBT_VEBT,N4: nat,Summary2: vEBT_VEBT] :
            ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,A22,TreeList2),Summary2) )
            & ! [X3: vEBT_VEBT] :
                ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList2))
               => aa_nat_o(vEBT_invar_vebt(X3),N4) )
            & aa_nat_o(vEBT_invar_vebt(Summary2),aa_nat_nat(suc,N4))
            & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N4)) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N4),aa_nat_nat(suc,N4)) )
            & ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),X_12)
            & ! [X3: vEBT_VEBT] :
                ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList2))
               => ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X_12) ) )
        | ? [TreeList2: list_VEBT_VEBT,N4: nat,Summary2: vEBT_VEBT,Mi2: nat,Ma2: nat] :
            ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),A22,TreeList2),Summary2) )
            & ! [X3: vEBT_VEBT] :
                ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList2))
               => aa_nat_o(vEBT_invar_vebt(X3),N4) )
            & aa_nat_o(vEBT_invar_vebt(Summary2),N4)
            & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N4),N4) )
            & ! [I: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4))
               => ( ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),I)),X_12)
                <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),I) ) )
            & ( ( Mi2 = Ma2 )
             => ! [X3: vEBT_VEBT] :
                  ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList2))
                 => ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X_12) ) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi2),Ma2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A22))
            & ( ( Mi2 != Ma2 )
             => ! [I: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4))
                 => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),N4) = I )
                     => aa_nat_o(vEBT_V8194947554948674370ptions(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),I)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma2),N4)) )
                    & ! [X3: nat] :
                        ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),N4) = I )
                          & aa_nat_o(vEBT_V8194947554948674370ptions(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),I)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),N4)) )
                       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi2),X3)
                          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Ma2) ) ) ) ) ) )
        | ? [TreeList2: list_VEBT_VEBT,N4: nat,Summary2: vEBT_VEBT,Mi2: nat,Ma2: nat] :
            ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi2),Ma2)),A22,TreeList2),Summary2) )
            & ! [X3: vEBT_VEBT] :
                ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList2))
               => aa_nat_o(vEBT_invar_vebt(X3),N4) )
            & aa_nat_o(vEBT_invar_vebt(Summary2),aa_nat_nat(suc,N4))
            & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N4)) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N4),aa_nat_nat(suc,N4)) )
            & ! [I: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N4)))
               => ( ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),I)),X_12)
                <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),I) ) )
            & ( ( Mi2 = Ma2 )
             => ! [X3: vEBT_VEBT] :
                  ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList2))
                 => ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X_12) ) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi2),Ma2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A22))
            & ( ( Mi2 != Ma2 )
             => ! [I: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N4)))
                 => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),N4) = I )
                     => aa_nat_o(vEBT_V8194947554948674370ptions(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),I)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma2),N4)) )
                    & ! [X3: nat] :
                        ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),N4) = I )
                          & aa_nat_o(vEBT_V8194947554948674370ptions(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),I)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),N4)) )
                       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi2),X3)
                          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Ma2) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
tff(fact_2499_invar__vebt_Ocases,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( aa_nat_o(vEBT_invar_vebt(A1),A22)
     => ( ( ? [A3: $o,B2: $o] : A1 = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B2))
         => ( A22 != aa_nat_nat(suc,zero_zero_nat) ) )
       => ( ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat] :
              ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList),Summary) )
             => ( ( A22 = Deg )
               => ( ! [X2: vEBT_VEBT] :
                      ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(TreeList))
                     => aa_nat_o(vEBT_invar_vebt(X2),N) )
                 => ( aa_nat_o(vEBT_invar_vebt(Summary),M2)
                   => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2) )
                     => ( ( M2 = N )
                       => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M2) )
                         => ( ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X_1)
                           => ~ ! [X2: vEBT_VEBT] :
                                  ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(TreeList))
                                 => ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X2),X_1) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat] :
                ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList),Summary) )
               => ( ( A22 = Deg )
                 => ( ! [X2: vEBT_VEBT] :
                        ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(TreeList))
                       => aa_nat_o(vEBT_invar_vebt(X2),N) )
                   => ( aa_nat_o(vEBT_invar_vebt(Summary),M2)
                     => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2) )
                       => ( ( M2 = aa_nat_nat(suc,N) )
                         => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M2) )
                           => ( ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X_1)
                             => ~ ! [X2: vEBT_VEBT] :
                                    ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(TreeList))
                                   => ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X2),X_1) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat,Mi3: nat,Ma3: nat] :
                  ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi3),Ma3)),Deg,TreeList),Summary) )
                 => ( ( A22 = Deg )
                   => ( ! [X2: vEBT_VEBT] :
                          ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(TreeList))
                         => aa_nat_o(vEBT_invar_vebt(X2),N) )
                     => ( aa_nat_o(vEBT_invar_vebt(Summary),M2)
                       => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2) )
                         => ( ( M2 = N )
                           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M2) )
                             => ( ! [I4: nat] :
                                    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2))
                                   => ( ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),I4)),X_12)
                                    <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summary),I4) ) )
                               => ( ( ( Mi3 = Ma3 )
                                   => ! [X2: vEBT_VEBT] :
                                        ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(TreeList))
                                       => ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X2),X_1) ) )
                                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi3),Ma3)
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
                                     => ~ ( ( Mi3 != Ma3 )
                                         => ! [I4: nat] :
                                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2))
                                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma3),N) = I4 )
                                                 => aa_nat_o(vEBT_V8194947554948674370ptions(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma3),N)) )
                                                & ! [X2: nat] :
                                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),N) = I4 )
                                                      & aa_nat_o(vEBT_V8194947554948674370ptions(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),N)) )
                                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi3),X2)
                                                      & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Ma3) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat,Mi3: nat,Ma3: nat] :
                    ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi3),Ma3)),Deg,TreeList),Summary) )
                   => ( ( A22 = Deg )
                     => ( ! [X2: vEBT_VEBT] :
                            ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(TreeList))
                           => aa_nat_o(vEBT_invar_vebt(X2),N) )
                       => ( aa_nat_o(vEBT_invar_vebt(Summary),M2)
                         => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2) )
                           => ( ( M2 = aa_nat_nat(suc,N) )
                             => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M2) )
                               => ( ! [I4: nat] :
                                      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2))
                                     => ( ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),I4)),X_12)
                                      <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summary),I4) ) )
                                 => ( ( ( Mi3 = Ma3 )
                                     => ! [X2: vEBT_VEBT] :
                                          ( member_VEBT_VEBT(X2,set_VEBT_VEBT2(TreeList))
                                         => ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X2),X_1) ) )
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi3),Ma3)
                                     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
                                       => ~ ( ( Mi3 != Ma3 )
                                           => ! [I4: nat] :
                                                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2))
                                               => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma3),N) = I4 )
                                                   => aa_nat_o(vEBT_V8194947554948674370ptions(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma3),N)) )
                                                  & ! [X2: nat] :
                                                      ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),N) = I4 )
                                                        & aa_nat_o(vEBT_V8194947554948674370ptions(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),N)) )
                                                     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi3),X2)
                                                        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Ma3) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
tff(fact_2500_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Q)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),Q))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Q)),Nb) ) ) ).

% less_eq_div_iff_mult_less_eq
tff(fact_2501_split__div,axiom,
    ! [P: fun_nat_o,Ma: nat,Nb: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),Nb))
    <=> ( ( ( Nb = zero_zero_nat )
         => aa_nat_o(P,zero_zero_nat) )
        & ( ( Nb != zero_zero_nat )
         => ! [I: nat,J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),Nb)
             => ( ( Ma = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),I)),J3) )
               => aa_nat_o(P,I) ) ) ) ) ) ).

% split_div
tff(fact_2502_dividend__less__div__times,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),Nb)),Nb))) ) ).

% dividend_less_div_times
tff(fact_2503_dividend__less__times__div,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),Nb)))) ) ).

% dividend_less_times_div
tff(fact_2504_split__mod,axiom,
    ! [P: fun_nat_o,Ma: nat,Nb: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),Nb))
    <=> ( ( ( Nb = zero_zero_nat )
         => aa_nat_o(P,Ma) )
        & ( ( Nb != zero_zero_nat )
         => ! [I: nat,J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),Nb)
             => ( ( Ma = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),I)),J3) )
               => aa_nat_o(P,J3) ) ) ) ) ) ).

% split_mod
tff(fact_2505_Collect__subset,axiom,
    ! [A2: set_real,P: fun_real_o] : ord_less_eq_set_real(collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_az(A2),P)),A2) ).

% Collect_subset
tff(fact_2506_Collect__subset,axiom,
    ! [A2: set_complex,P: fun_complex_o] : ord_le211207098394363844omplex(collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_ba(A2),P)),A2) ).

% Collect_subset
tff(fact_2507_Collect__subset,axiom,
    ! [A2: set_list_nat,P: fun_list_nat_o] : ord_le6045566169113846134st_nat(collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_bb(A2),P)),A2) ).

% Collect_subset
tff(fact_2508_Collect__subset,axiom,
    ! [A2: set_set_nat,P: fun_set_nat_o] : ord_le6893508408891458716et_nat(collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_bc(A2),P)),A2) ).

% Collect_subset
tff(fact_2509_Collect__subset,axiom,
    ! [A2: set_nat,P: fun_nat_o] : aa_set_nat_o(ord_less_eq_set_nat(collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_bd(A2),P))),A2) ).

% Collect_subset
tff(fact_2510_Collect__subset,axiom,
    ! [A2: set_int,P: fun_int_o] : ord_less_eq_set_int(collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_be(A2),P)),A2) ).

% Collect_subset
tff(fact_2511_less__eq__set__def,axiom,
    ! [A2: set_complex,B3: set_complex] :
      ( ord_le211207098394363844omplex(A2,B3)
    <=> ord_le7644322562001619171plex_o(aTP_Lamm_ac(A2),aTP_Lamm_ac(B3)) ) ).

% less_eq_set_def
tff(fact_2512_less__eq__set__def,axiom,
    ! [A2: set_real,B3: set_real] :
      ( ord_less_eq_set_real(A2,B3)
    <=> ord_le3605727316730466277real_o(aTP_Lamm_ab(A2),aTP_Lamm_ab(B3)) ) ).

% less_eq_set_def
tff(fact_2513_less__eq__set__def,axiom,
    ! [A2: set_set_nat,B3: set_set_nat] :
      ( ord_le6893508408891458716et_nat(A2,B3)
    <=> ord_le6468474429079055883_nat_o(aTP_Lamm_ae(A2),aTP_Lamm_ae(B3)) ) ).

% less_eq_set_def
tff(fact_2514_less__eq__set__def,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B3)
    <=> ord_le7240017405068453697_nat_o(aTP_Lamm_af(A2),aTP_Lamm_af(B3)) ) ).

% less_eq_set_def
tff(fact_2515_less__eq__set__def,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ord_less_eq_set_int(A2,B3)
    <=> ord_le5418975059635765733_int_o(aTP_Lamm_ag(A2),aTP_Lamm_ag(B3)) ) ).

% less_eq_set_def
tff(fact_2516_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeListb: list_VEBT_VEBT,Nb: nat,Summaryb: vEBT_VEBT,Ma: nat,Degb: nat,Mi: nat,Maa: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeListb))
         => aa_nat_o(vEBT_invar_vebt(X4),Nb) )
     => ( aa_nat_o(vEBT_invar_vebt(Summaryb),Ma)
       => ( ( size_s6755466524823107622T_VEBT(TreeListb) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma) )
         => ( ( Ma = aa_nat_nat(suc,Nb) )
           => ( ( Degb = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Ma) )
             => ( ! [I3: nat] :
                    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma))
                   => ( ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeListb),I3)),X_12)
                    <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summaryb),I3) ) )
               => ( ( ( Mi = Maa )
                   => ! [X4: vEBT_VEBT] :
                        ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeListb))
                       => ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X_13) ) )
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi),Maa)
                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Degb))
                     => ( ( ( Mi != Maa )
                         => ! [I3: nat] :
                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma))
                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Maa),Nb) = I3 )
                                 => aa_nat_o(vEBT_V8194947554948674370ptions(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeListb),I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Maa),Nb)) )
                                & ! [X4: nat] :
                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X4),Nb) = I3 )
                                      & aa_nat_o(vEBT_V8194947554948674370ptions(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeListb),I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X4),Nb)) )
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),X4)
                                      & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Maa) ) ) ) ) )
                       => aa_nat_o(vEBT_invar_vebt(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Maa)),Degb,TreeListb),Summaryb)),Degb) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
tff(fact_2517_convex__bound__lt,axiom,
    ! [X: real,A: real,Y2: real,U: real,V: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),U)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),V)
           => ( ( aa_real_real(plus_plus_real(U),V) = one_one_real )
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(aa_real_real(times_times_real(U),X)),aa_real_real(times_times_real(V),Y2))),A) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_2518_convex__bound__lt,axiom,
    ! [X: rat,A: rat,Y2: rat,U: rat,V: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y2),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),U)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),V)
           => ( ( aa_rat_rat(plus_plus_rat(U),V) = one_one_rat )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(U),X)),aa_rat_rat(times_times_rat(V),Y2))),A) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_2519_convex__bound__lt,axiom,
    ! [X: int,A: int,Y2: int,U: int,V: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y2),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),U)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),V)
           => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,U),V) = one_one_int )
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,U),X)),aa_int_int(aa_int_fun_int_int(times_times_int,V),Y2))),A) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_2520_divide__le__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(B),C)),aa_num_real(numeral_numeral_real,W))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,W)),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,W)),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_num_real(numeral_numeral_real,W))) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_2521_divide__le__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C)),aa_num_rat(numeral_numeral_rat,W))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,W))) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_2522_le__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,W)),aa_real_real(divide_divide_real(B),C))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,W)),C)),B),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,W)),C)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,W)),zero_zero_real)) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_2523_le__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,W)),aa_rat_rat(divide_divide_rat(B),C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,W)),zero_zero_rat)) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_2524_half__gt__zero,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(divide_divide_real(A),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ).

% half_gt_zero
tff(fact_2525_half__gt__zero,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(A),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ) ).

% half_gt_zero
tff(fact_2526_half__gt__zero__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(divide_divide_real(A),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% half_gt_zero_iff
tff(fact_2527_half__gt__zero__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(A),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% half_gt_zero_iff
tff(fact_2528_zero__le__power2,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% zero_le_power2
tff(fact_2529_zero__le__power2,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% zero_le_power2
tff(fact_2530_zero__le__power2,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% zero_le_power2
tff(fact_2531_power2__eq__imp__eq,axiom,
    ! [X: real,Y2: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y2)
         => ( X = Y2 ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_2532_power2__eq__imp__eq,axiom,
    ! [X: rat,Y2: rat] :
      ( ( aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y2)
         => ( X = Y2 ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_2533_power2__eq__imp__eq,axiom,
    ! [X: nat,Y2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),X)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y2)
         => ( X = Y2 ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_2534_power2__eq__imp__eq,axiom,
    ! [X: int,Y2: int] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y2)
         => ( X = Y2 ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_2535_power2__le__imp__le,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y2) ) ) ).

% power2_le_imp_le
tff(fact_2536_power2__le__imp__le,axiom,
    ! [X: rat,Y2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y2) ) ) ).

% power2_le_imp_le
tff(fact_2537_power2__le__imp__le,axiom,
    ! [X: nat,Y2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y2) ) ) ).

% power2_le_imp_le
tff(fact_2538_power2__le__imp__le,axiom,
    ! [X: int,Y2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y2) ) ) ).

% power2_le_imp_le
tff(fact_2539_power2__less__0,axiom,
    ! [A: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_real) ).

% power2_less_0
tff(fact_2540_power2__less__0,axiom,
    ! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_rat) ).

% power2_less_0
tff(fact_2541_power2__less__0,axiom,
    ! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_int) ).

% power2_less_0
tff(fact_2542_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
     => ( modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),modulo364778990260209775nteger(divide6298287555418463151nteger(A,B),C)),modulo364778990260209775nteger(A,B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_2543_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_2544_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_2545_exp__add__not__zero__imp__left,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb)) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma) != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_left
tff(fact_2546_exp__add__not__zero__imp__left,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb)) != zero_zero_int )
     => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Ma) != zero_zero_int ) ) ).

% exp_add_not_zero_imp_left
tff(fact_2547_exp__add__not__zero__imp__right,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb)) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_right
tff(fact_2548_exp__add__not__zero__imp__right,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb)) != zero_zero_int )
     => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb) != zero_zero_int ) ) ).

% exp_add_not_zero_imp_right
tff(fact_2549_power__odd__eq,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_complex_complex(times_times_complex(A),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_2550_power__odd__eq,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_real_real(times_times_real(A),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_2551_power__odd__eq,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(aa_nat_rat(power_power_rat(A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_2552_power__odd__eq,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_2553_power__odd__eq,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_2554_power2__less__imp__less,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2) ) ) ).

% power2_less_imp_less
tff(fact_2555_power2__less__imp__less,axiom,
    ! [X: rat,Y2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y2) ) ) ).

% power2_less_imp_less
tff(fact_2556_power2__less__imp__less,axiom,
    ! [X: nat,Y2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y2) ) ) ).

% power2_less_imp_less
tff(fact_2557_power2__less__imp__less,axiom,
    ! [X: int,Y2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y2) ) ) ).

% power2_less_imp_less
tff(fact_2558_sum__power2__ge__zero,axiom,
    ! [X: real,Y2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_power2_ge_zero
tff(fact_2559_sum__power2__ge__zero,axiom,
    ! [X: rat,Y2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_power2_ge_zero
tff(fact_2560_sum__power2__ge__zero,axiom,
    ! [X: int,Y2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_power2_ge_zero
tff(fact_2561_sum__power2__le__zero__iff,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_real)
    <=> ( ( X = zero_zero_real )
        & ( Y2 = zero_zero_real ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_2562_sum__power2__le__zero__iff,axiom,
    ! [X: rat,Y2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_rat)
    <=> ( ( X = zero_zero_rat )
        & ( Y2 = zero_zero_rat ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_2563_sum__power2__le__zero__iff,axiom,
    ! [X: int,Y2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_int)
    <=> ( ( X = zero_zero_int )
        & ( Y2 = zero_zero_int ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_2564_divmod__digit__0_I2_J,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))),B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ) ) ) ).

% divmod_digit_0(2)
tff(fact_2565_divmod__digit__0_I2_J,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),B)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ) ) ) ).

% divmod_digit_0(2)
tff(fact_2566_divmod__digit__0_I2_J,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B))),B)
       => ( modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B)) = modulo364778990260209775nteger(A,B) ) ) ) ).

% divmod_digit_0(2)
tff(fact_2567_not__sum__power2__lt__zero,axiom,
    ! [X: real,Y2: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_real) ).

% not_sum_power2_lt_zero
tff(fact_2568_not__sum__power2__lt__zero,axiom,
    ! [X: rat,Y2: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_rat) ).

% not_sum_power2_lt_zero
tff(fact_2569_not__sum__power2__lt__zero,axiom,
    ! [X: int,Y2: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_int) ).

% not_sum_power2_lt_zero
tff(fact_2570_sum__power2__gt__zero__iff,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( ( X != zero_zero_real )
        | ( Y2 != zero_zero_real ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_2571_sum__power2__gt__zero__iff,axiom,
    ! [X: rat,Y2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( ( X != zero_zero_rat )
        | ( Y2 != zero_zero_rat ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_2572_sum__power2__gt__zero__iff,axiom,
    ! [X: int,Y2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( ( X != zero_zero_int )
        | ( Y2 != zero_zero_int ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_2573_bits__stable__imp__add__self,axiom,
    ! [A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = A )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_nat ) ) ).

% bits_stable_imp_add_self
tff(fact_2574_bits__stable__imp__add__self,axiom,
    ! [A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) = zero_zero_int ) ) ).

% bits_stable_imp_add_self
tff(fact_2575_bits__stable__imp__add__self,axiom,
    ! [A: code_integer] :
      ( ( divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = A )
     => ( plus_p5714425477246183910nteger(A,modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))) = zero_z3403309356797280102nteger ) ) ).

% bits_stable_imp_add_self
tff(fact_2576_zero__le__even__power_H,axiom,
    ! [A: real,Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% zero_le_even_power'
tff(fact_2577_zero__le__even__power_H,axiom,
    ! [A: rat,Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% zero_le_even_power'
tff(fact_2578_zero__le__even__power_H,axiom,
    ! [A: int,Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% zero_le_even_power'
tff(fact_2579_verit__le__mono__div,axiom,
    ! [A2: nat,B3: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,
              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),Nb)),
                $ite(aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B3),Nb) = zero_zero_nat,one_one_nat,zero_zero_nat))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B3),Nb)) ) ) ).

% verit_le_mono_div
tff(fact_2580_divmod__digit__0_I1_J,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))),B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ) ).

% divmod_digit_0(1)
tff(fact_2581_divmod__digit__0_I1_J,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),B)
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ) ).

% divmod_digit_0(1)
tff(fact_2582_divmod__digit__0_I1_J,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B))),B)
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B))) = divide6298287555418463151nteger(A,B) ) ) ) ).

% divmod_digit_0(1)
tff(fact_2583_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeListb: list_VEBT_VEBT,Nb: nat,Summaryb: vEBT_VEBT,Ma: nat,Degb: nat,Mi: nat,Maa: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeListb))
         => aa_nat_o(vEBT_invar_vebt(X4),Nb) )
     => ( aa_nat_o(vEBT_invar_vebt(Summaryb),Ma)
       => ( ( size_s6755466524823107622T_VEBT(TreeListb) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma) )
         => ( ( Ma = Nb )
           => ( ( Degb = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Ma) )
             => ( ! [I3: nat] :
                    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma))
                   => ( ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeListb),I3)),X_12)
                    <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summaryb),I3) ) )
               => ( ( ( Mi = Maa )
                   => ! [X4: vEBT_VEBT] :
                        ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeListb))
                       => ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X_13) ) )
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi),Maa)
                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Degb))
                     => ( ( ( Mi != Maa )
                         => ! [I3: nat] :
                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma))
                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Maa),Nb) = I3 )
                                 => aa_nat_o(vEBT_V8194947554948674370ptions(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeListb),I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Maa),Nb)) )
                                & ! [X4: nat] :
                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X4),Nb) = I3 )
                                      & aa_nat_o(vEBT_V8194947554948674370ptions(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeListb),I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X4),Nb)) )
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),X4)
                                      & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Maa) ) ) ) ) )
                       => aa_nat_o(vEBT_invar_vebt(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Maa)),Degb,TreeListb),Summaryb)),Degb) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
tff(fact_2584_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [X: nat,Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Ma)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ma)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma)) ) ) ) ).

% VEBT_internal.exp_split_high_low(1)
tff(fact_2585_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [X: nat,Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Ma)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ma)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
tff(fact_2586_not__mod__2__eq__0__eq__1,axiom,
    ! [A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != zero_zero_nat )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_2587_not__mod__2__eq__0__eq__1,axiom,
    ! [A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != zero_zero_int )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_2588_not__mod__2__eq__0__eq__1,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) != zero_z3403309356797280102nteger )
    <=> ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = one_one_Code_integer ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_2589_not__mod__2__eq__1__eq__0,axiom,
    ! [A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != one_one_nat )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_2590_not__mod__2__eq__1__eq__0,axiom,
    ! [A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != one_one_int )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_2591_not__mod__2__eq__1__eq__0,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) != one_one_Code_integer )
    <=> ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_2592_not__mod2__eq__Suc__0__eq__0,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != aa_nat_nat(suc,zero_zero_nat) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) ) ).

% not_mod2_eq_Suc_0_eq_0
tff(fact_2593_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ~ aa_nat_o(vEBT_VEBT_membermima(X),Xa)
     => ( ! [Uu2: $o,Uv2: $o] : X != aa_o_VEBT_VEBT(vEBT_Leaf((Uu2)),(Uv2))
       => ( ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : X != aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux),Uy)
         => ( ! [Mi3: nat,Ma3: nat] :
                ( ? [Va2: list_VEBT_VEBT,Vb: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi3),Ma3)),zero_zero_nat,Va2),Vb)
               => ( ( Xa = Mi3 )
                  | ( Xa = Ma3 ) ) )
           => ( ! [Mi3: nat,Ma3: nat,V2: nat,TreeList: list_VEBT_VEBT] :
                  ( ? [Vc: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi3),Ma3)),aa_nat_nat(suc,V2),TreeList),Vc)
                 => ( ( Xa = Mi3 )
                    | ( Xa = Ma3 )
                    | $let(
                        pos: nat,
                        pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) )
             => ~ ! [V2: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Vd: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList),Vd)
                   => $let(
                        pos: nat,
                        pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
tff(fact_2594_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y2: $o] :
      ( ( aa_nat_o(vEBT_VEBT_membermima(X),Xa)
      <=> (Y2) )
     => ( ( ? [Uu2: $o,Uv2: $o] : X = aa_o_VEBT_VEBT(vEBT_Leaf((Uu2)),(Uv2))
         => (Y2) )
       => ( ( ? [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux),Uy)
           => (Y2) )
         => ( ! [Mi3: nat,Ma3: nat] :
                ( ? [Va2: list_VEBT_VEBT,Vb: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi3),Ma3)),zero_zero_nat,Va2),Vb)
               => ( (Y2)
                <=> ~ ( ( Xa = Mi3 )
                      | ( Xa = Ma3 ) ) ) )
           => ( ! [Mi3: nat,Ma3: nat,V2: nat,TreeList: list_VEBT_VEBT] :
                  ( ? [Vc: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi3),Ma3)),aa_nat_nat(suc,V2),TreeList),Vc)
                 => ( (Y2)
                  <=> ~ ( ( Xa = Mi3 )
                        | ( Xa = Ma3 )
                        | $let(
                            pos: nat,
                            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) )
             => ~ ! [V2: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Vd: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList),Vd)
                   => ( (Y2)
                    <=> ~ $let(
                            pos: nat,
                            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
tff(fact_2595_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ~ aa_nat_o(vEBT_V5719532721284313246member(X),Xa)
     => ( ! [A3: $o,B2: $o] :
            ( ( X = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B2)) )
           => $ite(
                Xa = zero_zero_nat,
                (A3),
                $ite(Xa = one_one_nat,(B2),$false) ) )
       => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw: vEBT_VEBT] : X != aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,zero_zero_nat,Uv2),Uw)
         => ~ ! [Uy: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT] :
                ( ? [S2: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy,aa_nat_nat(suc,V2),TreeList),S2)
               => $let(
                    pos: nat,
                    pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_V5719532721284313246member(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
tff(fact_2596_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( aa_nat_o(vEBT_V5719532721284313246member(X),Xa)
     => ( ! [A3: $o,B2: $o] :
            ( ( X = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B2)) )
           => ~ $ite(
                  Xa = zero_zero_nat,
                  (A3),
                  $ite(Xa = one_one_nat,(B2),$false) ) )
       => ~ ! [Uy: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT] :
              ( ? [S2: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy,aa_nat_nat(suc,V2),TreeList),S2)
             => ~ $let(
                    pos: nat,
                    pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_V5719532721284313246member(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
tff(fact_2597_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y2: $o] :
      ( ( aa_nat_o(vEBT_V5719532721284313246member(X),Xa)
      <=> (Y2) )
     => ( ! [A3: $o,B2: $o] :
            ( ( X = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B2)) )
           => ( (Y2)
            <=> ~ $ite(
                    Xa = zero_zero_nat,
                    (A3),
                    $ite(Xa = one_one_nat,(B2),$false) ) ) )
       => ( ( ? [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,zero_zero_nat,Uv2),Uw)
           => (Y2) )
         => ~ ! [Uy: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT] :
                ( ? [S2: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy,aa_nat_nat(suc,V2),TreeList),S2)
               => ( (Y2)
                <=> ~ $let(
                        pos: nat,
                        pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_V5719532721284313246member(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
tff(fact_2598_buildup__gives__valid,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(vEBT_invar_vebt(aa_nat_VEBT_VEBT(vEBT_vebt_buildup,Nb)),Nb) ) ).

% buildup_gives_valid
tff(fact_2599_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( aa_nat_o(vEBT_VEBT_membermima(X),Xa)
     => ( ! [Mi3: nat,Ma3: nat] :
            ( ? [Va2: list_VEBT_VEBT,Vb: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi3),Ma3)),zero_zero_nat,Va2),Vb)
           => ~ ( ( Xa = Mi3 )
                | ( Xa = Ma3 ) ) )
       => ( ! [Mi3: nat,Ma3: nat,V2: nat,TreeList: list_VEBT_VEBT] :
              ( ? [Vc: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi3),Ma3)),aa_nat_nat(suc,V2),TreeList),Vc)
             => ~ ( ( Xa = Mi3 )
                  | ( Xa = Ma3 )
                  | $let(
                      pos: nat,
                      pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) )
         => ~ ! [V2: nat,TreeList: list_VEBT_VEBT] :
                ( ? [Vd: vEBT_VEBT] : X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList),Vd)
               => ~ $let(
                      pos: nat,
                      pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
tff(fact_2600_not__None__eq,axiom,
    ! [X: option4927543243414619207at_nat] :
      ( ( X != none_P5556105721700978146at_nat )
    <=> ? [Y: product_prod_nat_nat] : X = some_P7363390416028606310at_nat(Y) ) ).

% not_None_eq
tff(fact_2601_not__None__eq,axiom,
    ! [X: option_num] :
      ( ( X != none_num )
    <=> ? [Y: num] : X = some_num(Y) ) ).

% not_None_eq
tff(fact_2602_buildup__nothing__in__min__max,axiom,
    ! [Nb: nat,X: nat] : ~ aa_nat_o(vEBT_VEBT_membermima(aa_nat_VEBT_VEBT(vEBT_vebt_buildup,Nb)),X) ).

% buildup_nothing_in_min_max
tff(fact_2603_buildup__nothing__in__leaf,axiom,
    ! [Nb: nat,X: nat] : ~ aa_nat_o(vEBT_V5719532721284313246member(aa_nat_VEBT_VEBT(vEBT_vebt_buildup,Nb)),X) ).

% buildup_nothing_in_leaf
tff(fact_2604_both__member__options__def,axiom,
    ! [T: vEBT_VEBT,X: nat] :
      ( aa_nat_o(vEBT_V8194947554948674370ptions(T),X)
    <=> ( aa_nat_o(vEBT_V5719532721284313246member(T),X)
        | aa_nat_o(vEBT_VEBT_membermima(T),X) ) ) ).

% both_member_options_def
tff(fact_2605_not__Some__eq,axiom,
    ! [X: option4927543243414619207at_nat] :
      ( ! [Y: product_prod_nat_nat] : X != some_P7363390416028606310at_nat(Y)
    <=> ( X = none_P5556105721700978146at_nat ) ) ).

% not_Some_eq
tff(fact_2606_not__Some__eq,axiom,
    ! [X: option_num] :
      ( ! [Y: num] : X != some_num(Y)
    <=> ( X = none_num ) ) ).

% not_Some_eq
tff(fact_2607_verit__la__generic,axiom,
    ! [A: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),X)
      | ( A = X )
      | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),A) ) ).

% verit_la_generic
tff(fact_2608_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o,Uw2: nat] : ~ aa_nat_o(vEBT_VEBT_membermima(aa_o_VEBT_VEBT(vEBT_Leaf((Uu)),(Uv))),Uw2) ).

% VEBT_internal.membermima.simps(1)
tff(fact_2609_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
    ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] : ~ aa_nat_o(vEBT_V5719532721284313246member(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu,zero_zero_nat,Uv),Uw2)),Ux2) ).

% VEBT_internal.naive_member.simps(2)
tff(fact_2610_vebt__buildup_Osimps_I1_J,axiom,
    aa_nat_VEBT_VEBT(vEBT_vebt_buildup,zero_zero_nat) = aa_o_VEBT_VEBT(vEBT_Leaf($false),$false) ).

% vebt_buildup.simps(1)
tff(fact_2611_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
    ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz: nat] : ~ aa_nat_o(vEBT_VEBT_membermima(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2),Uy2)),Uz) ).

% VEBT_internal.membermima.simps(2)
tff(fact_2612_vebt__buildup_Osimps_I2_J,axiom,
    aa_nat_VEBT_VEBT(vEBT_vebt_buildup,aa_nat_nat(suc,zero_zero_nat)) = aa_o_VEBT_VEBT(vEBT_Leaf($false),$false) ).

% vebt_buildup.simps(2)
tff(fact_2613_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,X: nat] :
      ( aa_nat_o(vEBT_V5719532721284313246member(aa_o_VEBT_VEBT(vEBT_Leaf((A)),(B))),X)
    <=> $ite(
          X = zero_zero_nat,
          (A),
          $ite(X = one_one_nat,(B),$false) ) ) ).

% VEBT_internal.naive_member.simps(1)
tff(fact_2614_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT,X: nat] :
      ( aa_nat_o(vEBT_VEBT_membermima(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),zero_zero_nat,Va3),Vb2)),X)
    <=> ( ( X = Mi )
        | ( X = Ma ) ) ) ).

% VEBT_internal.membermima.simps(3)
tff(fact_2615_option_Osize_I3_J,axiom,
    size_s170228958280169651at_nat(none_P5556105721700978146at_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_2616_option_Osize_I3_J,axiom,
    size_size_option_num(none_num) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_2617_combine__options__cases,axiom,
    ! [X: option4927543243414619207at_nat,P: fun_op1221328669798608686_nat_o,Y2: option4927543243414619207at_nat] :
      ( ( ( X = none_P5556105721700978146at_nat )
       => aa_opt7112393969400857165_nat_o(aa_opt9039718665754746463_nat_o(P,X),Y2) )
     => ( ( ( Y2 = none_P5556105721700978146at_nat )
         => aa_opt7112393969400857165_nat_o(aa_opt9039718665754746463_nat_o(P,X),Y2) )
       => ( ! [A3: product_prod_nat_nat,B2: product_prod_nat_nat] :
              ( ( X = some_P7363390416028606310at_nat(A3) )
             => ( ( Y2 = some_P7363390416028606310at_nat(B2) )
               => aa_opt7112393969400857165_nat_o(aa_opt9039718665754746463_nat_o(P,X),Y2) ) )
         => aa_opt7112393969400857165_nat_o(aa_opt9039718665754746463_nat_o(P,X),Y2) ) ) ) ).

% combine_options_cases
tff(fact_2618_combine__options__cases,axiom,
    ! [X: option4927543243414619207at_nat,P: fun_op7653269905477231939_num_o,Y2: option_num] :
      ( ( ( X = none_P5556105721700978146at_nat )
       => aa_option_num_o(aa_opt7868583491048818492_num_o(P,X),Y2) )
     => ( ( ( Y2 = none_num )
         => aa_option_num_o(aa_opt7868583491048818492_num_o(P,X),Y2) )
       => ( ! [A3: product_prod_nat_nat,B2: num] :
              ( ( X = some_P7363390416028606310at_nat(A3) )
             => ( ( Y2 = some_num(B2) )
               => aa_option_num_o(aa_opt7868583491048818492_num_o(P,X),Y2) ) )
         => aa_option_num_o(aa_opt7868583491048818492_num_o(P,X),Y2) ) ) ) ).

% combine_options_cases
tff(fact_2619_combine__options__cases,axiom,
    ! [X: option_num,P: fun_op5373868528577402299_nat_o,Y2: option4927543243414619207at_nat] :
      ( ( ( X = none_num )
       => aa_opt7112393969400857165_nat_o(aa_opt1662640216596507900_nat_o(P,X),Y2) )
     => ( ( ( Y2 = none_P5556105721700978146at_nat )
         => aa_opt7112393969400857165_nat_o(aa_opt1662640216596507900_nat_o(P,X),Y2) )
       => ( ! [A3: num,B2: product_prod_nat_nat] :
              ( ( X = some_num(A3) )
             => ( ( Y2 = some_P7363390416028606310at_nat(B2) )
               => aa_opt7112393969400857165_nat_o(aa_opt1662640216596507900_nat_o(P,X),Y2) ) )
         => aa_opt7112393969400857165_nat_o(aa_opt1662640216596507900_nat_o(P,X),Y2) ) ) ) ).

% combine_options_cases
tff(fact_2620_combine__options__cases,axiom,
    ! [X: option_num,P: fun_op493924057549976310_num_o,Y2: option_num] :
      ( ( ( X = none_num )
       => aa_option_num_o(aa_opt8674010985164783839_num_o(P,X),Y2) )
     => ( ( ( Y2 = none_num )
         => aa_option_num_o(aa_opt8674010985164783839_num_o(P,X),Y2) )
       => ( ! [A3: num,B2: num] :
              ( ( X = some_num(A3) )
             => ( ( Y2 = some_num(B2) )
               => aa_option_num_o(aa_opt8674010985164783839_num_o(P,X),Y2) ) )
         => aa_option_num_o(aa_opt8674010985164783839_num_o(P,X),Y2) ) ) ) ).

% combine_options_cases
tff(fact_2621_split__option__all,axiom,
    ! [P: fun_op8028897214545316308_nat_o] :
      ( ! [X_12: option4927543243414619207at_nat] : aa_opt7112393969400857165_nat_o(P,X_12)
    <=> ( aa_opt7112393969400857165_nat_o(P,none_P5556105721700978146at_nat)
        & ! [X3: product_prod_nat_nat] : aa_opt7112393969400857165_nat_o(P,some_P7363390416028606310at_nat(X3)) ) ) ).

% split_option_all
tff(fact_2622_split__option__all,axiom,
    ! [P: fun_option_num_o] :
      ( ! [X_12: option_num] : aa_option_num_o(P,X_12)
    <=> ( aa_option_num_o(P,none_num)
        & ! [X3: num] : aa_option_num_o(P,some_num(X3)) ) ) ).

% split_option_all
tff(fact_2623_split__option__ex,axiom,
    ! [P: fun_op8028897214545316308_nat_o] :
      ( ? [X_12: option4927543243414619207at_nat] : aa_opt7112393969400857165_nat_o(P,X_12)
    <=> ( aa_opt7112393969400857165_nat_o(P,none_P5556105721700978146at_nat)
        | ? [X3: product_prod_nat_nat] : aa_opt7112393969400857165_nat_o(P,some_P7363390416028606310at_nat(X3)) ) ) ).

% split_option_ex
tff(fact_2624_split__option__ex,axiom,
    ! [P: fun_option_num_o] :
      ( ? [X_12: option_num] : aa_option_num_o(P,X_12)
    <=> ( aa_option_num_o(P,none_num)
        | ? [X3: num] : aa_option_num_o(P,some_num(X3)) ) ) ).

% split_option_ex
tff(fact_2625_option_Oexhaust,axiom,
    ! [Y2: option4927543243414619207at_nat] :
      ( ( Y2 != none_P5556105721700978146at_nat )
     => ~ ! [X24: product_prod_nat_nat] : Y2 != some_P7363390416028606310at_nat(X24) ) ).

% option.exhaust
tff(fact_2626_option_Oexhaust,axiom,
    ! [Y2: option_num] :
      ( ( Y2 != none_num )
     => ~ ! [X24: num] : Y2 != some_num(X24) ) ).

% option.exhaust
tff(fact_2627_option_OdiscI,axiom,
    ! [Option: option4927543243414619207at_nat,X23: product_prod_nat_nat] :
      ( ( Option = some_P7363390416028606310at_nat(X23) )
     => ( Option != none_P5556105721700978146at_nat ) ) ).

% option.discI
tff(fact_2628_option_OdiscI,axiom,
    ! [Option: option_num,X23: num] :
      ( ( Option = some_num(X23) )
     => ( Option != none_num ) ) ).

% option.discI
tff(fact_2629_option_Odistinct_I1_J,axiom,
    ! [X23: product_prod_nat_nat] : none_P5556105721700978146at_nat != some_P7363390416028606310at_nat(X23) ).

% option.distinct(1)
tff(fact_2630_option_Odistinct_I1_J,axiom,
    ! [X23: num] : none_num != some_num(X23) ).

% option.distinct(1)
tff(fact_2631_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V: nat,TreeListb: list_VEBT_VEBT,Vd2: vEBT_VEBT,X: nat] :
      ( aa_nat_o(vEBT_VEBT_membermima(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V),TreeListb),Vd2)),X)
    <=> $let(
          pos: nat,
          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeListb)),aa_nat_o(vEBT_VEBT_membermima(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeListb),pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ).

% VEBT_internal.membermima.simps(5)
tff(fact_2632_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy2: option4927543243414619207at_nat,V: nat,TreeListb: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
      ( aa_nat_o(vEBT_V5719532721284313246member(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V),TreeListb),S)),X)
    <=> $let(
          pos: nat,
          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeListb)),aa_nat_o(vEBT_V5719532721284313246member(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeListb),pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ).

% VEBT_internal.naive_member.simps(3)
tff(fact_2633_double__not__eq__Suc__double,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma) != aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% double_not_eq_Suc_double
tff(fact_2634_Suc__double__not__eq__double,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma)) != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ).

% Suc_double_not_eq_double
tff(fact_2635_nat__induct2,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( aa_nat_o(P,one_one_nat)
       => ( ! [N: nat] :
              ( aa_nat_o(P,N)
             => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
         => aa_nat_o(P,Nb) ) ) ) ).

% nat_induct2
tff(fact_2636_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mi: nat,Ma: nat,V: nat,TreeListb: list_VEBT_VEBT,Vc2: vEBT_VEBT,X: nat] :
      ( aa_nat_o(vEBT_VEBT_membermima(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi),Ma)),aa_nat_nat(suc,V),TreeListb),Vc2)),X)
    <=> ( ( X = Mi )
        | ( X = Ma )
        | $let(
            pos: nat,
            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeListb)),aa_nat_o(vEBT_VEBT_membermima(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeListb),pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ).

% VEBT_internal.membermima.simps(4)
tff(fact_2637_zle__add1__eq__le,axiom,
    ! [W: int,Z2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W),aa_int_int(aa_int_fun_int_int(plus_plus_int,Z2),one_one_int))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Z2) ) ).

% zle_add1_eq_le
tff(fact_2638_set__bit__0,axiom,
    ! [A: int] : aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,zero_zero_nat),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ).

% set_bit_0
tff(fact_2639_set__bit__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,zero_zero_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% set_bit_0
tff(fact_2640_incr__mult__lemma,axiom,
    ! [D: int,P: fun_int_o,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ! [X4: int] :
            ( aa_int_o(P,X4)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D)) )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
         => ! [X2: int] :
              ( aa_int_o(P,X2)
             => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D))) ) ) ) ) ).

% incr_mult_lemma
tff(fact_2641_le__imp__0__less,axiom,
    ! [Z2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Z2)) ) ).

% le_imp_0_less
tff(fact_2642_double__eq__0__iff,axiom,
    ! [A: real] :
      ( ( aa_real_real(plus_plus_real(A),A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% double_eq_0_iff
tff(fact_2643_double__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% double_eq_0_iff
tff(fact_2644_double__eq__0__iff,axiom,
    ! [A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% double_eq_0_iff
tff(fact_2645_unset__bit__0,axiom,
    ! [A: int] : aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,zero_zero_nat),A) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% unset_bit_0
tff(fact_2646_unset__bit__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,zero_zero_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% unset_bit_0
tff(fact_2647_zless__imp__add1__zle,axiom,
    ! [W: int,Z2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,W),one_one_int)),Z2) ) ).

% zless_imp_add1_zle
tff(fact_2648_add1__zle__eq,axiom,
    ! [W: int,Z2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,W),one_one_int)),Z2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z2) ) ).

% add1_zle_eq
tff(fact_2649_odd__less__0__iff,axiom,
    ! [Z2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Z2)),Z2)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),zero_zero_int) ) ).

% odd_less_0_iff
tff(fact_2650_unset__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Nb),K2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2) ) ).

% unset_bit_nonnegative_int_iff
tff(fact_2651_set__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,Nb),K2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2) ) ).

% set_bit_nonnegative_int_iff
tff(fact_2652_unset__bit__negative__int__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Nb),K2)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int) ) ).

% unset_bit_negative_int_iff
tff(fact_2653_set__bit__negative__int__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,Nb),K2)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int) ) ).

% set_bit_negative_int_iff
tff(fact_2654_unset__bit__less__eq,axiom,
    ! [Nb: nat,K2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Nb),K2)),K2) ).

% unset_bit_less_eq
tff(fact_2655_set__bit__greater__eq,axiom,
    ! [K2: int,Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,Nb),K2)) ).

% set_bit_greater_eq
tff(fact_2656_minf_I7_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Z)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,T),X2) ) ).

% minf(7)
tff(fact_2657_minf_I7_J,axiom,
    ! [T: rat] :
    ? [Z: rat] :
    ! [X2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Z)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,T),X2) ) ).

% minf(7)
tff(fact_2658_minf_I7_J,axiom,
    ! [T: num] :
    ? [Z: num] :
    ! [X2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X2),Z)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,T),X2) ) ).

% minf(7)
tff(fact_2659_minf_I7_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Z)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,T),X2) ) ).

% minf(7)
tff(fact_2660_minf_I7_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Z)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,T),X2) ) ).

% minf(7)
tff(fact_2661_minf_I5_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Z)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,X2),T) ) ).

% minf(5)
tff(fact_2662_minf_I5_J,axiom,
    ! [T: rat] :
    ? [Z: rat] :
    ! [X2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Z)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),T) ) ).

% minf(5)
tff(fact_2663_minf_I5_J,axiom,
    ! [T: num] :
    ? [Z: num] :
    ! [X2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X2),Z)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,X2),T) ) ).

% minf(5)
tff(fact_2664_minf_I5_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Z)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),T) ) ).

% minf(5)
tff(fact_2665_minf_I5_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Z)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,X2),T) ) ).

% minf(5)
tff(fact_2666_minf_I4_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Z)
     => ( X2 != T ) ) ).

% minf(4)
tff(fact_2667_minf_I4_J,axiom,
    ! [T: rat] :
    ? [Z: rat] :
    ! [X2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Z)
     => ( X2 != T ) ) ).

% minf(4)
tff(fact_2668_minf_I4_J,axiom,
    ! [T: num] :
    ? [Z: num] :
    ! [X2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X2),Z)
     => ( X2 != T ) ) ).

% minf(4)
tff(fact_2669_minf_I4_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Z)
     => ( X2 != T ) ) ).

% minf(4)
tff(fact_2670_minf_I4_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Z)
     => ( X2 != T ) ) ).

% minf(4)
tff(fact_2671_minf_I3_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Z)
     => ( X2 != T ) ) ).

% minf(3)
tff(fact_2672_minf_I3_J,axiom,
    ! [T: rat] :
    ? [Z: rat] :
    ! [X2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Z)
     => ( X2 != T ) ) ).

% minf(3)
tff(fact_2673_minf_I3_J,axiom,
    ! [T: num] :
    ? [Z: num] :
    ! [X2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X2),Z)
     => ( X2 != T ) ) ).

% minf(3)
tff(fact_2674_minf_I3_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Z)
     => ( X2 != T ) ) ).

% minf(3)
tff(fact_2675_minf_I3_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Z)
     => ( X2 != T ) ) ).

% minf(3)
tff(fact_2676_minf_I2_J,axiom,
    ! [P: fun_real_o,P3: fun_real_o,Q2: fun_real_o,Q6: fun_real_o] :
      ( ? [Z3: real] :
        ! [X4: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Z3)
         => ( aa_real_o(P,X4)
          <=> aa_real_o(P3,X4) ) )
     => ( ? [Z3: real] :
          ! [X4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Z3)
           => ( aa_real_o(Q2,X4)
            <=> aa_real_o(Q6,X4) ) )
       => ? [Z: real] :
          ! [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Z)
           => ( ( aa_real_o(P,X2)
                | aa_real_o(Q2,X2) )
            <=> ( aa_real_o(P3,X2)
                | aa_real_o(Q6,X2) ) ) ) ) ) ).

% minf(2)
tff(fact_2677_minf_I2_J,axiom,
    ! [P: fun_rat_o,P3: fun_rat_o,Q2: fun_rat_o,Q6: fun_rat_o] :
      ( ? [Z3: rat] :
        ! [X4: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Z3)
         => ( aa_rat_o(P,X4)
          <=> aa_rat_o(P3,X4) ) )
     => ( ? [Z3: rat] :
          ! [X4: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Z3)
           => ( aa_rat_o(Q2,X4)
            <=> aa_rat_o(Q6,X4) ) )
       => ? [Z: rat] :
          ! [X2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Z)
           => ( ( aa_rat_o(P,X2)
                | aa_rat_o(Q2,X2) )
            <=> ( aa_rat_o(P3,X2)
                | aa_rat_o(Q6,X2) ) ) ) ) ) ).

% minf(2)
tff(fact_2678_minf_I2_J,axiom,
    ! [P: fun_num_o,P3: fun_num_o,Q2: fun_num_o,Q6: fun_num_o] :
      ( ? [Z3: num] :
        ! [X4: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,X4),Z3)
         => ( aa_num_o(P,X4)
          <=> aa_num_o(P3,X4) ) )
     => ( ? [Z3: num] :
          ! [X4: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,X4),Z3)
           => ( aa_num_o(Q2,X4)
            <=> aa_num_o(Q6,X4) ) )
       => ? [Z: num] :
          ! [X2: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,X2),Z)
           => ( ( aa_num_o(P,X2)
                | aa_num_o(Q2,X2) )
            <=> ( aa_num_o(P3,X2)
                | aa_num_o(Q6,X2) ) ) ) ) ) ).

% minf(2)
tff(fact_2679_minf_I2_J,axiom,
    ! [P: fun_nat_o,P3: fun_nat_o,Q2: fun_nat_o,Q6: fun_nat_o] :
      ( ? [Z3: nat] :
        ! [X4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Z3)
         => ( aa_nat_o(P,X4)
          <=> aa_nat_o(P3,X4) ) )
     => ( ? [Z3: nat] :
          ! [X4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Z3)
           => ( aa_nat_o(Q2,X4)
            <=> aa_nat_o(Q6,X4) ) )
       => ? [Z: nat] :
          ! [X2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Z)
           => ( ( aa_nat_o(P,X2)
                | aa_nat_o(Q2,X2) )
            <=> ( aa_nat_o(P3,X2)
                | aa_nat_o(Q6,X2) ) ) ) ) ) ).

% minf(2)
tff(fact_2680_minf_I2_J,axiom,
    ! [P: fun_int_o,P3: fun_int_o,Q2: fun_int_o,Q6: fun_int_o] :
      ( ? [Z3: int] :
        ! [X4: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Z3)
         => ( aa_int_o(P,X4)
          <=> aa_int_o(P3,X4) ) )
     => ( ? [Z3: int] :
          ! [X4: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Z3)
           => ( aa_int_o(Q2,X4)
            <=> aa_int_o(Q6,X4) ) )
       => ? [Z: int] :
          ! [X2: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Z)
           => ( ( aa_int_o(P,X2)
                | aa_int_o(Q2,X2) )
            <=> ( aa_int_o(P3,X2)
                | aa_int_o(Q6,X2) ) ) ) ) ) ).

% minf(2)
tff(fact_2681_minf_I1_J,axiom,
    ! [P: fun_real_o,P3: fun_real_o,Q2: fun_real_o,Q6: fun_real_o] :
      ( ? [Z3: real] :
        ! [X4: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Z3)
         => ( aa_real_o(P,X4)
          <=> aa_real_o(P3,X4) ) )
     => ( ? [Z3: real] :
          ! [X4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Z3)
           => ( aa_real_o(Q2,X4)
            <=> aa_real_o(Q6,X4) ) )
       => ? [Z: real] :
          ! [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Z)
           => ( ( aa_real_o(P,X2)
                & aa_real_o(Q2,X2) )
            <=> ( aa_real_o(P3,X2)
                & aa_real_o(Q6,X2) ) ) ) ) ) ).

% minf(1)
tff(fact_2682_minf_I1_J,axiom,
    ! [P: fun_rat_o,P3: fun_rat_o,Q2: fun_rat_o,Q6: fun_rat_o] :
      ( ? [Z3: rat] :
        ! [X4: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Z3)
         => ( aa_rat_o(P,X4)
          <=> aa_rat_o(P3,X4) ) )
     => ( ? [Z3: rat] :
          ! [X4: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Z3)
           => ( aa_rat_o(Q2,X4)
            <=> aa_rat_o(Q6,X4) ) )
       => ? [Z: rat] :
          ! [X2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Z)
           => ( ( aa_rat_o(P,X2)
                & aa_rat_o(Q2,X2) )
            <=> ( aa_rat_o(P3,X2)
                & aa_rat_o(Q6,X2) ) ) ) ) ) ).

% minf(1)
tff(fact_2683_minf_I1_J,axiom,
    ! [P: fun_num_o,P3: fun_num_o,Q2: fun_num_o,Q6: fun_num_o] :
      ( ? [Z3: num] :
        ! [X4: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,X4),Z3)
         => ( aa_num_o(P,X4)
          <=> aa_num_o(P3,X4) ) )
     => ( ? [Z3: num] :
          ! [X4: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,X4),Z3)
           => ( aa_num_o(Q2,X4)
            <=> aa_num_o(Q6,X4) ) )
       => ? [Z: num] :
          ! [X2: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,X2),Z)
           => ( ( aa_num_o(P,X2)
                & aa_num_o(Q2,X2) )
            <=> ( aa_num_o(P3,X2)
                & aa_num_o(Q6,X2) ) ) ) ) ) ).

% minf(1)
tff(fact_2684_minf_I1_J,axiom,
    ! [P: fun_nat_o,P3: fun_nat_o,Q2: fun_nat_o,Q6: fun_nat_o] :
      ( ? [Z3: nat] :
        ! [X4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Z3)
         => ( aa_nat_o(P,X4)
          <=> aa_nat_o(P3,X4) ) )
     => ( ? [Z3: nat] :
          ! [X4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Z3)
           => ( aa_nat_o(Q2,X4)
            <=> aa_nat_o(Q6,X4) ) )
       => ? [Z: nat] :
          ! [X2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Z)
           => ( ( aa_nat_o(P,X2)
                & aa_nat_o(Q2,X2) )
            <=> ( aa_nat_o(P3,X2)
                & aa_nat_o(Q6,X2) ) ) ) ) ) ).

% minf(1)
tff(fact_2685_minf_I1_J,axiom,
    ! [P: fun_int_o,P3: fun_int_o,Q2: fun_int_o,Q6: fun_int_o] :
      ( ? [Z3: int] :
        ! [X4: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Z3)
         => ( aa_int_o(P,X4)
          <=> aa_int_o(P3,X4) ) )
     => ( ? [Z3: int] :
          ! [X4: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Z3)
           => ( aa_int_o(Q2,X4)
            <=> aa_int_o(Q6,X4) ) )
       => ? [Z: int] :
          ! [X2: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Z)
           => ( ( aa_int_o(P,X2)
                & aa_int_o(Q2,X2) )
            <=> ( aa_int_o(P3,X2)
                & aa_int_o(Q6,X2) ) ) ) ) ) ).

% minf(1)
tff(fact_2686_pinf_I7_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),X2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,T),X2) ) ).

% pinf(7)
tff(fact_2687_pinf_I7_J,axiom,
    ! [T: rat] :
    ? [Z: rat] :
    ! [X2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),X2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,T),X2) ) ).

% pinf(7)
tff(fact_2688_pinf_I7_J,axiom,
    ! [T: num] :
    ? [Z: num] :
    ! [X2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z),X2)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,T),X2) ) ).

% pinf(7)
tff(fact_2689_pinf_I7_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),X2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,T),X2) ) ).

% pinf(7)
tff(fact_2690_pinf_I7_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),X2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,T),X2) ) ).

% pinf(7)
tff(fact_2691_pinf_I5_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),X2)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,X2),T) ) ).

% pinf(5)
tff(fact_2692_pinf_I5_J,axiom,
    ! [T: rat] :
    ? [Z: rat] :
    ! [X2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),X2)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),T) ) ).

% pinf(5)
tff(fact_2693_pinf_I5_J,axiom,
    ! [T: num] :
    ? [Z: num] :
    ! [X2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z),X2)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X2),T) ) ).

% pinf(5)
tff(fact_2694_pinf_I5_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),X2)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),T) ) ).

% pinf(5)
tff(fact_2695_pinf_I5_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),X2)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X2),T) ) ).

% pinf(5)
tff(fact_2696_pinf_I4_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),X2)
     => ( X2 != T ) ) ).

% pinf(4)
tff(fact_2697_pinf_I4_J,axiom,
    ! [T: rat] :
    ? [Z: rat] :
    ! [X2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),X2)
     => ( X2 != T ) ) ).

% pinf(4)
tff(fact_2698_pinf_I4_J,axiom,
    ! [T: num] :
    ? [Z: num] :
    ! [X2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z),X2)
     => ( X2 != T ) ) ).

% pinf(4)
tff(fact_2699_pinf_I4_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),X2)
     => ( X2 != T ) ) ).

% pinf(4)
tff(fact_2700_pinf_I4_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),X2)
     => ( X2 != T ) ) ).

% pinf(4)
tff(fact_2701_pinf_I3_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),X2)
     => ( X2 != T ) ) ).

% pinf(3)
tff(fact_2702_pinf_I3_J,axiom,
    ! [T: rat] :
    ? [Z: rat] :
    ! [X2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),X2)
     => ( X2 != T ) ) ).

% pinf(3)
tff(fact_2703_pinf_I3_J,axiom,
    ! [T: num] :
    ? [Z: num] :
    ! [X2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z),X2)
     => ( X2 != T ) ) ).

% pinf(3)
tff(fact_2704_pinf_I3_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),X2)
     => ( X2 != T ) ) ).

% pinf(3)
tff(fact_2705_pinf_I3_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),X2)
     => ( X2 != T ) ) ).

% pinf(3)
tff(fact_2706_pinf_I2_J,axiom,
    ! [P: fun_real_o,P3: fun_real_o,Q2: fun_real_o,Q6: fun_real_o] :
      ( ? [Z3: real] :
        ! [X4: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X4)
         => ( aa_real_o(P,X4)
          <=> aa_real_o(P3,X4) ) )
     => ( ? [Z3: real] :
          ! [X4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X4)
           => ( aa_real_o(Q2,X4)
            <=> aa_real_o(Q6,X4) ) )
       => ? [Z: real] :
          ! [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),X2)
           => ( ( aa_real_o(P,X2)
                | aa_real_o(Q2,X2) )
            <=> ( aa_real_o(P3,X2)
                | aa_real_o(Q6,X2) ) ) ) ) ) ).

% pinf(2)
tff(fact_2707_pinf_I2_J,axiom,
    ! [P: fun_rat_o,P3: fun_rat_o,Q2: fun_rat_o,Q6: fun_rat_o] :
      ( ? [Z3: rat] :
        ! [X4: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X4)
         => ( aa_rat_o(P,X4)
          <=> aa_rat_o(P3,X4) ) )
     => ( ? [Z3: rat] :
          ! [X4: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X4)
           => ( aa_rat_o(Q2,X4)
            <=> aa_rat_o(Q6,X4) ) )
       => ? [Z: rat] :
          ! [X2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),X2)
           => ( ( aa_rat_o(P,X2)
                | aa_rat_o(Q2,X2) )
            <=> ( aa_rat_o(P3,X2)
                | aa_rat_o(Q6,X2) ) ) ) ) ) ).

% pinf(2)
tff(fact_2708_pinf_I2_J,axiom,
    ! [P: fun_num_o,P3: fun_num_o,Q2: fun_num_o,Q6: fun_num_o] :
      ( ? [Z3: num] :
        ! [X4: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X4)
         => ( aa_num_o(P,X4)
          <=> aa_num_o(P3,X4) ) )
     => ( ? [Z3: num] :
          ! [X4: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X4)
           => ( aa_num_o(Q2,X4)
            <=> aa_num_o(Q6,X4) ) )
       => ? [Z: num] :
          ! [X2: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z),X2)
           => ( ( aa_num_o(P,X2)
                | aa_num_o(Q2,X2) )
            <=> ( aa_num_o(P3,X2)
                | aa_num_o(Q6,X2) ) ) ) ) ) ).

% pinf(2)
tff(fact_2709_pinf_I2_J,axiom,
    ! [P: fun_nat_o,P3: fun_nat_o,Q2: fun_nat_o,Q6: fun_nat_o] :
      ( ? [Z3: nat] :
        ! [X4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X4)
         => ( aa_nat_o(P,X4)
          <=> aa_nat_o(P3,X4) ) )
     => ( ? [Z3: nat] :
          ! [X4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X4)
           => ( aa_nat_o(Q2,X4)
            <=> aa_nat_o(Q6,X4) ) )
       => ? [Z: nat] :
          ! [X2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),X2)
           => ( ( aa_nat_o(P,X2)
                | aa_nat_o(Q2,X2) )
            <=> ( aa_nat_o(P3,X2)
                | aa_nat_o(Q6,X2) ) ) ) ) ) ).

% pinf(2)
tff(fact_2710_pinf_I2_J,axiom,
    ! [P: fun_int_o,P3: fun_int_o,Q2: fun_int_o,Q6: fun_int_o] :
      ( ? [Z3: int] :
        ! [X4: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X4)
         => ( aa_int_o(P,X4)
          <=> aa_int_o(P3,X4) ) )
     => ( ? [Z3: int] :
          ! [X4: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X4)
           => ( aa_int_o(Q2,X4)
            <=> aa_int_o(Q6,X4) ) )
       => ? [Z: int] :
          ! [X2: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),X2)
           => ( ( aa_int_o(P,X2)
                | aa_int_o(Q2,X2) )
            <=> ( aa_int_o(P3,X2)
                | aa_int_o(Q6,X2) ) ) ) ) ) ).

% pinf(2)
tff(fact_2711_pinf_I1_J,axiom,
    ! [P: fun_real_o,P3: fun_real_o,Q2: fun_real_o,Q6: fun_real_o] :
      ( ? [Z3: real] :
        ! [X4: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X4)
         => ( aa_real_o(P,X4)
          <=> aa_real_o(P3,X4) ) )
     => ( ? [Z3: real] :
          ! [X4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X4)
           => ( aa_real_o(Q2,X4)
            <=> aa_real_o(Q6,X4) ) )
       => ? [Z: real] :
          ! [X2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),X2)
           => ( ( aa_real_o(P,X2)
                & aa_real_o(Q2,X2) )
            <=> ( aa_real_o(P3,X2)
                & aa_real_o(Q6,X2) ) ) ) ) ) ).

% pinf(1)
tff(fact_2712_pinf_I1_J,axiom,
    ! [P: fun_rat_o,P3: fun_rat_o,Q2: fun_rat_o,Q6: fun_rat_o] :
      ( ? [Z3: rat] :
        ! [X4: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X4)
         => ( aa_rat_o(P,X4)
          <=> aa_rat_o(P3,X4) ) )
     => ( ? [Z3: rat] :
          ! [X4: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X4)
           => ( aa_rat_o(Q2,X4)
            <=> aa_rat_o(Q6,X4) ) )
       => ? [Z: rat] :
          ! [X2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),X2)
           => ( ( aa_rat_o(P,X2)
                & aa_rat_o(Q2,X2) )
            <=> ( aa_rat_o(P3,X2)
                & aa_rat_o(Q6,X2) ) ) ) ) ) ).

% pinf(1)
tff(fact_2713_pinf_I1_J,axiom,
    ! [P: fun_num_o,P3: fun_num_o,Q2: fun_num_o,Q6: fun_num_o] :
      ( ? [Z3: num] :
        ! [X4: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X4)
         => ( aa_num_o(P,X4)
          <=> aa_num_o(P3,X4) ) )
     => ( ? [Z3: num] :
          ! [X4: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X4)
           => ( aa_num_o(Q2,X4)
            <=> aa_num_o(Q6,X4) ) )
       => ? [Z: num] :
          ! [X2: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z),X2)
           => ( ( aa_num_o(P,X2)
                & aa_num_o(Q2,X2) )
            <=> ( aa_num_o(P3,X2)
                & aa_num_o(Q6,X2) ) ) ) ) ) ).

% pinf(1)
tff(fact_2714_pinf_I1_J,axiom,
    ! [P: fun_nat_o,P3: fun_nat_o,Q2: fun_nat_o,Q6: fun_nat_o] :
      ( ? [Z3: nat] :
        ! [X4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X4)
         => ( aa_nat_o(P,X4)
          <=> aa_nat_o(P3,X4) ) )
     => ( ? [Z3: nat] :
          ! [X4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X4)
           => ( aa_nat_o(Q2,X4)
            <=> aa_nat_o(Q6,X4) ) )
       => ? [Z: nat] :
          ! [X2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),X2)
           => ( ( aa_nat_o(P,X2)
                & aa_nat_o(Q2,X2) )
            <=> ( aa_nat_o(P3,X2)
                & aa_nat_o(Q6,X2) ) ) ) ) ) ).

% pinf(1)
tff(fact_2715_pinf_I1_J,axiom,
    ! [P: fun_int_o,P3: fun_int_o,Q2: fun_int_o,Q6: fun_int_o] :
      ( ? [Z3: int] :
        ! [X4: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X4)
         => ( aa_int_o(P,X4)
          <=> aa_int_o(P3,X4) ) )
     => ( ? [Z3: int] :
          ! [X4: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X4)
           => ( aa_int_o(Q2,X4)
            <=> aa_int_o(Q6,X4) ) )
       => ? [Z: int] :
          ! [X2: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),X2)
           => ( ( aa_int_o(P,X2)
                & aa_int_o(Q2,X2) )
            <=> ( aa_int_o(P3,X2)
                & aa_int_o(Q6,X2) ) ) ) ) ) ).

% pinf(1)
tff(fact_2716_minf_I8_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Z)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T),X2) ) ).

% minf(8)
tff(fact_2717_minf_I8_J,axiom,
    ! [T: rat] :
    ? [Z: rat] :
    ! [X2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Z)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,T),X2) ) ).

% minf(8)
tff(fact_2718_minf_I8_J,axiom,
    ! [T: num] :
    ? [Z: num] :
    ! [X2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X2),Z)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,T),X2) ) ).

% minf(8)
tff(fact_2719_minf_I8_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Z)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,T),X2) ) ).

% minf(8)
tff(fact_2720_minf_I8_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Z)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,T),X2) ) ).

% minf(8)
tff(fact_2721_minf_I6_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Z)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),T) ) ).

% minf(6)
tff(fact_2722_minf_I6_J,axiom,
    ! [T: rat] :
    ? [Z: rat] :
    ! [X2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Z)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),T) ) ).

% minf(6)
tff(fact_2723_minf_I6_J,axiom,
    ! [T: num] :
    ? [Z: num] :
    ! [X2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X2),Z)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),T) ) ).

% minf(6)
tff(fact_2724_minf_I6_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Z)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),T) ) ).

% minf(6)
tff(fact_2725_minf_I6_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Z)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),T) ) ).

% minf(6)
tff(fact_2726_pinf_I8_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),X2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T),X2) ) ).

% pinf(8)
tff(fact_2727_pinf_I8_J,axiom,
    ! [T: rat] :
    ? [Z: rat] :
    ! [X2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),X2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,T),X2) ) ).

% pinf(8)
tff(fact_2728_pinf_I8_J,axiom,
    ! [T: num] :
    ? [Z: num] :
    ! [X2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z),X2)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,T),X2) ) ).

% pinf(8)
tff(fact_2729_pinf_I8_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),X2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,T),X2) ) ).

% pinf(8)
tff(fact_2730_pinf_I8_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),X2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,T),X2) ) ).

% pinf(8)
tff(fact_2731_pinf_I6_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),X2)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),T) ) ).

% pinf(6)
tff(fact_2732_pinf_I6_J,axiom,
    ! [T: rat] :
    ? [Z: rat] :
    ! [X2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),X2)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X2),T) ) ).

% pinf(6)
tff(fact_2733_pinf_I6_J,axiom,
    ! [T: num] :
    ? [Z: num] :
    ! [X2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z),X2)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X2),T) ) ).

% pinf(6)
tff(fact_2734_pinf_I6_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),X2)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),T) ) ).

% pinf(6)
tff(fact_2735_pinf_I6_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),X2)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),T) ) ).

% pinf(6)
tff(fact_2736_conj__le__cong,axiom,
    ! [X: int,X6: int,P: $o,P3: $o] :
      ( ( X = X6 )
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X6)
         => ( (P)
          <=> (P3) ) )
       => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
            & (P) )
        <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X6)
            & (P3) ) ) ) ) ).

% conj_le_cong
tff(fact_2737_imp__le__cong,axiom,
    ! [X: int,X6: int,P: $o,P3: $o] :
      ( ( X = X6 )
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X6)
         => ( (P)
          <=> (P3) ) )
       => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
           => (P) )
        <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X6)
           => (P3) ) ) ) ) ).

% imp_le_cong
tff(fact_2738_less__eq__int__code_I1_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),zero_zero_int) ).

% less_eq_int_code(1)
tff(fact_2739_times__int__code_I1_J,axiom,
    ! [K2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,K2),zero_zero_int) = zero_zero_int ).

% times_int_code(1)
tff(fact_2740_times__int__code_I2_J,axiom,
    ! [L: int] : aa_int_int(aa_int_fun_int_int(times_times_int,zero_zero_int),L) = zero_zero_int ).

% times_int_code(2)
tff(fact_2741_int__distrib_I1_J,axiom,
    ! [Z1: int,Z22: int,W: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z1),Z22)),W) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z1),W)),aa_int_int(aa_int_fun_int_int(times_times_int,Z22),W)) ).

% int_distrib(1)
tff(fact_2742_int__distrib_I2_J,axiom,
    ! [W: int,Z1: int,Z22: int] : aa_int_int(aa_int_fun_int_int(times_times_int,W),aa_int_int(aa_int_fun_int_int(plus_plus_int,Z1),Z22)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z1)),aa_int_int(aa_int_fun_int_int(times_times_int,W),Z22)) ).

% int_distrib(2)
tff(fact_2743_zmult__zless__mono2,axiom,
    ! [I2: int,J: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,I2),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,K2),I2)),aa_int_int(aa_int_fun_int_int(times_times_int,K2),J)) ) ) ).

% zmult_zless_mono2
tff(fact_2744_odd__nonzero,axiom,
    ! [Z2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Z2)),Z2) != zero_zero_int ).

% odd_nonzero
tff(fact_2745_int__ge__induct,axiom,
    ! [K2: int,I2: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),I2)
     => ( aa_int_o(P,K2)
       => ( ! [I3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),I3)
             => ( aa_int_o(P,I3)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I3),one_one_int)) ) )
         => aa_int_o(P,I2) ) ) ) ).

% int_ge_induct
tff(fact_2746_int__gr__induct,axiom,
    ! [K2: int,I2: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),I2)
     => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),one_one_int))
       => ( ! [I3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),I3)
             => ( aa_int_o(P,I3)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I3),one_one_int)) ) )
         => aa_int_o(P,I2) ) ) ) ).

% int_gr_induct
tff(fact_2747_zless__add1__eq,axiom,
    ! [W: int,Z2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W),aa_int_int(aa_int_fun_int_int(plus_plus_int,Z2),one_one_int))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z2)
        | ( W = Z2 ) ) ) ).

% zless_add1_eq
tff(fact_2748_int__one__le__iff__zero__less,axiom,
    ! [Z2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Z2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z2) ) ).

% int_one_le_iff_zero_less
tff(fact_2749_set__bit__Suc,axiom,
    ! [Nb: nat,A: code_integer] : bit_se2793503036327961859nteger(aa_nat_nat(suc,Nb),A) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se2793503036327961859nteger(Nb,divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_2750_set__bit__Suc,axiom,
    ! [Nb: nat,A: int] : aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,aa_nat_nat(suc,Nb)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,Nb),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_2751_set__bit__Suc,axiom,
    ! [Nb: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,aa_nat_nat(suc,Nb)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_2752_unset__bit__Suc,axiom,
    ! [Nb: nat,A: code_integer] : bit_se8260200283734997820nteger(aa_nat_nat(suc,Nb),A) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se8260200283734997820nteger(Nb,divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_2753_unset__bit__Suc,axiom,
    ! [Nb: nat,A: int] : aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,aa_nat_nat(suc,Nb)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Nb),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_2754_unset__bit__Suc,axiom,
    ! [Nb: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,aa_nat_nat(suc,Nb)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_2755_pos__zmult__eq__1__iff,axiom,
    ! [Ma: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ma)
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Ma),Nb) = one_one_int )
      <=> ( ( Ma = one_one_int )
          & ( Nb = one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff
tff(fact_2756_flip__bit__Suc,axiom,
    ! [Nb: nat,A: code_integer] : bit_se1345352211410354436nteger(aa_nat_nat(suc,Nb),A) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se1345352211410354436nteger(Nb,divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_2757_flip__bit__Suc,axiom,
    ! [Nb: nat,A: int] : aa_int_int(bit_se2159334234014336723it_int(aa_nat_nat(suc,Nb)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se2159334234014336723it_int(Nb),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_2758_flip__bit__Suc,axiom,
    ! [Nb: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,aa_nat_nat(suc,Nb)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_2759_mult__le__cancel__iff1,axiom,
    ! [Z2: real,X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(X),Z2)),aa_real_real(times_times_real(Y2),Z2))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y2) ) ) ).

% mult_le_cancel_iff1
tff(fact_2760_mult__le__cancel__iff1,axiom,
    ! [Z2: rat,X: rat,Y2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(X),Z2)),aa_rat_rat(times_times_rat(Y2),Z2))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y2) ) ) ).

% mult_le_cancel_iff1
tff(fact_2761_mult__le__cancel__iff1,axiom,
    ! [Z2: int,X: int,Y2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),Z2)),aa_int_int(aa_int_fun_int_int(times_times_int,Y2),Z2))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y2) ) ) ).

% mult_le_cancel_iff1
tff(fact_2762_mult__le__cancel__iff2,axiom,
    ! [Z2: real,X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(Z2),X)),aa_real_real(times_times_real(Z2),Y2))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y2) ) ) ).

% mult_le_cancel_iff2
tff(fact_2763_mult__le__cancel__iff2,axiom,
    ! [Z2: rat,X: rat,Y2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Z2),X)),aa_rat_rat(times_times_rat(Z2),Y2))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y2) ) ) ).

% mult_le_cancel_iff2
tff(fact_2764_mult__le__cancel__iff2,axiom,
    ! [Z2: int,X: int,Y2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z2),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Z2),Y2))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y2) ) ) ).

% mult_le_cancel_iff2
tff(fact_2765_product__nth,axiom,
    ! [Nb: nat,Xs: list_num,Ys: list_num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_num(Xs)),size_size_list_num(Ys)))
     => ( nth_Pr6456567536196504476um_num(product_num_num(Xs,Ys),Nb) = aa_num3156681643174609981um_num(aa_num8681986212680877308um_num(product_Pair_num_num,aa_nat_num(nth_num(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_num(Ys)))),aa_nat_num(nth_num(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_num(Ys)))) ) ) ).

% product_nth
tff(fact_2766_product__nth,axiom,
    ! [Nb: nat,Xs: list_Code_integer,Ys: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s3445333598471063425nteger(Xs)),size_size_list_o(Ys)))
     => ( nth_Pr8522763379788166057eger_o(produc3607205314601156340eger_o(Xs,Ys),Nb) = aa_o_P4192039665577170892eger_o(aa_Cod3473573201699832512eger_o(produc6677183202524767010eger_o,aa_nat_Code_integer(nth_Code_integer(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_o(Ys)))),aa_nat_o(nth_o(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_o(Ys)))) ) ) ).

% product_nth
tff(fact_2767_product__nth,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_s6755466524823107622T_VEBT(Ys)))
     => ( nth_Pr4953567300277697838T_VEBT(produc4743750530478302277T_VEBT(Xs,Ys),Nb) = produc537772716801021591T_VEBT(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_s6755466524823107622T_VEBT(Ys))),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_s6755466524823107622T_VEBT(Ys)))) ) ) ).

% product_nth
tff(fact_2768_product__nth,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,Ys: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_o(Ys)))
     => ( nth_Pr4606735188037164562VEBT_o(product_VEBT_VEBT_o(Xs,Ys),Nb) = produc8721562602347293563VEBT_o(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_o(Ys))),aa_nat_o(nth_o(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_o(Ys)))) ) ) ).

% product_nth
tff(fact_2769_product__nth,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,Ys: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_nat(Ys)))
     => ( nth_Pr1791586995822124652BT_nat(produc7295137177222721919BT_nat(Xs,Ys),Nb) = produc738532404422230701BT_nat(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_nat(Ys))),aa_nat_nat(nth_nat(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_nat(Ys)))) ) ) ).

% product_nth
tff(fact_2770_product__nth,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,Ys: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_int(Ys)))
     => ( nth_Pr6837108013167703752BT_int(produc7292646706713671643BT_int(Xs,Ys),Nb) = produc736041933913180425BT_int(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_int(Ys))),aa_nat_int(nth_int(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_int(Ys)))) ) ) ).

% product_nth
tff(fact_2771_product__nth,axiom,
    ! [Nb: nat,Xs: list_o,Ys: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_s6755466524823107622T_VEBT(Ys)))
     => ( nth_Pr6777367263587873994T_VEBT(product_o_VEBT_VEBT(Xs,Ys),Nb) = produc2982872950893828659T_VEBT(aa_nat_o(nth_o(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_s6755466524823107622T_VEBT(Ys))),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_s6755466524823107622T_VEBT(Ys)))) ) ) ).

% product_nth
tff(fact_2772_product__nth,axiom,
    ! [Nb: nat,Xs: list_o,Ys: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_size_list_o(Ys)))
     => ( nth_Product_prod_o_o(product_o_o(Xs,Ys),Nb) = product_Pair_o_o(aa_nat_o(nth_o(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_o(Ys))),aa_nat_o(nth_o(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_o(Ys)))) ) ) ).

% product_nth
tff(fact_2773_product__nth,axiom,
    ! [Nb: nat,Xs: list_o,Ys: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_size_list_nat(Ys)))
     => ( nth_Pr5826913651314560976_o_nat(product_o_nat(Xs,Ys),Nb) = product_Pair_o_nat(aa_nat_o(nth_o(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_nat(Ys))),aa_nat_nat(nth_nat(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_nat(Ys)))) ) ) ).

% product_nth
tff(fact_2774_product__nth,axiom,
    ! [Nb: nat,Xs: list_o,Ys: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_size_list_int(Ys)))
     => ( nth_Pr1649062631805364268_o_int(product_o_int(Xs,Ys),Nb) = product_Pair_o_int(aa_nat_o(nth_o(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_int(Ys))),aa_nat_int(nth_int(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_int(Ys)))) ) ) ).

% product_nth
tff(fact_2775_gcd__nat__induct,axiom,
    ! [P: fun_nat_fun_nat_o,Ma: nat,Nb: nat] :
      ( ! [M2: nat] : aa_nat_o(aa_nat_fun_nat_o(P,M2),zero_zero_nat)
     => ( ! [M2: nat,N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
           => ( aa_nat_o(aa_nat_fun_nat_o(P,N),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),N))
             => aa_nat_o(aa_nat_fun_nat_o(P,M2),N) ) )
       => aa_nat_o(aa_nat_fun_nat_o(P,Ma),Nb) ) ) ).

% gcd_nat_induct
tff(fact_2776_concat__bit__Suc,axiom,
    ! [Nb: nat,K2: int,L: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(aa_nat_nat(suc,Nb)),K2),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_concat_bit(Nb),aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),L))) ).

% concat_bit_Suc
tff(fact_2777_dual__order_Orefl,axiom,
    ! [A: set_int] : ord_less_eq_set_int(A,A) ).

% dual_order.refl
tff(fact_2778_dual__order_Orefl,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),A) ).

% dual_order.refl
tff(fact_2779_dual__order_Orefl,axiom,
    ! [A: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),A) ).

% dual_order.refl
tff(fact_2780_dual__order_Orefl,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),A) ).

% dual_order.refl
tff(fact_2781_dual__order_Orefl,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),A) ).

% dual_order.refl
tff(fact_2782_order__refl,axiom,
    ! [X: set_int] : ord_less_eq_set_int(X,X) ).

% order_refl
tff(fact_2783_order__refl,axiom,
    ! [X: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),X) ).

% order_refl
tff(fact_2784_order__refl,axiom,
    ! [X: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),X) ).

% order_refl
tff(fact_2785_order__refl,axiom,
    ! [X: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),X) ).

% order_refl
tff(fact_2786_order__refl,axiom,
    ! [X: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),X) ).

% order_refl
tff(fact_2787_flip__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se2159334234014336723it_int(Nb),K2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2) ) ).

% flip_bit_nonnegative_int_iff
tff(fact_2788_flip__bit__negative__int__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se2159334234014336723it_int(Nb),K2)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int) ) ).

% flip_bit_negative_int_iff
tff(fact_2789_concat__bit__0,axiom,
    ! [K2: int,L: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(zero_zero_nat),K2),L) = L ).

% concat_bit_0
tff(fact_2790_concat__bit__nonnegative__iff,axiom,
    ! [Nb: nat,K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_concat_bit(Nb),K2),L))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) ) ).

% concat_bit_nonnegative_iff
tff(fact_2791_concat__bit__negative__iff,axiom,
    ! [Nb: nat,K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_concat_bit(Nb),K2),L)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ).

% concat_bit_negative_iff
tff(fact_2792_length__product,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] : size_s7466405169056248089T_VEBT(produc4743750530478302277T_VEBT(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_s6755466524823107622T_VEBT(Ys)) ).

% length_product
tff(fact_2793_length__product,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_o] : size_s9168528473962070013VEBT_o(product_VEBT_VEBT_o(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_o(Ys)) ).

% length_product
tff(fact_2794_length__product,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_nat] : size_s6152045936467909847BT_nat(produc7295137177222721919BT_nat(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_nat(Ys)) ).

% length_product
tff(fact_2795_length__product,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_int] : size_s3661962791536183091BT_int(produc7292646706713671643BT_int(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_int(Ys)) ).

% length_product
tff(fact_2796_length__product,axiom,
    ! [Xs: list_o,Ys: list_VEBT_VEBT] : size_s4313452262239582901T_VEBT(product_o_VEBT_VEBT(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_s6755466524823107622T_VEBT(Ys)) ).

% length_product
tff(fact_2797_length__product,axiom,
    ! [Xs: list_o,Ys: list_o] : size_s1515746228057227161od_o_o(product_o_o(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_size_list_o(Ys)) ).

% length_product
tff(fact_2798_length__product,axiom,
    ! [Xs: list_o,Ys: list_nat] : size_s5443766701097040955_o_nat(product_o_nat(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_size_list_nat(Ys)) ).

% length_product
tff(fact_2799_length__product,axiom,
    ! [Xs: list_o,Ys: list_int] : size_s2953683556165314199_o_int(product_o_int(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_size_list_int(Ys)) ).

% length_product
tff(fact_2800_length__product,axiom,
    ! [Xs: list_nat,Ys: list_VEBT_VEBT] : size_s4762443039079500285T_VEBT(produc7156399406898700509T_VEBT(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_nat(Xs)),size_s6755466524823107622T_VEBT(Ys)) ).

% length_product
tff(fact_2801_length__product,axiom,
    ! [Xs: list_nat,Ys: list_o] : size_s6491369823275344609_nat_o(product_nat_o(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_nat(Xs)),size_size_list_o(Ys)) ).

% length_product
tff(fact_2802_concat__bit__assoc,axiom,
    ! [Nb: nat,K2: int,Ma: nat,L: int,R: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(Nb),K2),aa_int_int(aa_int_fun_int_int(bit_concat_bit(Ma),L),R)) = aa_int_int(aa_int_fun_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb)),aa_int_int(aa_int_fun_int_int(bit_concat_bit(Nb),K2),L)),R) ).

% concat_bit_assoc
tff(fact_2803_VEBT__internal_Ovalid_H_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv2: $o,D2: nat] : X != produc738532404422230701BT_nat(aa_o_VEBT_VEBT(vEBT_Leaf((Uu2)),(Uv2)),D2)
     => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg2: nat] : X != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg,TreeList),Summary),Deg2) ) ).

% VEBT_internal.valid'.cases
tff(fact_2804_order__antisym__conv,axiom,
    ! [Y2: set_int,X: set_int] :
      ( ord_less_eq_set_int(Y2,X)
     => ( ord_less_eq_set_int(X,Y2)
      <=> ( X = Y2 ) ) ) ).

% order_antisym_conv
tff(fact_2805_order__antisym__conv,axiom,
    ! [Y2: rat,X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y2),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y2)
      <=> ( X = Y2 ) ) ) ).

% order_antisym_conv
tff(fact_2806_order__antisym__conv,axiom,
    ! [Y2: num,X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y2),X)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y2)
      <=> ( X = Y2 ) ) ) ).

% order_antisym_conv
tff(fact_2807_order__antisym__conv,axiom,
    ! [Y2: nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y2),X)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y2)
      <=> ( X = Y2 ) ) ) ).

% order_antisym_conv
tff(fact_2808_order__antisym__conv,axiom,
    ! [Y2: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y2),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y2)
      <=> ( X = Y2 ) ) ) ).

% order_antisym_conv
tff(fact_2809_linorder__le__cases,axiom,
    ! [X: rat,Y2: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y2),X) ) ).

% linorder_le_cases
tff(fact_2810_linorder__le__cases,axiom,
    ! [X: num,Y2: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y2)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y2),X) ) ).

% linorder_le_cases
tff(fact_2811_linorder__le__cases,axiom,
    ! [X: nat,Y2: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y2),X) ) ).

% linorder_le_cases
tff(fact_2812_linorder__le__cases,axiom,
    ! [X: int,Y2: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y2),X) ) ).

% linorder_le_cases
tff(fact_2813_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( ( aa_rat_rat(F,B) = C )
       => ( ! [X4: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_2814_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( ( aa_rat_num(F,B) = C )
       => ( ! [X4: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Y4)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X4)),aa_rat_num(F,Y4)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_2815_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( ( aa_rat_nat(F,B) = C )
       => ( ! [X4: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Y4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y4)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_2816_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( ( aa_rat_int(F,B) = C )
       => ( ! [X4: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Y4)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X4)),aa_rat_int(F,Y4)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_2817_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: fun_num_rat,C: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( ( aa_num_rat(F,B) = C )
       => ( ! [X4: num,Y4: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X4),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_2818_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: fun_num_num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( ( aa_num_num(F,B) = C )
       => ( ! [X4: num,Y4: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X4),Y4)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X4)),aa_num_num(F,Y4)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_2819_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: fun_num_nat,C: nat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( ( aa_num_nat(F,B) = C )
       => ( ! [X4: num,Y4: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X4),Y4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X4)),aa_num_nat(F,Y4)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_2820_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: fun_num_int,C: int] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( ( aa_num_int(F,B) = C )
       => ( ! [X4: num,Y4: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X4),Y4)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X4)),aa_num_int(F,Y4)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_2821_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: fun_nat_rat,C: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( ( aa_nat_rat(F,B) = C )
       => ( ! [X4: nat,Y4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X4)),aa_nat_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_2822_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: fun_nat_num,C: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( ( aa_nat_num(F,B) = C )
       => ( ! [X4: nat,Y4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Y4)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,X4)),aa_nat_num(F,Y4)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_2823_ord__eq__le__subst,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( ( A = aa_rat_rat(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X4: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_2824_ord__eq__le__subst,axiom,
    ! [A: num,F: fun_rat_num,B: rat,C: rat] :
      ( ( A = aa_rat_num(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X4: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Y4)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X4)),aa_rat_num(F,Y4)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_rat_num(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_2825_ord__eq__le__subst,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,C: rat] :
      ( ( A = aa_rat_nat(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X4: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Y4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y4)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_rat_nat(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_2826_ord__eq__le__subst,axiom,
    ! [A: int,F: fun_rat_int,B: rat,C: rat] :
      ( ( A = aa_rat_int(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X4: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Y4)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X4)),aa_rat_int(F,Y4)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_rat_int(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_2827_ord__eq__le__subst,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C: num] :
      ( ( A = aa_num_rat(F,B) )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X4: num,Y4: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X4),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_num_rat(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_2828_ord__eq__le__subst,axiom,
    ! [A: num,F: fun_num_num,B: num,C: num] :
      ( ( A = aa_num_num(F,B) )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X4: num,Y4: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X4),Y4)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X4)),aa_num_num(F,Y4)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_num_num(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_2829_ord__eq__le__subst,axiom,
    ! [A: nat,F: fun_num_nat,B: num,C: num] :
      ( ( A = aa_num_nat(F,B) )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X4: num,Y4: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X4),Y4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X4)),aa_num_nat(F,Y4)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_num_nat(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_2830_ord__eq__le__subst,axiom,
    ! [A: int,F: fun_num_int,B: num,C: num] :
      ( ( A = aa_num_int(F,B) )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X4: num,Y4: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X4),Y4)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X4)),aa_num_int(F,Y4)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_num_int(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_2831_ord__eq__le__subst,axiom,
    ! [A: rat,F: fun_nat_rat,B: nat,C: nat] :
      ( ( A = aa_nat_rat(F,B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => ( ! [X4: nat,Y4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X4)),aa_nat_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_nat_rat(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_2832_ord__eq__le__subst,axiom,
    ! [A: num,F: fun_nat_num,B: nat,C: nat] :
      ( ( A = aa_nat_num(F,B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => ( ! [X4: nat,Y4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Y4)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,X4)),aa_nat_num(F,Y4)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_nat_num(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_2833_linorder__linear,axiom,
    ! [X: rat,Y2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y2)
      | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y2),X) ) ).

% linorder_linear
tff(fact_2834_linorder__linear,axiom,
    ! [X: num,Y2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y2)
      | aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y2),X) ) ).

% linorder_linear
tff(fact_2835_linorder__linear,axiom,
    ! [X: nat,Y2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y2)
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y2),X) ) ).

% linorder_linear
tff(fact_2836_linorder__linear,axiom,
    ! [X: int,Y2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y2)
      | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y2),X) ) ).

% linorder_linear
tff(fact_2837_order__eq__refl,axiom,
    ! [X: set_int,Y2: set_int] :
      ( ( X = Y2 )
     => ord_less_eq_set_int(X,Y2) ) ).

% order_eq_refl
tff(fact_2838_order__eq__refl,axiom,
    ! [X: rat,Y2: rat] :
      ( ( X = Y2 )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y2) ) ).

% order_eq_refl
tff(fact_2839_order__eq__refl,axiom,
    ! [X: num,Y2: num] :
      ( ( X = Y2 )
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y2) ) ).

% order_eq_refl
tff(fact_2840_order__eq__refl,axiom,
    ! [X: nat,Y2: nat] :
      ( ( X = Y2 )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y2) ) ).

% order_eq_refl
tff(fact_2841_order__eq__refl,axiom,
    ! [X: int,Y2: int] :
      ( ( X = Y2 )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y2) ) ).

% order_eq_refl
tff(fact_2842_order__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,B)),C)
       => ( ! [X4: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_2843_order__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,B)),C)
       => ( ! [X4: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Y4)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X4)),aa_rat_num(F,Y4)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_2844_order__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,B)),C)
       => ( ! [X4: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Y4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y4)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_2845_order__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,B)),C)
       => ( ! [X4: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Y4)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X4)),aa_rat_int(F,Y4)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_2846_order__subst2,axiom,
    ! [A: num,B: num,F: fun_num_rat,C: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,B)),C)
       => ( ! [X4: num,Y4: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X4),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_2847_order__subst2,axiom,
    ! [A: num,B: num,F: fun_num_num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,B)),C)
       => ( ! [X4: num,Y4: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X4),Y4)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X4)),aa_num_num(F,Y4)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_2848_order__subst2,axiom,
    ! [A: num,B: num,F: fun_num_nat,C: nat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,B)),C)
       => ( ! [X4: num,Y4: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X4),Y4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X4)),aa_num_nat(F,Y4)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_2849_order__subst2,axiom,
    ! [A: num,B: num,F: fun_num_int,C: int] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,B)),C)
       => ( ! [X4: num,Y4: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X4),Y4)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X4)),aa_num_int(F,Y4)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_2850_order__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_rat,C: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,B)),C)
       => ( ! [X4: nat,Y4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X4)),aa_nat_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_2851_order__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_num,C: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,B)),C)
       => ( ! [X4: nat,Y4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Y4)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,X4)),aa_nat_num(F,Y4)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_2852_order__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X4: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(F,C)) ) ) ) ).

% order_subst1
tff(fact_2853_order__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_num_rat(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X4: num,Y4: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X4),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_num_rat(F,C)) ) ) ) ).

% order_subst1
tff(fact_2854_order__subst1,axiom,
    ! [A: rat,F: fun_nat_rat,B: nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_nat_rat(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => ( ! [X4: nat,Y4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X4)),aa_nat_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_nat_rat(F,C)) ) ) ) ).

% order_subst1
tff(fact_2855_order__subst1,axiom,
    ! [A: rat,F: fun_int_rat,B: int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_int_rat(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => ( ! [X4: int,Y4: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X4),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X4)),aa_int_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_int_rat(F,C)) ) ) ) ).

% order_subst1
tff(fact_2856_order__subst1,axiom,
    ! [A: num,F: fun_rat_num,B: rat,C: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_rat_num(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X4: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Y4)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X4)),aa_rat_num(F,Y4)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_rat_num(F,C)) ) ) ) ).

% order_subst1
tff(fact_2857_order__subst1,axiom,
    ! [A: num,F: fun_num_num,B: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_num_num(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X4: num,Y4: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X4),Y4)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X4)),aa_num_num(F,Y4)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_num_num(F,C)) ) ) ) ).

% order_subst1
tff(fact_2858_order__subst1,axiom,
    ! [A: num,F: fun_nat_num,B: nat,C: nat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_nat_num(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => ( ! [X4: nat,Y4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Y4)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,X4)),aa_nat_num(F,Y4)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_nat_num(F,C)) ) ) ) ).

% order_subst1
tff(fact_2859_order__subst1,axiom,
    ! [A: num,F: fun_int_num,B: int,C: int] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_int_num(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => ( ! [X4: int,Y4: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X4),Y4)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_int_num(F,X4)),aa_int_num(F,Y4)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_int_num(F,C)) ) ) ) ).

% order_subst1
tff(fact_2860_order__subst1,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,C: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_rat_nat(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X4: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Y4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y4)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_rat_nat(F,C)) ) ) ) ).

% order_subst1
tff(fact_2861_order__subst1,axiom,
    ! [A: nat,F: fun_num_nat,B: num,C: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_num_nat(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X4: num,Y4: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X4),Y4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X4)),aa_num_nat(F,Y4)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_num_nat(F,C)) ) ) ) ).

% order_subst1
tff(fact_2862_Orderings_Oorder__eq__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( A = B )
    <=> ( ord_less_eq_set_int(A,B)
        & ord_less_eq_set_int(B,A) ) ) ).

% Orderings.order_eq_iff
tff(fact_2863_Orderings_Oorder__eq__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).

% Orderings.order_eq_iff
tff(fact_2864_Orderings_Oorder__eq__iff,axiom,
    ! [A: num,B: num] :
      ( ( A = B )
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A) ) ) ).

% Orderings.order_eq_iff
tff(fact_2865_Orderings_Oorder__eq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ) ).

% Orderings.order_eq_iff
tff(fact_2866_Orderings_Oorder__eq__iff,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ).

% Orderings.order_eq_iff
tff(fact_2867_antisym,axiom,
    ! [A: set_int,B: set_int] :
      ( ord_less_eq_set_int(A,B)
     => ( ord_less_eq_set_int(B,A)
       => ( A = B ) ) ) ).

% antisym
tff(fact_2868_antisym,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
       => ( A = B ) ) ) ).

% antisym
tff(fact_2869_antisym,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
       => ( A = B ) ) ) ).

% antisym
tff(fact_2870_antisym,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
       => ( A = B ) ) ) ).

% antisym
tff(fact_2871_antisym,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
       => ( A = B ) ) ) ).

% antisym
tff(fact_2872_dual__order_Otrans,axiom,
    ! [B: set_int,A: set_int,C: set_int] :
      ( ord_less_eq_set_int(B,A)
     => ( ord_less_eq_set_int(C,B)
       => ord_less_eq_set_int(C,A) ) ) ).

% dual_order.trans
tff(fact_2873_dual__order_Otrans,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A) ) ) ).

% dual_order.trans
tff(fact_2874_dual__order_Otrans,axiom,
    ! [B: num,A: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),B)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),A) ) ) ).

% dual_order.trans
tff(fact_2875_dual__order_Otrans,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A) ) ) ).

% dual_order.trans
tff(fact_2876_dual__order_Otrans,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A) ) ) ).

% dual_order.trans
tff(fact_2877_dual__order_Oantisym,axiom,
    ! [B: set_int,A: set_int] :
      ( ord_less_eq_set_int(B,A)
     => ( ord_less_eq_set_int(A,B)
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_2878_dual__order_Oantisym,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_2879_dual__order_Oantisym,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_2880_dual__order_Oantisym,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_2881_dual__order_Oantisym,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_2882_dual__order_Oeq__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( A = B )
    <=> ( ord_less_eq_set_int(B,A)
        & ord_less_eq_set_int(A,B) ) ) ).

% dual_order.eq_iff
tff(fact_2883_dual__order_Oeq__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% dual_order.eq_iff
tff(fact_2884_dual__order_Oeq__iff,axiom,
    ! [A: num,B: num] :
      ( ( A = B )
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B) ) ) ).

% dual_order.eq_iff
tff(fact_2885_dual__order_Oeq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).

% dual_order.eq_iff
tff(fact_2886_dual__order_Oeq__iff,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% dual_order.eq_iff
tff(fact_2887_linorder__wlog,axiom,
    ! [P: fun_rat_fun_rat_o,A: rat,B: rat] :
      ( ! [A3: rat,B2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A3),B2)
         => aa_rat_o(aa_rat_fun_rat_o(P,A3),B2) )
     => ( ! [A3: rat,B2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(P,B2),A3)
           => aa_rat_o(aa_rat_fun_rat_o(P,A3),B2) )
       => aa_rat_o(aa_rat_fun_rat_o(P,A),B) ) ) ).

% linorder_wlog
tff(fact_2888_linorder__wlog,axiom,
    ! [P: fun_num_fun_num_o,A: num,B: num] :
      ( ! [A3: num,B2: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A3),B2)
         => aa_num_o(aa_num_fun_num_o(P,A3),B2) )
     => ( ! [A3: num,B2: num] :
            ( aa_num_o(aa_num_fun_num_o(P,B2),A3)
           => aa_num_o(aa_num_fun_num_o(P,A3),B2) )
       => aa_num_o(aa_num_fun_num_o(P,A),B) ) ) ).

% linorder_wlog
tff(fact_2889_linorder__wlog,axiom,
    ! [P: fun_nat_fun_nat_o,A: nat,B: nat] :
      ( ! [A3: nat,B2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A3),B2)
         => aa_nat_o(aa_nat_fun_nat_o(P,A3),B2) )
     => ( ! [A3: nat,B2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(P,B2),A3)
           => aa_nat_o(aa_nat_fun_nat_o(P,A3),B2) )
       => aa_nat_o(aa_nat_fun_nat_o(P,A),B) ) ) ).

% linorder_wlog
tff(fact_2890_linorder__wlog,axiom,
    ! [P: fun_int_fun_int_o,A: int,B: int] :
      ( ! [A3: int,B2: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A3),B2)
         => aa_int_o(aa_int_fun_int_o(P,A3),B2) )
     => ( ! [A3: int,B2: int] :
            ( aa_int_o(aa_int_fun_int_o(P,B2),A3)
           => aa_int_o(aa_int_fun_int_o(P,A3),B2) )
       => aa_int_o(aa_int_fun_int_o(P,A),B) ) ) ).

% linorder_wlog
tff(fact_2891_order__trans,axiom,
    ! [X: set_int,Y2: set_int,Z2: set_int] :
      ( ord_less_eq_set_int(X,Y2)
     => ( ord_less_eq_set_int(Y2,Z2)
       => ord_less_eq_set_int(X,Z2) ) ) ).

% order_trans
tff(fact_2892_order__trans,axiom,
    ! [X: rat,Y2: rat,Z2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y2),Z2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Z2) ) ) ).

% order_trans
tff(fact_2893_order__trans,axiom,
    ! [X: num,Y2: num,Z2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y2)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y2),Z2)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Z2) ) ) ).

% order_trans
tff(fact_2894_order__trans,axiom,
    ! [X: nat,Y2: nat,Z2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y2),Z2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Z2) ) ) ).

% order_trans
tff(fact_2895_order__trans,axiom,
    ! [X: int,Y2: int,Z2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y2),Z2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Z2) ) ) ).

% order_trans
tff(fact_2896_order_Otrans,axiom,
    ! [A: set_int,B: set_int,C: set_int] :
      ( ord_less_eq_set_int(A,B)
     => ( ord_less_eq_set_int(B,C)
       => ord_less_eq_set_int(A,C) ) ) ).

% order.trans
tff(fact_2897_order_Otrans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),C) ) ) ).

% order.trans
tff(fact_2898_order_Otrans,axiom,
    ! [A: num,B: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),C) ) ) ).

% order.trans
tff(fact_2899_order_Otrans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C) ) ) ).

% order.trans
tff(fact_2900_order_Otrans,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C) ) ) ).

% order.trans
tff(fact_2901_order__antisym,axiom,
    ! [X: set_int,Y2: set_int] :
      ( ord_less_eq_set_int(X,Y2)
     => ( ord_less_eq_set_int(Y2,X)
       => ( X = Y2 ) ) ) ).

% order_antisym
tff(fact_2902_order__antisym,axiom,
    ! [X: rat,Y2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y2),X)
       => ( X = Y2 ) ) ) ).

% order_antisym
tff(fact_2903_order__antisym,axiom,
    ! [X: num,Y2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y2)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y2),X)
       => ( X = Y2 ) ) ) ).

% order_antisym
tff(fact_2904_order__antisym,axiom,
    ! [X: nat,Y2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y2),X)
       => ( X = Y2 ) ) ) ).

% order_antisym
tff(fact_2905_order__antisym,axiom,
    ! [X: int,Y2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y2),X)
       => ( X = Y2 ) ) ) ).

% order_antisym
tff(fact_2906_ord__le__eq__trans,axiom,
    ! [A: set_int,B: set_int,C: set_int] :
      ( ord_less_eq_set_int(A,B)
     => ( ( B = C )
       => ord_less_eq_set_int(A,C) ) ) ).

% ord_le_eq_trans
tff(fact_2907_ord__le__eq__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( ( B = C )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),C) ) ) ).

% ord_le_eq_trans
tff(fact_2908_ord__le__eq__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( ( B = C )
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),C) ) ) ).

% ord_le_eq_trans
tff(fact_2909_ord__le__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( ( B = C )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C) ) ) ).

% ord_le_eq_trans
tff(fact_2910_ord__le__eq__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( ( B = C )
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C) ) ) ).

% ord_le_eq_trans
tff(fact_2911_ord__eq__le__trans,axiom,
    ! [A: set_int,B: set_int,C: set_int] :
      ( ( A = B )
     => ( ord_less_eq_set_int(B,C)
       => ord_less_eq_set_int(A,C) ) ) ).

% ord_eq_le_trans
tff(fact_2912_ord__eq__le__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = B )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),C) ) ) ).

% ord_eq_le_trans
tff(fact_2913_ord__eq__le__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( A = B )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),C) ) ) ).

% ord_eq_le_trans
tff(fact_2914_ord__eq__le__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C) ) ) ).

% ord_eq_le_trans
tff(fact_2915_ord__eq__le__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A = B )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C) ) ) ).

% ord_eq_le_trans
tff(fact_2916_order__class_Oorder__eq__iff,axiom,
    ! [X: set_int,Y2: set_int] :
      ( ( X = Y2 )
    <=> ( ord_less_eq_set_int(X,Y2)
        & ord_less_eq_set_int(Y2,X) ) ) ).

% order_class.order_eq_iff
tff(fact_2917_order__class_Oorder__eq__iff,axiom,
    ! [X: rat,Y2: rat] :
      ( ( X = Y2 )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y2)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y2),X) ) ) ).

% order_class.order_eq_iff
tff(fact_2918_order__class_Oorder__eq__iff,axiom,
    ! [X: num,Y2: num] :
      ( ( X = Y2 )
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y2)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y2),X) ) ) ).

% order_class.order_eq_iff
tff(fact_2919_order__class_Oorder__eq__iff,axiom,
    ! [X: nat,Y2: nat] :
      ( ( X = Y2 )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y2),X) ) ) ).

% order_class.order_eq_iff
tff(fact_2920_order__class_Oorder__eq__iff,axiom,
    ! [X: int,Y2: int] :
      ( ( X = Y2 )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y2)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y2),X) ) ) ).

% order_class.order_eq_iff
tff(fact_2921_le__cases3,axiom,
    ! [X: rat,Y2: rat,Z2: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y2)
       => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y2),Z2) )
     => ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y2),X)
         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Z2) )
       => ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Z2)
           => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z2),Y2) )
         => ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z2),Y2)
             => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y2),X) )
           => ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y2),Z2)
               => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z2),X) )
             => ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z2),X)
                 => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y2) ) ) ) ) ) ) ).

% le_cases3
tff(fact_2922_le__cases3,axiom,
    ! [X: num,Y2: num,Z2: num] :
      ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y2)
       => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y2),Z2) )
     => ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y2),X)
         => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Z2) )
       => ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Z2)
           => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z2),Y2) )
         => ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z2),Y2)
             => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y2),X) )
           => ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y2),Z2)
               => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z2),X) )
             => ~ ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z2),X)
                 => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y2) ) ) ) ) ) ) ).

% le_cases3
tff(fact_2923_le__cases3,axiom,
    ! [X: nat,Y2: nat,Z2: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y2)
       => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y2),Z2) )
     => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y2),X)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Z2) )
       => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Z2)
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z2),Y2) )
         => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z2),Y2)
             => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y2),X) )
           => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y2),Z2)
               => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z2),X) )
             => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z2),X)
                 => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y2) ) ) ) ) ) ) ).

% le_cases3
tff(fact_2924_le__cases3,axiom,
    ! [X: int,Y2: int,Z2: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y2)
       => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y2),Z2) )
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y2),X)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Z2) )
       => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Z2)
           => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z2),Y2) )
         => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z2),Y2)
             => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y2),X) )
           => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y2),Z2)
               => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z2),X) )
             => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z2),X)
                 => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y2) ) ) ) ) ) ) ).

% le_cases3
tff(fact_2925_nle__le,axiom,
    ! [A: rat,B: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
        & ( B != A ) ) ) ).

% nle_le
tff(fact_2926_nle__le,axiom,
    ! [A: num,B: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
        & ( B != A ) ) ) ).

% nle_le
tff(fact_2927_nle__le,axiom,
    ! [A: nat,B: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
        & ( B != A ) ) ) ).

% nle_le
tff(fact_2928_nle__le,axiom,
    ! [A: int,B: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
        & ( B != A ) ) ) ).

% nle_le
tff(fact_2929_order__less__imp__not__less,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),X) ) ).

% order_less_imp_not_less
tff(fact_2930_order__less__imp__not__less,axiom,
    ! [X: rat,Y2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y2)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y2),X) ) ).

% order_less_imp_not_less
tff(fact_2931_order__less__imp__not__less,axiom,
    ! [X: num,Y2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y2)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Y2),X) ) ).

% order_less_imp_not_less
tff(fact_2932_order__less__imp__not__less,axiom,
    ! [X: nat,Y2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y2)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y2),X) ) ).

% order_less_imp_not_less
tff(fact_2933_order__less__imp__not__less,axiom,
    ! [X: int,Y2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y2)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y2),X) ) ).

% order_less_imp_not_less
tff(fact_2934_order__less__imp__not__eq2,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2)
     => ( Y2 != X ) ) ).

% order_less_imp_not_eq2
tff(fact_2935_order__less__imp__not__eq2,axiom,
    ! [X: rat,Y2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y2)
     => ( Y2 != X ) ) ).

% order_less_imp_not_eq2
tff(fact_2936_order__less__imp__not__eq2,axiom,
    ! [X: num,Y2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y2)
     => ( Y2 != X ) ) ).

% order_less_imp_not_eq2
tff(fact_2937_order__less__imp__not__eq2,axiom,
    ! [X: nat,Y2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y2)
     => ( Y2 != X ) ) ).

% order_less_imp_not_eq2
tff(fact_2938_order__less__imp__not__eq2,axiom,
    ! [X: int,Y2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y2)
     => ( Y2 != X ) ) ).

% order_less_imp_not_eq2
tff(fact_2939_order__less__imp__not__eq,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2)
     => ( X != Y2 ) ) ).

% order_less_imp_not_eq
tff(fact_2940_order__less__imp__not__eq,axiom,
    ! [X: rat,Y2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y2)
     => ( X != Y2 ) ) ).

% order_less_imp_not_eq
tff(fact_2941_order__less__imp__not__eq,axiom,
    ! [X: num,Y2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y2)
     => ( X != Y2 ) ) ).

% order_less_imp_not_eq
tff(fact_2942_order__less__imp__not__eq,axiom,
    ! [X: nat,Y2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y2)
     => ( X != Y2 ) ) ).

% order_less_imp_not_eq
tff(fact_2943_order__less__imp__not__eq,axiom,
    ! [X: int,Y2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y2)
     => ( X != Y2 ) ) ).

% order_less_imp_not_eq
tff(fact_2944_linorder__less__linear,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2)
      | ( X = Y2 )
      | aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),X) ) ).

% linorder_less_linear
tff(fact_2945_linorder__less__linear,axiom,
    ! [X: rat,Y2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y2)
      | ( X = Y2 )
      | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y2),X) ) ).

% linorder_less_linear
tff(fact_2946_linorder__less__linear,axiom,
    ! [X: num,Y2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y2)
      | ( X = Y2 )
      | aa_num_o(aa_num_fun_num_o(ord_less_num,Y2),X) ) ).

% linorder_less_linear
tff(fact_2947_linorder__less__linear,axiom,
    ! [X: nat,Y2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y2)
      | ( X = Y2 )
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y2),X) ) ).

% linorder_less_linear
tff(fact_2948_linorder__less__linear,axiom,
    ! [X: int,Y2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y2)
      | ( X = Y2 )
      | aa_int_o(aa_int_fun_int_o(ord_less_int,Y2),X) ) ).

% linorder_less_linear
tff(fact_2949_order__less__imp__triv,axiom,
    ! [X: real,Y2: real,P: $o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),X)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_2950_order__less__imp__triv,axiom,
    ! [X: rat,Y2: rat,P: $o] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y2),X)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_2951_order__less__imp__triv,axiom,
    ! [X: num,Y2: num,P: $o] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y2)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Y2),X)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_2952_order__less__imp__triv,axiom,
    ! [X: nat,Y2: nat,P: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y2),X)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_2953_order__less__imp__triv,axiom,
    ! [X: int,Y2: int,P: $o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y2),X)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_2954_order__less__not__sym,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),X) ) ).

% order_less_not_sym
tff(fact_2955_order__less__not__sym,axiom,
    ! [X: rat,Y2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y2)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y2),X) ) ).

% order_less_not_sym
tff(fact_2956_order__less__not__sym,axiom,
    ! [X: num,Y2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y2)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Y2),X) ) ).

% order_less_not_sym
tff(fact_2957_order__less__not__sym,axiom,
    ! [X: nat,Y2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y2)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y2),X) ) ).

% order_less_not_sym
tff(fact_2958_order__less__not__sym,axiom,
    ! [X: int,Y2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y2)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y2),X) ) ).

% order_less_not_sym
tff(fact_2959_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,B)),C)
       => ( ! [X4: real,Y4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Y4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X4)),aa_real_real(F,Y4)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_2960_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_rat,C: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,B)),C)
       => ( ! [X4: real,Y4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X4)),aa_real_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_2961_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_num,C: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,B)),C)
       => ( ! [X4: real,Y4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Y4)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,X4)),aa_real_num(F,Y4)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_2962_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_nat,C: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,B)),C)
       => ( ! [X4: real,Y4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Y4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,X4)),aa_real_nat(F,Y4)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_2963_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_int,C: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,B)),C)
       => ( ! [X4: real,Y4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Y4)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,X4)),aa_real_int(F,Y4)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_2964_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_real,C: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,B)),C)
       => ( ! [X4: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Y4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X4)),aa_rat_real(F,Y4)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_2965_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,B)),C)
       => ( ! [X4: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_2966_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,B)),C)
       => ( ! [X4: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Y4)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,X4)),aa_rat_num(F,Y4)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_2967_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,B)),C)
       => ( ! [X4: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Y4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y4)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_2968_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,B)),C)
       => ( ! [X4: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Y4)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,X4)),aa_rat_int(F,Y4)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_2969_order__less__subst1,axiom,
    ! [A: real,F: fun_real_real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(F,B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X4: real,Y4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Y4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X4)),aa_real_real(F,Y4)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_2970_order__less__subst1,axiom,
    ! [A: real,F: fun_rat_real,B: rat,C: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_rat_real(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X4: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Y4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X4)),aa_rat_real(F,Y4)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_rat_real(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_2971_order__less__subst1,axiom,
    ! [A: real,F: fun_num_real,B: num,C: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_num_real(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C)
       => ( ! [X4: num,Y4: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X4),Y4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,X4)),aa_num_real(F,Y4)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_num_real(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_2972_order__less__subst1,axiom,
    ! [A: real,F: fun_nat_real,B: nat,C: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_nat_real(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => ( ! [X4: nat,Y4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Y4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,X4)),aa_nat_real(F,Y4)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_nat_real(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_2973_order__less__subst1,axiom,
    ! [A: real,F: fun_int_real,B: int,C: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_int_real(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => ( ! [X4: int,Y4: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Y4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,X4)),aa_int_real(F,Y4)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_int_real(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_2974_order__less__subst1,axiom,
    ! [A: rat,F: fun_real_rat,B: real,C: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_real_rat(F,B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X4: real,Y4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X4)),aa_real_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_real_rat(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_2975_order__less__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X4: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_2976_order__less__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_num_rat(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C)
       => ( ! [X4: num,Y4: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X4),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_num_rat(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_2977_order__less__subst1,axiom,
    ! [A: rat,F: fun_nat_rat,B: nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_nat_rat(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => ( ! [X4: nat,Y4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X4)),aa_nat_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_nat_rat(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_2978_order__less__subst1,axiom,
    ! [A: rat,F: fun_int_rat,B: int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_int_rat(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => ( ! [X4: int,Y4: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X4)),aa_int_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_int_rat(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_2979_order__less__irrefl,axiom,
    ! [X: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,X),X) ).

% order_less_irrefl
tff(fact_2980_order__less__irrefl,axiom,
    ! [X: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),X) ).

% order_less_irrefl
tff(fact_2981_order__less__irrefl,axiom,
    ! [X: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X),X) ).

% order_less_irrefl
tff(fact_2982_order__less__irrefl,axiom,
    ! [X: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),X) ).

% order_less_irrefl
tff(fact_2983_order__less__irrefl,axiom,
    ! [X: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),X) ).

% order_less_irrefl
tff(fact_2984_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ( aa_real_real(F,B) = C )
       => ( ! [X4: real,Y4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Y4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X4)),aa_real_real(F,Y4)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_2985_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_rat,C: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ( aa_real_rat(F,B) = C )
       => ( ! [X4: real,Y4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X4)),aa_real_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_2986_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_num,C: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ( aa_real_num(F,B) = C )
       => ( ! [X4: real,Y4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Y4)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,X4)),aa_real_num(F,Y4)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_2987_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_nat,C: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ( aa_real_nat(F,B) = C )
       => ( ! [X4: real,Y4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Y4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,X4)),aa_real_nat(F,Y4)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_2988_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_int,C: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ( aa_real_int(F,B) = C )
       => ( ! [X4: real,Y4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Y4)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,X4)),aa_real_int(F,Y4)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_2989_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_real,C: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ( aa_rat_real(F,B) = C )
       => ( ! [X4: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Y4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X4)),aa_rat_real(F,Y4)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_2990_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ( aa_rat_rat(F,B) = C )
       => ( ! [X4: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_2991_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ( aa_rat_num(F,B) = C )
       => ( ! [X4: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Y4)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,X4)),aa_rat_num(F,Y4)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_2992_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ( aa_rat_nat(F,B) = C )
       => ( ! [X4: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Y4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y4)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_2993_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ( aa_rat_int(F,B) = C )
       => ( ! [X4: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Y4)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,X4)),aa_rat_int(F,Y4)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_2994_ord__eq__less__subst,axiom,
    ! [A: real,F: fun_real_real,B: real,C: real] :
      ( ( A = aa_real_real(F,B) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X4: real,Y4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Y4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X4)),aa_real_real(F,Y4)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_2995_ord__eq__less__subst,axiom,
    ! [A: rat,F: fun_real_rat,B: real,C: real] :
      ( ( A = aa_real_rat(F,B) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X4: real,Y4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X4)),aa_real_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_real_rat(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_2996_ord__eq__less__subst,axiom,
    ! [A: num,F: fun_real_num,B: real,C: real] :
      ( ( A = aa_real_num(F,B) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X4: real,Y4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Y4)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,X4)),aa_real_num(F,Y4)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_real_num(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_2997_ord__eq__less__subst,axiom,
    ! [A: nat,F: fun_real_nat,B: real,C: real] :
      ( ( A = aa_real_nat(F,B) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X4: real,Y4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Y4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,X4)),aa_real_nat(F,Y4)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_real_nat(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_2998_ord__eq__less__subst,axiom,
    ! [A: int,F: fun_real_int,B: real,C: real] :
      ( ( A = aa_real_int(F,B) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X4: real,Y4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Y4)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,X4)),aa_real_int(F,Y4)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_real_int(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_2999_ord__eq__less__subst,axiom,
    ! [A: real,F: fun_rat_real,B: rat,C: rat] :
      ( ( A = aa_rat_real(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X4: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Y4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X4)),aa_rat_real(F,Y4)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_rat_real(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_3000_ord__eq__less__subst,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( ( A = aa_rat_rat(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X4: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_3001_ord__eq__less__subst,axiom,
    ! [A: num,F: fun_rat_num,B: rat,C: rat] :
      ( ( A = aa_rat_num(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X4: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Y4)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,X4)),aa_rat_num(F,Y4)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_rat_num(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_3002_ord__eq__less__subst,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,C: rat] :
      ( ( A = aa_rat_nat(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X4: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Y4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y4)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_rat_nat(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_3003_ord__eq__less__subst,axiom,
    ! [A: int,F: fun_rat_int,B: rat,C: rat] :
      ( ( A = aa_rat_int(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X4: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Y4)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,X4)),aa_rat_int(F,Y4)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_rat_int(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_3004_order__less__trans,axiom,
    ! [X: real,Y2: real,Z2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),Z2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z2) ) ) ).

% order_less_trans
tff(fact_3005_order__less__trans,axiom,
    ! [X: rat,Y2: rat,Z2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y2),Z2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z2) ) ) ).

% order_less_trans
tff(fact_3006_order__less__trans,axiom,
    ! [X: num,Y2: num,Z2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y2)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Y2),Z2)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,X),Z2) ) ) ).

% order_less_trans
tff(fact_3007_order__less__trans,axiom,
    ! [X: nat,Y2: nat,Z2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y2),Z2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z2) ) ) ).

% order_less_trans
tff(fact_3008_order__less__trans,axiom,
    ! [X: int,Y2: int,Z2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y2),Z2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z2) ) ) ).

% order_less_trans
tff(fact_3009_order__less__asym_H,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ).

% order_less_asym'
tff(fact_3010_order__less__asym_H,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ).

% order_less_asym'
tff(fact_3011_order__less__asym_H,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,B),A) ) ).

% order_less_asym'
tff(fact_3012_order__less__asym_H,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A) ) ).

% order_less_asym'
tff(fact_3013_order__less__asym_H,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ).

% order_less_asym'
tff(fact_3014_linorder__neq__iff,axiom,
    ! [X: real,Y2: real] :
      ( ( X != Y2 )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2)
        | aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),X) ) ) ).

% linorder_neq_iff
tff(fact_3015_linorder__neq__iff,axiom,
    ! [X: rat,Y2: rat] :
      ( ( X != Y2 )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y2)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y2),X) ) ) ).

% linorder_neq_iff
tff(fact_3016_linorder__neq__iff,axiom,
    ! [X: num,Y2: num] :
      ( ( X != Y2 )
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y2)
        | aa_num_o(aa_num_fun_num_o(ord_less_num,Y2),X) ) ) ).

% linorder_neq_iff
tff(fact_3017_linorder__neq__iff,axiom,
    ! [X: nat,Y2: nat] :
      ( ( X != Y2 )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y2)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y2),X) ) ) ).

% linorder_neq_iff
tff(fact_3018_linorder__neq__iff,axiom,
    ! [X: int,Y2: int] :
      ( ( X != Y2 )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y2)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,Y2),X) ) ) ).

% linorder_neq_iff
tff(fact_3019_order__less__asym,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),X) ) ).

% order_less_asym
tff(fact_3020_order__less__asym,axiom,
    ! [X: rat,Y2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y2)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y2),X) ) ).

% order_less_asym
tff(fact_3021_order__less__asym,axiom,
    ! [X: num,Y2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y2)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Y2),X) ) ).

% order_less_asym
tff(fact_3022_order__less__asym,axiom,
    ! [X: nat,Y2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y2)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y2),X) ) ).

% order_less_asym
tff(fact_3023_order__less__asym,axiom,
    ! [X: int,Y2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y2)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y2),X) ) ).

% order_less_asym
tff(fact_3024_linorder__neqE,axiom,
    ! [X: real,Y2: real] :
      ( ( X != Y2 )
     => ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),X) ) ) ).

% linorder_neqE
tff(fact_3025_linorder__neqE,axiom,
    ! [X: rat,Y2: rat] :
      ( ( X != Y2 )
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y2),X) ) ) ).

% linorder_neqE
tff(fact_3026_linorder__neqE,axiom,
    ! [X: num,Y2: num] :
      ( ( X != Y2 )
     => ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y2)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Y2),X) ) ) ).

% linorder_neqE
tff(fact_3027_linorder__neqE,axiom,
    ! [X: nat,Y2: nat] :
      ( ( X != Y2 )
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y2),X) ) ) ).

% linorder_neqE
tff(fact_3028_linorder__neqE,axiom,
    ! [X: int,Y2: int] :
      ( ( X != Y2 )
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Y2),X) ) ) ).

% linorder_neqE
tff(fact_3029_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_3030_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_3031_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_3032_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_3033_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_3034_order_Ostrict__implies__not__eq,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_3035_order_Ostrict__implies__not__eq,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_3036_order_Ostrict__implies__not__eq,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_3037_order_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_3038_order_Ostrict__implies__not__eq,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_3039_dual__order_Ostrict__trans,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,C),A) ) ) ).

% dual_order.strict_trans
tff(fact_3040_dual__order_Ostrict__trans,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),A) ) ) ).

% dual_order.strict_trans
tff(fact_3041_dual__order_Ostrict__trans,axiom,
    ! [B: num,A: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,C),B)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,C),A) ) ) ).

% dual_order.strict_trans
tff(fact_3042_dual__order_Ostrict__trans,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A) ) ) ).

% dual_order.strict_trans
tff(fact_3043_dual__order_Ostrict__trans,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,C),A) ) ) ).

% dual_order.strict_trans
tff(fact_3044_not__less__iff__gr__or__eq,axiom,
    ! [X: real,Y2: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),X)
        | ( X = Y2 ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_3045_not__less__iff__gr__or__eq,axiom,
    ! [X: rat,Y2: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y2)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y2),X)
        | ( X = Y2 ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_3046_not__less__iff__gr__or__eq,axiom,
    ! [X: num,Y2: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y2)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Y2),X)
        | ( X = Y2 ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_3047_not__less__iff__gr__or__eq,axiom,
    ! [X: nat,Y2: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y2)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y2),X)
        | ( X = Y2 ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_3048_not__less__iff__gr__or__eq,axiom,
    ! [X: int,Y2: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y2)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y2),X)
        | ( X = Y2 ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_3049_order_Ostrict__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),C) ) ) ).

% order.strict_trans
tff(fact_3050_order_Ostrict__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C) ) ) ).

% order.strict_trans
tff(fact_3051_order_Ostrict__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),C) ) ) ).

% order.strict_trans
tff(fact_3052_order_Ostrict__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C) ) ) ).

% order.strict_trans
tff(fact_3053_order_Ostrict__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),C) ) ) ).

% order.strict_trans
tff(fact_3054_linorder__less__wlog,axiom,
    ! [P: fun_real_fun_real_o,A: real,B: real] :
      ( ! [A3: real,B2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,A3),B2)
         => aa_real_o(aa_real_fun_real_o(P,A3),B2) )
     => ( ! [A3: real] : aa_real_o(aa_real_fun_real_o(P,A3),A3)
       => ( ! [A3: real,B2: real] :
              ( aa_real_o(aa_real_fun_real_o(P,B2),A3)
             => aa_real_o(aa_real_fun_real_o(P,A3),B2) )
         => aa_real_o(aa_real_fun_real_o(P,A),B) ) ) ) ).

% linorder_less_wlog
tff(fact_3055_linorder__less__wlog,axiom,
    ! [P: fun_rat_fun_rat_o,A: rat,B: rat] :
      ( ! [A3: rat,B2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A3),B2)
         => aa_rat_o(aa_rat_fun_rat_o(P,A3),B2) )
     => ( ! [A3: rat] : aa_rat_o(aa_rat_fun_rat_o(P,A3),A3)
       => ( ! [A3: rat,B2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(P,B2),A3)
             => aa_rat_o(aa_rat_fun_rat_o(P,A3),B2) )
         => aa_rat_o(aa_rat_fun_rat_o(P,A),B) ) ) ) ).

% linorder_less_wlog
tff(fact_3056_linorder__less__wlog,axiom,
    ! [P: fun_num_fun_num_o,A: num,B: num] :
      ( ! [A3: num,B2: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,A3),B2)
         => aa_num_o(aa_num_fun_num_o(P,A3),B2) )
     => ( ! [A3: num] : aa_num_o(aa_num_fun_num_o(P,A3),A3)
       => ( ! [A3: num,B2: num] :
              ( aa_num_o(aa_num_fun_num_o(P,B2),A3)
             => aa_num_o(aa_num_fun_num_o(P,A3),B2) )
         => aa_num_o(aa_num_fun_num_o(P,A),B) ) ) ) ).

% linorder_less_wlog
tff(fact_3057_linorder__less__wlog,axiom,
    ! [P: fun_nat_fun_nat_o,A: nat,B: nat] :
      ( ! [A3: nat,B2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A3),B2)
         => aa_nat_o(aa_nat_fun_nat_o(P,A3),B2) )
     => ( ! [A3: nat] : aa_nat_o(aa_nat_fun_nat_o(P,A3),A3)
       => ( ! [A3: nat,B2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(P,B2),A3)
             => aa_nat_o(aa_nat_fun_nat_o(P,A3),B2) )
         => aa_nat_o(aa_nat_fun_nat_o(P,A),B) ) ) ) ).

% linorder_less_wlog
tff(fact_3058_linorder__less__wlog,axiom,
    ! [P: fun_int_fun_int_o,A: int,B: int] :
      ( ! [A3: int,B2: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,A3),B2)
         => aa_int_o(aa_int_fun_int_o(P,A3),B2) )
     => ( ! [A3: int] : aa_int_o(aa_int_fun_int_o(P,A3),A3)
       => ( ! [A3: int,B2: int] :
              ( aa_int_o(aa_int_fun_int_o(P,B2),A3)
             => aa_int_o(aa_int_fun_int_o(P,A3),B2) )
         => aa_int_o(aa_int_fun_int_o(P,A),B) ) ) ) ).

% linorder_less_wlog
tff(fact_3059_exists__least__iff,axiom,
    ! [P: fun_nat_o] :
      ( ? [X_12: nat] : aa_nat_o(P,X_12)
    <=> ? [N4: nat] :
          ( aa_nat_o(P,N4)
          & ! [M3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),N4)
             => ~ aa_nat_o(P,M3) ) ) ) ).

% exists_least_iff
tff(fact_3060_dual__order_Oirrefl,axiom,
    ! [A: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),A) ).

% dual_order.irrefl
tff(fact_3061_dual__order_Oirrefl,axiom,
    ! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),A) ).

% dual_order.irrefl
tff(fact_3062_dual__order_Oirrefl,axiom,
    ! [A: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A),A) ).

% dual_order.irrefl
tff(fact_3063_dual__order_Oirrefl,axiom,
    ! [A: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),A) ).

% dual_order.irrefl
tff(fact_3064_dual__order_Oirrefl,axiom,
    ! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),A) ).

% dual_order.irrefl
tff(fact_3065_dual__order_Oasym,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% dual_order.asym
tff(fact_3066_dual__order_Oasym,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% dual_order.asym
tff(fact_3067_dual__order_Oasym,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A),B) ) ).

% dual_order.asym
tff(fact_3068_dual__order_Oasym,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% dual_order.asym
tff(fact_3069_dual__order_Oasym,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% dual_order.asym
tff(fact_3070_linorder__cases,axiom,
    ! [X: real,Y2: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2)
     => ( ( X != Y2 )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),X) ) ) ).

% linorder_cases
tff(fact_3071_linorder__cases,axiom,
    ! [X: rat,Y2: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y2)
     => ( ( X != Y2 )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y2),X) ) ) ).

% linorder_cases
tff(fact_3072_linorder__cases,axiom,
    ! [X: num,Y2: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y2)
     => ( ( X != Y2 )
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Y2),X) ) ) ).

% linorder_cases
tff(fact_3073_linorder__cases,axiom,
    ! [X: nat,Y2: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y2)
     => ( ( X != Y2 )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y2),X) ) ) ).

% linorder_cases
tff(fact_3074_linorder__cases,axiom,
    ! [X: int,Y2: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y2)
     => ( ( X != Y2 )
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Y2),X) ) ) ).

% linorder_cases
tff(fact_3075_antisym__conv3,axiom,
    ! [Y2: real,X: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),X)
     => ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2)
      <=> ( X = Y2 ) ) ) ).

% antisym_conv3
tff(fact_3076_antisym__conv3,axiom,
    ! [Y2: rat,X: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y2),X)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y2)
      <=> ( X = Y2 ) ) ) ).

% antisym_conv3
tff(fact_3077_antisym__conv3,axiom,
    ! [Y2: num,X: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Y2),X)
     => ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y2)
      <=> ( X = Y2 ) ) ) ).

% antisym_conv3
tff(fact_3078_antisym__conv3,axiom,
    ! [Y2: nat,X: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y2),X)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y2)
      <=> ( X = Y2 ) ) ) ).

% antisym_conv3
tff(fact_3079_antisym__conv3,axiom,
    ! [Y2: int,X: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y2),X)
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y2)
      <=> ( X = Y2 ) ) ) ).

% antisym_conv3
tff(fact_3080_less__induct,axiom,
    ! [P: fun_nat_o,A: nat] :
      ( ! [X4: nat] :
          ( ! [Y5: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y5),X4)
             => aa_nat_o(P,Y5) )
         => aa_nat_o(P,X4) )
     => aa_nat_o(P,A) ) ).

% less_induct
tff(fact_3081_ord__less__eq__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ( B = C )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),C) ) ) ).

% ord_less_eq_trans
tff(fact_3082_ord__less__eq__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ( B = C )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C) ) ) ).

% ord_less_eq_trans
tff(fact_3083_ord__less__eq__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( ( B = C )
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),C) ) ) ).

% ord_less_eq_trans
tff(fact_3084_ord__less__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( ( B = C )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C) ) ) ).

% ord_less_eq_trans
tff(fact_3085_ord__less__eq__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( ( B = C )
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),C) ) ) ).

% ord_less_eq_trans
tff(fact_3086_ord__eq__less__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A = B )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),C) ) ) ).

% ord_eq_less_trans
tff(fact_3087_ord__eq__less__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = B )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C) ) ) ).

% ord_eq_less_trans
tff(fact_3088_ord__eq__less__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( A = B )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),C) ) ) ).

% ord_eq_less_trans
tff(fact_3089_ord__eq__less__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C) ) ) ).

% ord_eq_less_trans
tff(fact_3090_ord__eq__less__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A = B )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),C) ) ) ).

% ord_eq_less_trans
tff(fact_3091_order_Oasym,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ).

% order.asym
tff(fact_3092_order_Oasym,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ).

% order.asym
tff(fact_3093_order_Oasym,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,B),A) ) ).

% order.asym
tff(fact_3094_order_Oasym,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A) ) ).

% order.asym
tff(fact_3095_order_Oasym,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ).

% order.asym
tff(fact_3096_less__imp__neq,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2)
     => ( X != Y2 ) ) ).

% less_imp_neq
tff(fact_3097_less__imp__neq,axiom,
    ! [X: rat,Y2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y2)
     => ( X != Y2 ) ) ).

% less_imp_neq
tff(fact_3098_less__imp__neq,axiom,
    ! [X: num,Y2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y2)
     => ( X != Y2 ) ) ).

% less_imp_neq
tff(fact_3099_less__imp__neq,axiom,
    ! [X: nat,Y2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y2)
     => ( X != Y2 ) ) ).

% less_imp_neq
tff(fact_3100_less__imp__neq,axiom,
    ! [X: int,Y2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y2)
     => ( X != Y2 ) ) ).

% less_imp_neq
tff(fact_3101_dense,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2)
     => ? [Z: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,Z),Y2) ) ) ).

% dense
tff(fact_3102_dense,axiom,
    ! [X: rat,Y2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y2)
     => ? [Z: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),Y2) ) ) ).

% dense
tff(fact_3103_gt__ex,axiom,
    ! [X: real] :
    ? [X_13: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,X),X_13) ).

% gt_ex
tff(fact_3104_gt__ex,axiom,
    ! [X: rat] :
    ? [X_13: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),X_13) ).

% gt_ex
tff(fact_3105_gt__ex,axiom,
    ! [X: nat] :
    ? [X_13: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),X_13) ).

% gt_ex
tff(fact_3106_gt__ex,axiom,
    ! [X: int] :
    ? [X_13: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,X),X_13) ).

% gt_ex
tff(fact_3107_lt__ex,axiom,
    ! [X: real] :
    ? [Y4: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y4),X) ).

% lt_ex
tff(fact_3108_lt__ex,axiom,
    ! [X: rat] :
    ? [Y4: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y4),X) ).

% lt_ex
tff(fact_3109_lt__ex,axiom,
    ! [X: int] :
    ? [Y4: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,Y4),X) ).

% lt_ex
tff(fact_3110_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B2: $o,X4: nat] : X != produc738532404422230701BT_nat(aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B2)),X4)
     => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] : X != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,zero_zero_nat,Uv2),Uw),Ux)
       => ~ ! [Uy: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S2: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy,aa_nat_nat(suc,V2),TreeList),S2),X4) ) ) ).

% VEBT_internal.naive_member.cases
tff(fact_3111_VEBT__internal_Oelim__dead_Ocases,axiom,
    ! [X: produc7272778201969148633d_enat] :
      ( ! [A3: $o,B2: $o,Uu2: extended_enat] : X != produc581526299967858633d_enat(aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B2)),Uu2)
     => ( ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] : X != produc581526299967858633d_enat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList),Summary),extend5688581933313929465d_enat)
       => ~ ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,L2: nat] : X != produc581526299967858633d_enat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList),Summary),aa_nat_Extended_enat(extended_enat2,L2)) ) ) ).

% VEBT_internal.elim_dead.cases
tff(fact_3112_order__le__imp__less__or__eq,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2)
        | ( X = Y2 ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_3113_order__le__imp__less__or__eq,axiom,
    ! [X: set_int,Y2: set_int] :
      ( ord_less_eq_set_int(X,Y2)
     => ( ord_less_set_int(X,Y2)
        | ( X = Y2 ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_3114_order__le__imp__less__or__eq,axiom,
    ! [X: rat,Y2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y2)
        | ( X = Y2 ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_3115_order__le__imp__less__or__eq,axiom,
    ! [X: num,Y2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y2)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y2)
        | ( X = Y2 ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_3116_order__le__imp__less__or__eq,axiom,
    ! [X: nat,Y2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y2)
        | ( X = Y2 ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_3117_order__le__imp__less__or__eq,axiom,
    ! [X: int,Y2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y2)
        | ( X = Y2 ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_3118_linorder__le__less__linear,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y2)
      | aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),X) ) ).

% linorder_le_less_linear
tff(fact_3119_linorder__le__less__linear,axiom,
    ! [X: rat,Y2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y2)
      | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y2),X) ) ).

% linorder_le_less_linear
tff(fact_3120_linorder__le__less__linear,axiom,
    ! [X: num,Y2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y2)
      | aa_num_o(aa_num_fun_num_o(ord_less_num,Y2),X) ) ).

% linorder_le_less_linear
tff(fact_3121_linorder__le__less__linear,axiom,
    ! [X: nat,Y2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y2)
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y2),X) ) ).

% linorder_le_less_linear
tff(fact_3122_linorder__le__less__linear,axiom,
    ! [X: int,Y2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y2)
      | aa_int_o(aa_int_fun_int_o(ord_less_int,Y2),X) ) ).

% linorder_le_less_linear
tff(fact_3123_order__less__le__subst2,axiom,
    ! [A: real,B: real,F: fun_real_real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,B)),C)
       => ( ! [X4: real,Y4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Y4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X4)),aa_real_real(F,Y4)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_3124_order__less__le__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_real,C: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_rat_real(F,B)),C)
       => ( ! [X4: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Y4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X4)),aa_rat_real(F,Y4)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_3125_order__less__le__subst2,axiom,
    ! [A: num,B: num,F: fun_num_real,C: real] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(F,B)),C)
       => ( ! [X4: num,Y4: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X4),Y4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,X4)),aa_num_real(F,Y4)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_3126_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_real,C: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,B)),C)
       => ( ! [X4: nat,Y4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Y4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,X4)),aa_nat_real(F,Y4)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_3127_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: fun_int_real,C: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,B)),C)
       => ( ! [X4: int,Y4: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Y4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,X4)),aa_int_real(F,Y4)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_3128_order__less__le__subst2,axiom,
    ! [A: real,B: real,F: fun_real_rat,C: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,B)),C)
       => ( ! [X4: real,Y4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X4)),aa_real_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_3129_order__less__le__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,B)),C)
       => ( ! [X4: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_3130_order__less__le__subst2,axiom,
    ! [A: num,B: num,F: fun_num_rat,C: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,B)),C)
       => ( ! [X4: num,Y4: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X4),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_3131_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_rat,C: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,B)),C)
       => ( ! [X4: nat,Y4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X4)),aa_nat_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_3132_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: fun_int_rat,C: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,B)),C)
       => ( ! [X4: int,Y4: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X4)),aa_int_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_3133_order__less__le__subst1,axiom,
    ! [A: real,F: fun_rat_real,B: rat,C: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_rat_real(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X4: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Y4)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_rat_real(F,X4)),aa_rat_real(F,Y4)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_rat_real(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_3134_order__less__le__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X4: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_3135_order__less__le__subst1,axiom,
    ! [A: num,F: fun_rat_num,B: rat,C: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_rat_num(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X4: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Y4)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X4)),aa_rat_num(F,Y4)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_rat_num(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_3136_order__less__le__subst1,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,C: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_rat_nat(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X4: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Y4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y4)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_rat_nat(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_3137_order__less__le__subst1,axiom,
    ! [A: int,F: fun_rat_int,B: rat,C: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_rat_int(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X4: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Y4)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X4)),aa_rat_int(F,Y4)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_rat_int(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_3138_order__less__le__subst1,axiom,
    ! [A: real,F: fun_num_real,B: num,C: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_num_real(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X4: num,Y4: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X4),Y4)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(F,X4)),aa_num_real(F,Y4)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_num_real(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_3139_order__less__le__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_num_rat(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X4: num,Y4: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X4),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_num_rat(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_3140_order__less__le__subst1,axiom,
    ! [A: num,F: fun_num_num,B: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_num_num(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X4: num,Y4: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X4),Y4)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X4)),aa_num_num(F,Y4)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_num_num(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_3141_order__less__le__subst1,axiom,
    ! [A: nat,F: fun_num_nat,B: num,C: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_num_nat(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X4: num,Y4: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X4),Y4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X4)),aa_num_nat(F,Y4)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_num_nat(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_3142_order__less__le__subst1,axiom,
    ! [A: int,F: fun_num_int,B: num,C: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_num_int(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X4: num,Y4: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X4),Y4)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X4)),aa_num_int(F,Y4)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_num_int(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_3143_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_real,C: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,B)),C)
       => ( ! [X4: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Y4)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_rat_real(F,X4)),aa_rat_real(F,Y4)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_3144_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,B)),C)
       => ( ! [X4: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_3145_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,B)),C)
       => ( ! [X4: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Y4)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X4)),aa_rat_num(F,Y4)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_3146_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,B)),C)
       => ( ! [X4: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Y4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X4)),aa_rat_nat(F,Y4)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_3147_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,B)),C)
       => ( ! [X4: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Y4)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X4)),aa_rat_int(F,Y4)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_3148_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_real,C: real] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,B)),C)
       => ( ! [X4: num,Y4: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X4),Y4)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(F,X4)),aa_num_real(F,Y4)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_3149_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_rat,C: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,B)),C)
       => ( ! [X4: num,Y4: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X4),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_3150_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(F,B)),C)
       => ( ! [X4: num,Y4: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X4),Y4)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X4)),aa_num_num(F,Y4)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_3151_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_nat,C: nat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(F,B)),C)
       => ( ! [X4: num,Y4: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X4),Y4)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X4)),aa_num_nat(F,Y4)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_3152_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_int,C: int] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(F,B)),C)
       => ( ! [X4: num,Y4: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X4),Y4)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X4)),aa_num_int(F,Y4)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_3153_order__le__less__subst1,axiom,
    ! [A: real,F: fun_real_real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(F,B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X4: real,Y4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Y4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X4)),aa_real_real(F,Y4)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_3154_order__le__less__subst1,axiom,
    ! [A: real,F: fun_rat_real,B: rat,C: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_rat_real(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X4: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Y4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X4)),aa_rat_real(F,Y4)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_rat_real(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_3155_order__le__less__subst1,axiom,
    ! [A: real,F: fun_num_real,B: num,C: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_num_real(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C)
       => ( ! [X4: num,Y4: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X4),Y4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,X4)),aa_num_real(F,Y4)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_num_real(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_3156_order__le__less__subst1,axiom,
    ! [A: real,F: fun_nat_real,B: nat,C: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_nat_real(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => ( ! [X4: nat,Y4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Y4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,X4)),aa_nat_real(F,Y4)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_nat_real(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_3157_order__le__less__subst1,axiom,
    ! [A: real,F: fun_int_real,B: int,C: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_int_real(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => ( ! [X4: int,Y4: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Y4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,X4)),aa_int_real(F,Y4)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_int_real(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_3158_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_real_rat,B: real,C: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_real_rat(F,B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X4: real,Y4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X4)),aa_real_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_real_rat(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_3159_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X4: rat,Y4: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X4)),aa_rat_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_3160_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_num_rat(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C)
       => ( ! [X4: num,Y4: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X4),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,X4)),aa_num_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_num_rat(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_3161_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_nat_rat,B: nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_nat_rat(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => ( ! [X4: nat,Y4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X4)),aa_nat_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_nat_rat(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_3162_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_int_rat,B: int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_int_rat(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => ( ! [X4: int,Y4: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Y4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X4)),aa_int_rat(F,Y4)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_int_rat(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_3163_order__less__le__trans,axiom,
    ! [X: real,Y2: real,Z2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),Z2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z2) ) ) ).

% order_less_le_trans
tff(fact_3164_order__less__le__trans,axiom,
    ! [X: set_int,Y2: set_int,Z2: set_int] :
      ( ord_less_set_int(X,Y2)
     => ( ord_less_eq_set_int(Y2,Z2)
       => ord_less_set_int(X,Z2) ) ) ).

% order_less_le_trans
tff(fact_3165_order__less__le__trans,axiom,
    ! [X: rat,Y2: rat,Z2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y2),Z2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z2) ) ) ).

% order_less_le_trans
tff(fact_3166_order__less__le__trans,axiom,
    ! [X: num,Y2: num,Z2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y2)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y2),Z2)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,X),Z2) ) ) ).

% order_less_le_trans
tff(fact_3167_order__less__le__trans,axiom,
    ! [X: nat,Y2: nat,Z2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y2),Z2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z2) ) ) ).

% order_less_le_trans
tff(fact_3168_order__less__le__trans,axiom,
    ! [X: int,Y2: int,Z2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y2),Z2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z2) ) ) ).

% order_less_le_trans
tff(fact_3169_order__le__less__trans,axiom,
    ! [X: real,Y2: real,Z2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),Z2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z2) ) ) ).

% order_le_less_trans
tff(fact_3170_order__le__less__trans,axiom,
    ! [X: set_int,Y2: set_int,Z2: set_int] :
      ( ord_less_eq_set_int(X,Y2)
     => ( ord_less_set_int(Y2,Z2)
       => ord_less_set_int(X,Z2) ) ) ).

% order_le_less_trans
tff(fact_3171_order__le__less__trans,axiom,
    ! [X: rat,Y2: rat,Z2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y2),Z2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z2) ) ) ).

% order_le_less_trans
tff(fact_3172_order__le__less__trans,axiom,
    ! [X: num,Y2: num,Z2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y2)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Y2),Z2)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,X),Z2) ) ) ).

% order_le_less_trans
tff(fact_3173_order__le__less__trans,axiom,
    ! [X: nat,Y2: nat,Z2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y2),Z2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z2) ) ) ).

% order_le_less_trans
tff(fact_3174_order__le__less__trans,axiom,
    ! [X: int,Y2: int,Z2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y2),Z2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z2) ) ) ).

% order_le_less_trans
tff(fact_3175_order__neq__le__trans,axiom,
    ! [A: real,B: real] :
      ( ( A != B )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% order_neq_le_trans
tff(fact_3176_order__neq__le__trans,axiom,
    ! [A: set_int,B: set_int] :
      ( ( A != B )
     => ( ord_less_eq_set_int(A,B)
       => ord_less_set_int(A,B) ) ) ).

% order_neq_le_trans
tff(fact_3177_order__neq__le__trans,axiom,
    ! [A: rat,B: rat] :
      ( ( A != B )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% order_neq_le_trans
tff(fact_3178_order__neq__le__trans,axiom,
    ! [A: num,B: num] :
      ( ( A != B )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),B) ) ) ).

% order_neq_le_trans
tff(fact_3179_order__neq__le__trans,axiom,
    ! [A: nat,B: nat] :
      ( ( A != B )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% order_neq_le_trans
tff(fact_3180_order__neq__le__trans,axiom,
    ! [A: int,B: int] :
      ( ( A != B )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% order_neq_le_trans
tff(fact_3181_order__le__neq__trans,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( ( A != B )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% order_le_neq_trans
tff(fact_3182_order__le__neq__trans,axiom,
    ! [A: set_int,B: set_int] :
      ( ord_less_eq_set_int(A,B)
     => ( ( A != B )
       => ord_less_set_int(A,B) ) ) ).

% order_le_neq_trans
tff(fact_3183_order__le__neq__trans,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( ( A != B )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% order_le_neq_trans
tff(fact_3184_order__le__neq__trans,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( ( A != B )
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),B) ) ) ).

% order_le_neq_trans
tff(fact_3185_order__le__neq__trans,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( ( A != B )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% order_le_neq_trans
tff(fact_3186_order__le__neq__trans,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( ( A != B )
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% order_le_neq_trans
tff(fact_3187_order__less__imp__le,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y2) ) ).

% order_less_imp_le
tff(fact_3188_order__less__imp__le,axiom,
    ! [X: set_int,Y2: set_int] :
      ( ord_less_set_int(X,Y2)
     => ord_less_eq_set_int(X,Y2) ) ).

% order_less_imp_le
tff(fact_3189_order__less__imp__le,axiom,
    ! [X: rat,Y2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y2) ) ).

% order_less_imp_le
tff(fact_3190_order__less__imp__le,axiom,
    ! [X: num,Y2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y2)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y2) ) ).

% order_less_imp_le
tff(fact_3191_order__less__imp__le,axiom,
    ! [X: nat,Y2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y2) ) ).

% order_less_imp_le
tff(fact_3192_order__less__imp__le,axiom,
    ! [X: int,Y2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y2) ) ).

% order_less_imp_le
tff(fact_3193_linorder__not__less,axiom,
    ! [X: real,Y2: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),X) ) ).

% linorder_not_less
tff(fact_3194_linorder__not__less,axiom,
    ! [X: rat,Y2: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y2),X) ) ).

% linorder_not_less
tff(fact_3195_linorder__not__less,axiom,
    ! [X: num,Y2: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y2)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y2),X) ) ).

% linorder_not_less
tff(fact_3196_linorder__not__less,axiom,
    ! [X: nat,Y2: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y2)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y2),X) ) ).

% linorder_not_less
tff(fact_3197_linorder__not__less,axiom,
    ! [X: int,Y2: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y2),X) ) ).

% linorder_not_less
tff(fact_3198_linorder__not__le,axiom,
    ! [X: real,Y2: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),X) ) ).

% linorder_not_le
tff(fact_3199_linorder__not__le,axiom,
    ! [X: rat,Y2: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y2),X) ) ).

% linorder_not_le
tff(fact_3200_linorder__not__le,axiom,
    ! [X: num,Y2: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y2)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Y2),X) ) ).

% linorder_not_le
tff(fact_3201_linorder__not__le,axiom,
    ! [X: nat,Y2: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y2)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y2),X) ) ).

% linorder_not_le
tff(fact_3202_linorder__not__le,axiom,
    ! [X: int,Y2: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Y2),X) ) ).

% linorder_not_le
tff(fact_3203_order__less__le,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y2)
        & ( X != Y2 ) ) ) ).

% order_less_le
tff(fact_3204_order__less__le,axiom,
    ! [X: set_int,Y2: set_int] :
      ( ord_less_set_int(X,Y2)
    <=> ( ord_less_eq_set_int(X,Y2)
        & ( X != Y2 ) ) ) ).

% order_less_le
tff(fact_3205_order__less__le,axiom,
    ! [X: rat,Y2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y2)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y2)
        & ( X != Y2 ) ) ) ).

% order_less_le
tff(fact_3206_order__less__le,axiom,
    ! [X: num,Y2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y2)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y2)
        & ( X != Y2 ) ) ) ).

% order_less_le
tff(fact_3207_order__less__le,axiom,
    ! [X: nat,Y2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y2)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y2)
        & ( X != Y2 ) ) ) ).

% order_less_le
tff(fact_3208_order__less__le,axiom,
    ! [X: int,Y2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y2)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y2)
        & ( X != Y2 ) ) ) ).

% order_less_le
tff(fact_3209_order__le__less,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y2)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2)
        | ( X = Y2 ) ) ) ).

% order_le_less
tff(fact_3210_order__le__less,axiom,
    ! [X: set_int,Y2: set_int] :
      ( ord_less_eq_set_int(X,Y2)
    <=> ( ord_less_set_int(X,Y2)
        | ( X = Y2 ) ) ) ).

% order_le_less
tff(fact_3211_order__le__less,axiom,
    ! [X: rat,Y2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y2)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y2)
        | ( X = Y2 ) ) ) ).

% order_le_less
tff(fact_3212_order__le__less,axiom,
    ! [X: num,Y2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y2)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y2)
        | ( X = Y2 ) ) ) ).

% order_le_less
tff(fact_3213_order__le__less,axiom,
    ! [X: nat,Y2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y2)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y2)
        | ( X = Y2 ) ) ) ).

% order_le_less
tff(fact_3214_order__le__less,axiom,
    ! [X: int,Y2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y2)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y2)
        | ( X = Y2 ) ) ) ).

% order_le_less
tff(fact_3215_dual__order_Ostrict__implies__order,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ).

% dual_order.strict_implies_order
tff(fact_3216_dual__order_Ostrict__implies__order,axiom,
    ! [B: set_int,A: set_int] :
      ( ord_less_set_int(B,A)
     => ord_less_eq_set_int(B,A) ) ).

% dual_order.strict_implies_order
tff(fact_3217_dual__order_Ostrict__implies__order,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ).

% dual_order.strict_implies_order
tff(fact_3218_dual__order_Ostrict__implies__order,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A) ) ).

% dual_order.strict_implies_order
tff(fact_3219_dual__order_Ostrict__implies__order,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ).

% dual_order.strict_implies_order
tff(fact_3220_dual__order_Ostrict__implies__order,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ).

% dual_order.strict_implies_order
tff(fact_3221_order_Ostrict__implies__order,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% order.strict_implies_order
tff(fact_3222_order_Ostrict__implies__order,axiom,
    ! [A: set_int,B: set_int] :
      ( ord_less_set_int(A,B)
     => ord_less_eq_set_int(A,B) ) ).

% order.strict_implies_order
tff(fact_3223_order_Ostrict__implies__order,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% order.strict_implies_order
tff(fact_3224_order_Ostrict__implies__order,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B) ) ).

% order.strict_implies_order
tff(fact_3225_order_Ostrict__implies__order,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% order.strict_implies_order
tff(fact_3226_order_Ostrict__implies__order,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% order.strict_implies_order
tff(fact_3227_dual__order_Ostrict__iff__not,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
        & ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_3228_dual__order_Ostrict__iff__not,axiom,
    ! [B: set_int,A: set_int] :
      ( ord_less_set_int(B,A)
    <=> ( ord_less_eq_set_int(B,A)
        & ~ ord_less_eq_set_int(A,B) ) ) ).

% dual_order.strict_iff_not
tff(fact_3229_dual__order_Ostrict__iff__not,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
        & ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_3230_dual__order_Ostrict__iff__not,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
        & ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_3231_dual__order_Ostrict__iff__not,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
        & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_3232_dual__order_Ostrict__iff__not,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
        & ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_3233_dual__order_Ostrict__trans2,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,C),A) ) ) ).

% dual_order.strict_trans2
tff(fact_3234_dual__order_Ostrict__trans2,axiom,
    ! [B: set_int,A: set_int,C: set_int] :
      ( ord_less_set_int(B,A)
     => ( ord_less_eq_set_int(C,B)
       => ord_less_set_int(C,A) ) ) ).

% dual_order.strict_trans2
tff(fact_3235_dual__order_Ostrict__trans2,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),A) ) ) ).

% dual_order.strict_trans2
tff(fact_3236_dual__order_Ostrict__trans2,axiom,
    ! [B: num,A: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),B)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,C),A) ) ) ).

% dual_order.strict_trans2
tff(fact_3237_dual__order_Ostrict__trans2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A) ) ) ).

% dual_order.strict_trans2
tff(fact_3238_dual__order_Ostrict__trans2,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,C),A) ) ) ).

% dual_order.strict_trans2
tff(fact_3239_dual__order_Ostrict__trans1,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,C),A) ) ) ).

% dual_order.strict_trans1
tff(fact_3240_dual__order_Ostrict__trans1,axiom,
    ! [B: set_int,A: set_int,C: set_int] :
      ( ord_less_eq_set_int(B,A)
     => ( ord_less_set_int(C,B)
       => ord_less_set_int(C,A) ) ) ).

% dual_order.strict_trans1
tff(fact_3241_dual__order_Ostrict__trans1,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),A) ) ) ).

% dual_order.strict_trans1
tff(fact_3242_dual__order_Ostrict__trans1,axiom,
    ! [B: num,A: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,C),B)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,C),A) ) ) ).

% dual_order.strict_trans1
tff(fact_3243_dual__order_Ostrict__trans1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A) ) ) ).

% dual_order.strict_trans1
tff(fact_3244_dual__order_Ostrict__trans1,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,C),A) ) ) ).

% dual_order.strict_trans1
tff(fact_3245_dual__order_Ostrict__iff__order,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_3246_dual__order_Ostrict__iff__order,axiom,
    ! [B: set_int,A: set_int] :
      ( ord_less_set_int(B,A)
    <=> ( ord_less_eq_set_int(B,A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_3247_dual__order_Ostrict__iff__order,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_3248_dual__order_Ostrict__iff__order,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_3249_dual__order_Ostrict__iff__order,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_3250_dual__order_Ostrict__iff__order,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_3251_dual__order_Oorder__iff__strict,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_3252_dual__order_Oorder__iff__strict,axiom,
    ! [B: set_int,A: set_int] :
      ( ord_less_eq_set_int(B,A)
    <=> ( ord_less_set_int(B,A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_3253_dual__order_Oorder__iff__strict,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_3254_dual__order_Oorder__iff__strict,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_3255_dual__order_Oorder__iff__strict,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_3256_dual__order_Oorder__iff__strict,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_3257_dense__le__bounded,axiom,
    ! [X: real,Y2: real,Z2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2)
     => ( ! [W2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),W2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,W2),Y2)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,W2),Z2) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),Z2) ) ) ).

% dense_le_bounded
tff(fact_3258_dense__le__bounded,axiom,
    ! [X: rat,Y2: rat,Z2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y2)
     => ( ! [W2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),W2)
           => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,W2),Y2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,W2),Z2) ) )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y2),Z2) ) ) ).

% dense_le_bounded
tff(fact_3259_dense__ge__bounded,axiom,
    ! [Z2: real,X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X)
     => ( ! [W2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),W2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,W2),X)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),W2) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),Z2) ) ) ).

% dense_ge_bounded
tff(fact_3260_dense__ge__bounded,axiom,
    ! [Z2: rat,X: rat,Y2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X)
     => ( ! [W2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),W2)
           => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,W2),X)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y2),W2) ) )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y2),Z2) ) ) ).

% dense_ge_bounded
tff(fact_3261_order_Ostrict__iff__not,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
        & ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ).

% order.strict_iff_not
tff(fact_3262_order_Ostrict__iff__not,axiom,
    ! [A: set_int,B: set_int] :
      ( ord_less_set_int(A,B)
    <=> ( ord_less_eq_set_int(A,B)
        & ~ ord_less_eq_set_int(B,A) ) ) ).

% order.strict_iff_not
tff(fact_3263_order_Ostrict__iff__not,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
        & ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).

% order.strict_iff_not
tff(fact_3264_order_Ostrict__iff__not,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
        & ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A) ) ) ).

% order.strict_iff_not
tff(fact_3265_order_Ostrict__iff__not,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
        & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ) ).

% order.strict_iff_not
tff(fact_3266_order_Ostrict__iff__not,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
        & ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ).

% order.strict_iff_not
tff(fact_3267_order_Ostrict__trans2,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),C) ) ) ).

% order.strict_trans2
tff(fact_3268_order_Ostrict__trans2,axiom,
    ! [A: set_int,B: set_int,C: set_int] :
      ( ord_less_set_int(A,B)
     => ( ord_less_eq_set_int(B,C)
       => ord_less_set_int(A,C) ) ) ).

% order.strict_trans2
tff(fact_3269_order_Ostrict__trans2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C) ) ) ).

% order.strict_trans2
tff(fact_3270_order_Ostrict__trans2,axiom,
    ! [A: num,B: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),C) ) ) ).

% order.strict_trans2
tff(fact_3271_order_Ostrict__trans2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C) ) ) ).

% order.strict_trans2
tff(fact_3272_order_Ostrict__trans2,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),C) ) ) ).

% order.strict_trans2
tff(fact_3273_order_Ostrict__trans1,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),C) ) ) ).

% order.strict_trans1
tff(fact_3274_order_Ostrict__trans1,axiom,
    ! [A: set_int,B: set_int,C: set_int] :
      ( ord_less_eq_set_int(A,B)
     => ( ord_less_set_int(B,C)
       => ord_less_set_int(A,C) ) ) ).

% order.strict_trans1
tff(fact_3275_order_Ostrict__trans1,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C) ) ) ).

% order.strict_trans1
tff(fact_3276_order_Ostrict__trans1,axiom,
    ! [A: num,B: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),C) ) ) ).

% order.strict_trans1
tff(fact_3277_order_Ostrict__trans1,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C) ) ) ).

% order.strict_trans1
tff(fact_3278_order_Ostrict__trans1,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),C) ) ) ).

% order.strict_trans1
tff(fact_3279_order_Ostrict__iff__order,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_3280_order_Ostrict__iff__order,axiom,
    ! [A: set_int,B: set_int] :
      ( ord_less_set_int(A,B)
    <=> ( ord_less_eq_set_int(A,B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_3281_order_Ostrict__iff__order,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_3282_order_Ostrict__iff__order,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_3283_order_Ostrict__iff__order,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_3284_order_Ostrict__iff__order,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_3285_order_Oorder__iff__strict,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_3286_order_Oorder__iff__strict,axiom,
    ! [A: set_int,B: set_int] :
      ( ord_less_eq_set_int(A,B)
    <=> ( ord_less_set_int(A,B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_3287_order_Oorder__iff__strict,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_3288_order_Oorder__iff__strict,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_3289_order_Oorder__iff__strict,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_3290_order_Oorder__iff__strict,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_3291_not__le__imp__less,axiom,
    ! [Y2: real,X: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2) ) ).

% not_le_imp_less
tff(fact_3292_not__le__imp__less,axiom,
    ! [Y2: rat,X: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y2),X)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y2) ) ).

% not_le_imp_less
tff(fact_3293_not__le__imp__less,axiom,
    ! [Y2: num,X: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y2),X)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y2) ) ).

% not_le_imp_less
tff(fact_3294_not__le__imp__less,axiom,
    ! [Y2: nat,X: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y2),X)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y2) ) ).

% not_le_imp_less
tff(fact_3295_not__le__imp__less,axiom,
    ! [Y2: int,X: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y2),X)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y2) ) ).

% not_le_imp_less
tff(fact_3296_less__le__not__le,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y2)
        & ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),X) ) ) ).

% less_le_not_le
tff(fact_3297_less__le__not__le,axiom,
    ! [X: set_int,Y2: set_int] :
      ( ord_less_set_int(X,Y2)
    <=> ( ord_less_eq_set_int(X,Y2)
        & ~ ord_less_eq_set_int(Y2,X) ) ) ).

% less_le_not_le
tff(fact_3298_less__le__not__le,axiom,
    ! [X: rat,Y2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y2)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y2)
        & ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y2),X) ) ) ).

% less_le_not_le
tff(fact_3299_less__le__not__le,axiom,
    ! [X: num,Y2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y2)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y2)
        & ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y2),X) ) ) ).

% less_le_not_le
tff(fact_3300_less__le__not__le,axiom,
    ! [X: nat,Y2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y2)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y2)
        & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y2),X) ) ) ).

% less_le_not_le
tff(fact_3301_less__le__not__le,axiom,
    ! [X: int,Y2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y2)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y2)
        & ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y2),X) ) ) ).

% less_le_not_le
tff(fact_3302_dense__le,axiom,
    ! [Y2: real,Z2: real] :
      ( ! [X4: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Y2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),Z2) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),Z2) ) ).

% dense_le
tff(fact_3303_dense__le,axiom,
    ! [Y2: rat,Z2: rat] :
      ( ! [X4: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Y2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Z2) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y2),Z2) ) ).

% dense_le
tff(fact_3304_dense__ge,axiom,
    ! [Z2: real,Y2: real] :
      ( ! [X4: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X4)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),X4) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),Z2) ) ).

% dense_ge
tff(fact_3305_dense__ge,axiom,
    ! [Z2: rat,Y2: rat] :
      ( ! [X4: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X4)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y2),X4) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y2),Z2) ) ).

% dense_ge
tff(fact_3306_antisym__conv2,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y2)
     => ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2)
      <=> ( X = Y2 ) ) ) ).

% antisym_conv2
tff(fact_3307_antisym__conv2,axiom,
    ! [X: set_int,Y2: set_int] :
      ( ord_less_eq_set_int(X,Y2)
     => ( ~ ord_less_set_int(X,Y2)
      <=> ( X = Y2 ) ) ) ).

% antisym_conv2
tff(fact_3308_antisym__conv2,axiom,
    ! [X: rat,Y2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y2)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y2)
      <=> ( X = Y2 ) ) ) ).

% antisym_conv2
tff(fact_3309_antisym__conv2,axiom,
    ! [X: num,Y2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y2)
     => ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y2)
      <=> ( X = Y2 ) ) ) ).

% antisym_conv2
tff(fact_3310_antisym__conv2,axiom,
    ! [X: nat,Y2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y2)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y2)
      <=> ( X = Y2 ) ) ) ).

% antisym_conv2
tff(fact_3311_antisym__conv2,axiom,
    ! [X: int,Y2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y2)
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y2)
      <=> ( X = Y2 ) ) ) ).

% antisym_conv2
tff(fact_3312_antisym__conv1,axiom,
    ! [X: real,Y2: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y2)
      <=> ( X = Y2 ) ) ) ).

% antisym_conv1
tff(fact_3313_antisym__conv1,axiom,
    ! [X: set_int,Y2: set_int] :
      ( ~ ord_less_set_int(X,Y2)
     => ( ord_less_eq_set_int(X,Y2)
      <=> ( X = Y2 ) ) ) ).

% antisym_conv1
tff(fact_3314_antisym__conv1,axiom,
    ! [X: rat,Y2: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y2)
      <=> ( X = Y2 ) ) ) ).

% antisym_conv1
tff(fact_3315_antisym__conv1,axiom,
    ! [X: num,Y2: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y2)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y2)
      <=> ( X = Y2 ) ) ) ).

% antisym_conv1
tff(fact_3316_antisym__conv1,axiom,
    ! [X: nat,Y2: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y2)
      <=> ( X = Y2 ) ) ) ).

% antisym_conv1
tff(fact_3317_antisym__conv1,axiom,
    ! [X: int,Y2: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y2)
      <=> ( X = Y2 ) ) ) ).

% antisym_conv1
tff(fact_3318_nless__le,axiom,
    ! [A: real,B: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
    <=> ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_3319_nless__le,axiom,
    ! [A: set_int,B: set_int] :
      ( ~ ord_less_set_int(A,B)
    <=> ( ~ ord_less_eq_set_int(A,B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_3320_nless__le,axiom,
    ! [A: rat,B: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
    <=> ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_3321_nless__le,axiom,
    ! [A: num,B: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
    <=> ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_3322_nless__le,axiom,
    ! [A: nat,B: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_3323_nless__le,axiom,
    ! [A: int,B: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
    <=> ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_3324_leI,axiom,
    ! [X: real,Y2: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),X) ) ).

% leI
tff(fact_3325_leI,axiom,
    ! [X: rat,Y2: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y2),X) ) ).

% leI
tff(fact_3326_leI,axiom,
    ! [X: num,Y2: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y2)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y2),X) ) ).

% leI
tff(fact_3327_leI,axiom,
    ! [X: nat,Y2: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y2),X) ) ).

% leI
tff(fact_3328_leI,axiom,
    ! [X: int,Y2: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y2),X) ) ).

% leI
tff(fact_3329_leD,axiom,
    ! [Y2: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),X)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2) ) ).

% leD
tff(fact_3330_leD,axiom,
    ! [Y2: set_int,X: set_int] :
      ( ord_less_eq_set_int(Y2,X)
     => ~ ord_less_set_int(X,Y2) ) ).

% leD
tff(fact_3331_leD,axiom,
    ! [Y2: rat,X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y2),X)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y2) ) ).

% leD
tff(fact_3332_leD,axiom,
    ! [Y2: num,X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y2),X)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y2) ) ).

% leD
tff(fact_3333_leD,axiom,
    ! [Y2: nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y2),X)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y2) ) ).

% leD
tff(fact_3334_leD,axiom,
    ! [Y2: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y2),X)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y2) ) ).

% leD
tff(fact_3335_Euclid__induct,axiom,
    ! [P: fun_nat_fun_nat_o,A: nat,B: nat] :
      ( ! [A3: nat,B2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(P,A3),B2)
        <=> aa_nat_o(aa_nat_fun_nat_o(P,B2),A3) )
     => ( ! [A3: nat] : aa_nat_o(aa_nat_fun_nat_o(P,A3),zero_zero_nat)
       => ( ! [A3: nat,B2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(P,A3),B2)
             => aa_nat_o(aa_nat_fun_nat_o(P,A3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A3),B2)) )
         => aa_nat_o(aa_nat_fun_nat_o(P,A),B) ) ) ) ).

% Euclid_induct
tff(fact_3336_VEBT__internal_Omembermima_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv2: $o,Uw: nat] : X != produc738532404422230701BT_nat(aa_o_VEBT_VEBT(vEBT_Leaf((Uu2)),(Uv2)),Uw)
     => ( ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT,Uz2: nat] : X != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux),Uy),Uz2)
       => ( ! [Mi3: nat,Ma3: nat,Va2: list_VEBT_VEBT,Vb: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi3),Ma3)),zero_zero_nat,Va2),Vb),X4)
         => ( ! [Mi3: nat,Ma3: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi3),Ma3)),aa_nat_nat(suc,V2),TreeList),Vc),X4)
           => ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd: vEBT_VEBT,X4: nat] : X != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList),Vd),X4) ) ) ) ) ).

% VEBT_internal.membermima.cases
tff(fact_3337_mult__less__iff1,axiom,
    ! [Z2: real,X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(X),Z2)),aa_real_real(times_times_real(Y2),Z2))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2) ) ) ).

% mult_less_iff1
tff(fact_3338_mult__less__iff1,axiom,
    ! [Z2: rat,X: rat,Y2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(X),Z2)),aa_rat_rat(times_times_rat(Y2),Z2))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y2) ) ) ).

% mult_less_iff1
tff(fact_3339_mult__less__iff1,axiom,
    ! [Z2: int,X: int,Y2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),Z2)),aa_int_int(aa_int_fun_int_int(times_times_int,Y2),Z2))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y2) ) ) ).

% mult_less_iff1
tff(fact_3340_pos__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q: int,R: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
     => ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q),R))
       => aa_Pro4592068669816069595_int_o(eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),R)))) ) ) ).

% pos_eucl_rel_int_mult_2
tff(fact_3341_VEBT__internal_Oelim__dead_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa: extended_enat,Y2: vEBT_VEBT] :
      ( ( vEBT_VEBT_elim_dead(X,Xa) = Y2 )
     => ( accp_P6183159247885693666d_enat(vEBT_V312737461966249ad_rel,produc581526299967858633d_enat(X,Xa))
       => ( ! [A3: $o,B2: $o] :
              ( ( X = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B2)) )
             => ( ( Y2 = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B2)) )
               => ~ accp_P6183159247885693666d_enat(vEBT_V312737461966249ad_rel,produc581526299967858633d_enat(aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B2)),Xa)) ) )
         => ( ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList),Summary) )
               => ( ( Xa = extend5688581933313929465d_enat )
                 => ( ( Y2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,map_VE8901447254227204932T_VEBT(aTP_Lamm_aa(Deg),TreeList)),vEBT_VEBT_elim_dead(Summary,extend5688581933313929465d_enat)) )
                   => ~ accp_P6183159247885693666d_enat(vEBT_V312737461966249ad_rel,produc581526299967858633d_enat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList),Summary),extend5688581933313929465d_enat)) ) ) )
           => ~ ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                  ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList),Summary) )
                 => ! [L2: nat] :
                      ( ( Xa = aa_nat_Extended_enat(extended_enat2,L2) )
                     => ( ( Y2 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,take_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,L2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),map_VE8901447254227204932T_VEBT(aTP_Lamm_aa(Deg),TreeList))),vEBT_VEBT_elim_dead(Summary,aa_nat_Extended_enat(extended_enat2,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,L2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))) )
                       => ~ accp_P6183159247885693666d_enat(vEBT_V312737461966249ad_rel,produc581526299967858633d_enat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeList),Summary),aa_nat_Extended_enat(extended_enat2,L2))) ) ) ) ) ) ) ) ).

% VEBT_internal.elim_dead.pelims
tff(fact_3342_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y2: $o] :
      ( ( aa_nat_o(vEBT_VEBT_membermima(X),Xa)
      <=> (Y2) )
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),produc738532404422230701BT_nat(X,Xa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X = aa_o_VEBT_VEBT(vEBT_Leaf((Uu2)),(Uv2)) )
             => ( ~ (Y2)
               => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),produc738532404422230701BT_nat(aa_o_VEBT_VEBT(vEBT_Leaf((Uu2)),(Uv2)),Xa)) ) )
         => ( ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
                ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux),Uy) )
               => ( ~ (Y2)
                 => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux),Uy),Xa)) ) )
           => ( ! [Mi3: nat,Ma3: nat,Va2: list_VEBT_VEBT,Vb: vEBT_VEBT] :
                  ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi3),Ma3)),zero_zero_nat,Va2),Vb) )
                 => ( ( (Y2)
                    <=> ( ( Xa = Mi3 )
                        | ( Xa = Ma3 ) ) )
                   => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi3),Ma3)),zero_zero_nat,Va2),Vb),Xa)) ) )
             => ( ! [Mi3: nat,Ma3: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                    ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi3),Ma3)),aa_nat_nat(suc,V2),TreeList),Vc) )
                   => ( ( (Y2)
                      <=> ( ( Xa = Mi3 )
                          | ( Xa = Ma3 )
                          | $let(
                              pos: nat,
                              pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) )
                     => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi3),Ma3)),aa_nat_nat(suc,V2),TreeList),Vc),Xa)) ) )
               => ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                      ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList),Vd) )
                     => ( ( (Y2)
                        <=> $let(
                              pos: nat,
                              pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) )
                       => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList),Vd),Xa)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
tff(fact_3343_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ~ aa_nat_o(vEBT_VEBT_membermima(X),Xa)
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),produc738532404422230701BT_nat(X,Xa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X = aa_o_VEBT_VEBT(vEBT_Leaf((Uu2)),(Uv2)) )
             => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),produc738532404422230701BT_nat(aa_o_VEBT_VEBT(vEBT_Leaf((Uu2)),(Uv2)),Xa)) )
         => ( ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
                ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux),Uy) )
               => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux),Uy),Xa)) )
           => ( ! [Mi3: nat,Ma3: nat,Va2: list_VEBT_VEBT,Vb: vEBT_VEBT] :
                  ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi3),Ma3)),zero_zero_nat,Va2),Vb) )
                 => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi3),Ma3)),zero_zero_nat,Va2),Vb),Xa))
                   => ( ( Xa = Mi3 )
                      | ( Xa = Ma3 ) ) ) )
             => ( ! [Mi3: nat,Ma3: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                    ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi3),Ma3)),aa_nat_nat(suc,V2),TreeList),Vc) )
                   => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi3),Ma3)),aa_nat_nat(suc,V2),TreeList),Vc),Xa))
                     => ( ( Xa = Mi3 )
                        | ( Xa = Ma3 )
                        | $let(
                            pos: nat,
                            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) )
               => ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                      ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList),Vd) )
                     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList),Vd),Xa))
                       => $let(
                            pos: nat,
                            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
tff(fact_3344_option_Osize__gen_I2_J,axiom,
    ! [X: fun_Pr2602117396471556196at_nat,X23: product_prod_nat_nat] : size_o8335143837870341156at_nat(X,some_P7363390416028606310at_nat(X23)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Pro4243173770044397973at_nat(X,X23)),aa_nat_nat(suc,zero_zero_nat)) ).

% option.size_gen(2)
tff(fact_3345_option_Osize__gen_I2_J,axiom,
    ! [X: fun_num_nat,X23: num] : size_option_num(X,some_num(X23)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(X,X23)),aa_nat_nat(suc,zero_zero_nat)) ).

% option.size_gen(2)
tff(fact_3346_even__succ__mod__exp,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ) ) ) ).

% even_succ_mod_exp
tff(fact_3347_even__succ__mod__exp,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))) ) ) ) ).

% even_succ_mod_exp
tff(fact_3348_even__succ__mod__exp,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( modulo364778990260209775nteger(plus_p5714425477246183910nteger(one_one_Code_integer,A),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)) = plus_p5714425477246183910nteger(one_one_Code_integer,modulo364778990260209775nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb))) ) ) ) ).

% even_succ_mod_exp
tff(fact_3349_even__succ__div__exp,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(one_one_Code_integer,A),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)) = divide6298287555418463151nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)) ) ) ) ).

% even_succ_div_exp
tff(fact_3350_even__succ__div__exp,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ) ) ) ).

% even_succ_div_exp
tff(fact_3351_even__succ__div__exp,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ) ) ) ).

% even_succ_div_exp
tff(fact_3352_signed__take__bit__Suc,axiom,
    ! [Nb: nat,A: code_integer] : bit_ri6519982836138164636nteger(aa_nat_nat(suc,Nb),A) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_ri6519982836138164636nteger(Nb,divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))))) ).

% signed_take_bit_Suc
tff(fact_3353_signed__take__bit__Suc,axiom,
    ! [Nb: nat,A: int] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Nb)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% signed_take_bit_Suc
tff(fact_3354_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( aa_nat_o(vEBT_VEBT_membermima(X),Xa)
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),produc738532404422230701BT_nat(X,Xa))
       => ( ! [Mi3: nat,Ma3: nat,Va2: list_VEBT_VEBT,Vb: vEBT_VEBT] :
              ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi3),Ma3)),zero_zero_nat,Va2),Vb) )
             => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi3),Ma3)),zero_zero_nat,Va2),Vb),Xa))
               => ~ ( ( Xa = Mi3 )
                    | ( Xa = Ma3 ) ) ) )
         => ( ! [Mi3: nat,Ma3: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi3),Ma3)),aa_nat_nat(suc,V2),TreeList),Vc) )
               => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Mi3),Ma3)),aa_nat_nat(suc,V2),TreeList),Vc),Xa))
                 => ~ ( ( Xa = Mi3 )
                      | ( Xa = Ma3 )
                      | $let(
                          pos: nat,
                          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) )
           => ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                  ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList),Vd) )
                 => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList),Vd),Xa))
                   => ~ $let(
                          pos: nat,
                          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
tff(fact_3355_nat__dvd__1__iff__1,axiom,
    ! [Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ma),one_one_nat)
    <=> ( Ma = one_one_nat ) ) ).

% nat_dvd_1_iff_1
tff(fact_3356_dvd__add__triv__left__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(A,B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ).

% dvd_add_triv_left_iff
tff(fact_3357_dvd__add__triv__left__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(plus_plus_real(A),B))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ).

% dvd_add_triv_left_iff
tff(fact_3358_dvd__add__triv__left__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(plus_plus_rat(A),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ).

% dvd_add_triv_left_iff
tff(fact_3359_dvd__add__triv__left__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% dvd_add_triv_left_iff
tff(fact_3360_dvd__add__triv__left__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% dvd_add_triv_left_iff
tff(fact_3361_dvd__add__triv__right__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(B,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ).

% dvd_add_triv_right_iff
tff(fact_3362_dvd__add__triv__right__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(plus_plus_real(B),A))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ).

% dvd_add_triv_right_iff
tff(fact_3363_dvd__add__triv__right__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(plus_plus_rat(B),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ).

% dvd_add_triv_right_iff
tff(fact_3364_dvd__add__triv__right__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% dvd_add_triv_right_iff
tff(fact_3365_dvd__add__triv__right__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% dvd_add_triv_right_iff
tff(fact_3366_dvd__1__iff__1,axiom,
    ! [Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ma),aa_nat_nat(suc,zero_zero_nat))
    <=> ( Ma = aa_nat_nat(suc,zero_zero_nat) ) ) ).

% dvd_1_iff_1
tff(fact_3367_dvd__1__left,axiom,
    ! [K2: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(suc,zero_zero_nat)),K2) ).

% dvd_1_left
tff(fact_3368_div__dvd__div,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,divide6298287555418463151nteger(B,A)),divide6298287555418463151nteger(C,A))
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C) ) ) ) ).

% div_dvd_div
tff(fact_3369_div__dvd__div,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),A))
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ) ) ).

% div_dvd_div
tff(fact_3370_div__dvd__div,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A)),aa_int_int(aa_int_fun_int_int(divide_divide_int,C),A))
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ) ) ).

% div_dvd_div
tff(fact_3371_nat__mult__dvd__cancel__disj,axiom,
    ! [K2: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Nb))
    <=> ( ( K2 = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ma),Nb) ) ) ).

% nat_mult_dvd_cancel_disj
tff(fact_3372_signed__take__bit__of__0,axiom,
    ! [Nb: nat] : aa_int_int(bit_ri631733984087533419it_int(Nb),zero_zero_int) = zero_zero_int ).

% signed_take_bit_of_0
tff(fact_3373_dvd__mult__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
    <=> ( ( C = zero_z3403309356797280102nteger )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_3374_dvd__mult__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( aa_complex_o(dvd_dvd_complex(aa_complex_complex(times_times_complex(C),A)),aa_complex_complex(times_times_complex(C),B))
    <=> ( ( C = zero_zero_complex )
        | aa_complex_o(dvd_dvd_complex(A),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_3375_dvd__mult__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B))
    <=> ( ( C = zero_zero_real )
        | aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_3376_dvd__mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B))
    <=> ( ( C = zero_zero_rat )
        | aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_3377_dvd__mult__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
    <=> ( ( C = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_3378_dvd__mult__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))
    <=> ( ( C = zero_z3403309356797280102nteger )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_3379_dvd__mult__cancel__right,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( aa_complex_o(dvd_dvd_complex(aa_complex_complex(times_times_complex(A),C)),aa_complex_complex(times_times_complex(B),C))
    <=> ( ( C = zero_zero_complex )
        | aa_complex_o(dvd_dvd_complex(A),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_3380_dvd__mult__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),C))
    <=> ( ( C = zero_zero_real )
        | aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_3381_dvd__mult__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C))
    <=> ( ( C = zero_zero_rat )
        | aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_3382_dvd__mult__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
    <=> ( ( C = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_3383_dvd__times__left__cancel__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_3384_dvd__times__left__cancel__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_3385_dvd__times__left__cancel__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_3386_dvd__times__right__cancel__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_3387_dvd__times__right__cancel__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_3388_dvd__times__right__cancel__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),A))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_3389_unit__prod,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),one_one_Code_integer) ) ) ).

% unit_prod
tff(fact_3390_unit__prod,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),one_one_nat) ) ) ).

% unit_prod
tff(fact_3391_unit__prod,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),one_one_int) ) ) ).

% unit_prod
tff(fact_3392_dvd__add__times__triv__left__iff,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A),B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_3393_dvd__add__times__triv__left__iff,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(plus_plus_real(aa_real_real(times_times_real(C),A)),B))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_3394_dvd__add__times__triv__left__iff,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(C),A)),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_3395_dvd__add__times__triv__left__iff,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_3396_dvd__add__times__triv__left__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),B))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_3397_dvd__add__times__triv__right__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(B,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_3398_dvd__add__times__triv__right__iff,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(plus_plus_real(B),aa_real_real(times_times_real(C),A)))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_3399_dvd__add__times__triv__right__iff,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(plus_plus_rat(B),aa_rat_rat(times_times_rat(C),A)))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_3400_dvd__add__times__triv__right__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_3401_dvd__add__times__triv__right__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(times_times_int,C),A)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_3402_dvd__div__mult__self,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(B,A)),A) = B ) ) ).

% dvd_div_mult_self
tff(fact_3403_dvd__div__mult__self,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A)),A) = B ) ) ).

% dvd_div_mult_self
tff(fact_3404_dvd__div__mult__self,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A)),A) = B ) ) ).

% dvd_div_mult_self
tff(fact_3405_dvd__mult__div__cancel,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),divide6298287555418463151nteger(B,A)) = B ) ) ).

% dvd_mult_div_cancel
tff(fact_3406_dvd__mult__div__cancel,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A)) = B ) ) ).

% dvd_mult_div_cancel
tff(fact_3407_dvd__mult__div__cancel,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A)) = B ) ) ).

% dvd_mult_div_cancel
tff(fact_3408_unit__div,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,divide6298287555418463151nteger(A,B)),one_one_Code_integer) ) ) ).

% unit_div
tff(fact_3409_unit__div,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),one_one_nat) ) ) ).

% unit_div
tff(fact_3410_unit__div,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),one_one_int) ) ) ).

% unit_div
tff(fact_3411_unit__div__1__unit,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,divide6298287555418463151nteger(one_one_Code_integer,A)),one_one_Code_integer) ) ).

% unit_div_1_unit
tff(fact_3412_unit__div__1__unit,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A)),one_one_nat) ) ).

% unit_div_1_unit
tff(fact_3413_unit__div__1__unit,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),A)),one_one_int) ) ).

% unit_div_1_unit
tff(fact_3414_unit__div__1__div__1,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( divide6298287555418463151nteger(one_one_Code_integer,divide6298287555418463151nteger(one_one_Code_integer,A)) = A ) ) ).

% unit_div_1_div_1
tff(fact_3415_unit__div__1__div__1,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A)) = A ) ) ).

% unit_div_1_div_1
tff(fact_3416_unit__div__1__div__1,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),A)) = A ) ) ).

% unit_div_1_div_1
tff(fact_3417_div__add,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
       => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,B),C) = plus_p5714425477246183910nteger(divide6298287555418463151nteger(A,C),divide6298287555418463151nteger(B,C)) ) ) ) ).

% div_add
tff(fact_3418_div__add,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) ) ) ) ).

% div_add
tff(fact_3419_div__add,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) ) ) ) ).

% div_add
tff(fact_3420_dvd__imp__mod__0,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),A) = zero_zero_nat ) ) ).

% dvd_imp_mod_0
tff(fact_3421_dvd__imp__mod__0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),A) = zero_zero_int ) ) ).

% dvd_imp_mod_0
tff(fact_3422_dvd__imp__mod__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( modulo364778990260209775nteger(B,A) = zero_z3403309356797280102nteger ) ) ).

% dvd_imp_mod_0
tff(fact_3423_signed__take__bit__Suc__1,axiom,
    ! [Nb: nat] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Nb)),one_one_int) = one_one_int ).

% signed_take_bit_Suc_1
tff(fact_3424_signed__take__bit__numeral__of__1,axiom,
    ! [K2: num] : aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(numeral_numeral_nat,K2)),one_one_int) = one_one_int ).

% signed_take_bit_numeral_of_1
tff(fact_3425_unit__mult__div__div,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(one_one_Code_integer,A)) = divide6298287555418463151nteger(B,A) ) ) ).

% unit_mult_div_div
tff(fact_3426_unit__mult__div__div,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) ) ) ).

% unit_mult_div_div
tff(fact_3427_unit__mult__div__div,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),A)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A) ) ) ).

% unit_mult_div_div
tff(fact_3428_unit__div__mult__self,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(B,A)),A) = B ) ) ).

% unit_div_mult_self
tff(fact_3429_unit__div__mult__self,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A)),A) = B ) ) ).

% unit_div_mult_self
tff(fact_3430_unit__div__mult__self,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A)),A) = B ) ) ).

% unit_div_mult_self
tff(fact_3431_even__Suc,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,Nb))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ) ).

% even_Suc
tff(fact_3432_even__Suc__Suc__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Nb)))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ) ).

% even_Suc_Suc_iff
tff(fact_3433_pow__divides__pow__iff,axiom,
    ! [Nb: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ) ).

% pow_divides_pow_iff
tff(fact_3434_pow__divides__pow__iff,axiom,
    ! [Nb: nat,A: int,B: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Nb))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ) ).

% pow_divides_pow_iff
tff(fact_3435_even__mult__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),B) ) ) ).

% even_mult_iff
tff(fact_3436_even__mult__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B) ) ) ).

% even_mult_iff
tff(fact_3437_even__mult__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B) ) ) ).

% even_mult_iff
tff(fact_3438_odd__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,B))
    <=> ~ ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
        <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),B) ) ) ).

% odd_add
tff(fact_3439_odd__add,axiom,
    ! [A: nat,B: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
    <=> ~ ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B) ) ) ).

% odd_add
tff(fact_3440_odd__add,axiom,
    ! [A: int,B: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))
    <=> ~ ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B) ) ) ).

% odd_add
tff(fact_3441_even__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,B))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),B) ) ) ).

% even_add
tff(fact_3442_even__add,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B) ) ) ).

% even_add
tff(fact_3443_even__add,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B) ) ) ).

% even_add
tff(fact_3444_even__mod__2__iff,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A) ) ).

% even_mod_2_iff
tff(fact_3445_even__mod__2__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) ) ).

% even_mod_2_iff
tff(fact_3446_even__mod__2__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A) ) ).

% even_mod_2_iff
tff(fact_3447_odd__Suc__div__two,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% odd_Suc_div_two
tff(fact_3448_even__Suc__div__two,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% even_Suc_div_two
tff(fact_3449_signed__take__bit__Suc__bit0,axiom,
    ! [Nb: nat,K2: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Nb)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,K2))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(Nb),aa_num_int(numeral_numeral_int,K2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% signed_take_bit_Suc_bit0
tff(fact_3450_zero__le__power__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,W)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_3451_zero__le__power__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_3452_zero__le__power__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,W)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_3453_power__less__zero__eq,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),zero_zero_real)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ) ).

% power_less_zero_eq
tff(fact_3454_power__less__zero__eq,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Nb)),zero_zero_rat)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ) ).

% power_less_zero_eq
tff(fact_3455_power__less__zero__eq,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)),zero_zero_int)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ) ).

% power_less_zero_eq
tff(fact_3456_power__less__zero__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_real)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        & aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ) ).

% power_less_zero_eq_numeral
tff(fact_3457_power__less__zero__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_rat)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ) ).

% power_less_zero_eq_numeral
tff(fact_3458_power__less__zero__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_int)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        & aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ) ).

% power_less_zero_eq_numeral
tff(fact_3459_even__plus__one__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,one_one_Code_integer))
    <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A) ) ).

% even_plus_one_iff
tff(fact_3460_even__plus__one__iff,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A) ) ).

% even_plus_one_iff
tff(fact_3461_even__plus__one__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int))
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) ) ).

% even_plus_one_iff
tff(fact_3462_zero__less__power__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,W)))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & ( A != zero_zero_real ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_3463_zero__less__power__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W)))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & ( A != zero_zero_rat ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_3464_zero__less__power__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,W)))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & ( A != zero_zero_int ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_3465_even__succ__div__2,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(one_one_Code_integer,A),numera6620942414471956472nteger(aa_num_num(bit0,one))) = divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) ) ) ).

% even_succ_div_2
tff(fact_3466_even__succ__div__2,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_2
tff(fact_3467_even__succ__div__2,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_2
tff(fact_3468_odd__succ__div__two,axiom,
    ! [A: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,one_one_Code_integer),numera6620942414471956472nteger(aa_num_num(bit0,one))) = plus_p5714425477246183910nteger(divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))),one_one_Code_integer) ) ) ).

% odd_succ_div_two
tff(fact_3469_odd__succ__div__two,axiom,
    ! [A: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat) ) ) ).

% odd_succ_div_two
tff(fact_3470_odd__succ__div__two,axiom,
    ! [A: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ) ) ).

% odd_succ_div_two
tff(fact_3471_even__succ__div__two,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,one_one_Code_integer),numera6620942414471956472nteger(aa_num_num(bit0,one))) = divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) ) ) ).

% even_succ_div_two
tff(fact_3472_even__succ__div__two,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_two
tff(fact_3473_even__succ__div__two,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_two
tff(fact_3474_even__power,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% even_power
tff(fact_3475_even__power,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% even_power
tff(fact_3476_even__power,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% even_power
tff(fact_3477_odd__two__times__div__two__succ,axiom,
    ! [A: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
     => ( plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))),one_one_Code_integer) = A ) ) ).

% odd_two_times_div_two_succ
tff(fact_3478_odd__two__times__div__two__succ,axiom,
    ! [A: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),one_one_nat) = A ) ) ).

% odd_two_times_div_two_succ
tff(fact_3479_odd__two__times__div__two__succ,axiom,
    ! [A: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),one_one_int) = A ) ) ).

% odd_two_times_div_two_succ
tff(fact_3480_power__le__zero__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_real)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W))
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
            & ( A = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_3481_power__le__zero__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_rat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W))
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_3482_power__le__zero__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_int)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W))
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_3483_dvd__field__iff,axiom,
    ! [A: complex,B: complex] :
      ( aa_complex_o(dvd_dvd_complex(A),B)
    <=> ( ( A = zero_zero_complex )
       => ( B = zero_zero_complex ) ) ) ).

% dvd_field_iff
tff(fact_3484_dvd__field__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B)
    <=> ( ( A = zero_zero_real )
       => ( B = zero_zero_real ) ) ) ).

% dvd_field_iff
tff(fact_3485_dvd__field__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B)
    <=> ( ( A = zero_zero_rat )
       => ( B = zero_zero_rat ) ) ) ).

% dvd_field_iff
tff(fact_3486_dvdE,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
     => ~ ! [K: code_integer] : A != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),K) ) ).

% dvdE
tff(fact_3487_dvdE,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A)
     => ~ ! [K: real] : A != aa_real_real(times_times_real(B),K) ) ).

% dvdE
tff(fact_3488_dvdE,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A)
     => ~ ! [K: rat] : A != aa_rat_rat(times_times_rat(B),K) ) ).

% dvdE
tff(fact_3489_dvdE,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
     => ~ ! [K: nat] : A != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K) ) ).

% dvdE
tff(fact_3490_dvdE,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => ~ ! [K: int] : A != aa_int_int(aa_int_fun_int_int(times_times_int,B),K) ) ).

% dvdE
tff(fact_3491_dvdI,axiom,
    ! [A: code_integer,B: code_integer,K2: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),K2) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A) ) ).

% dvdI
tff(fact_3492_dvdI,axiom,
    ! [A: real,B: real,K2: real] :
      ( ( A = aa_real_real(times_times_real(B),K2) )
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A) ) ).

% dvdI
tff(fact_3493_dvdI,axiom,
    ! [A: rat,B: rat,K2: rat] :
      ( ( A = aa_rat_rat(times_times_rat(B),K2) )
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A) ) ).

% dvdI
tff(fact_3494_dvdI,axiom,
    ! [A: nat,B: nat,K2: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K2) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A) ) ).

% dvdI
tff(fact_3495_dvdI,axiom,
    ! [A: int,B: int,K2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(times_times_int,B),K2) )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A) ) ).

% dvdI
tff(fact_3496_dvd__def,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
    <=> ? [K3: code_integer] : A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),K3) ) ).

% dvd_def
tff(fact_3497_dvd__def,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A)
    <=> ? [K3: real] : A = aa_real_real(times_times_real(B),K3) ) ).

% dvd_def
tff(fact_3498_dvd__def,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A)
    <=> ? [K3: rat] : A = aa_rat_rat(times_times_rat(B),K3) ) ).

% dvd_def
tff(fact_3499_dvd__def,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
    <=> ? [K3: nat] : A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K3) ) ).

% dvd_def
tff(fact_3500_dvd__def,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
    <=> ? [K3: int] : A = aa_int_int(aa_int_fun_int_int(times_times_int,B),K3) ) ).

% dvd_def
tff(fact_3501_dvd__mult,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ).

% dvd_mult
tff(fact_3502_dvd__mult,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),C)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(times_times_real(B),C)) ) ).

% dvd_mult
tff(fact_3503_dvd__mult,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(times_times_rat(B),C)) ) ).

% dvd_mult
tff(fact_3504_dvd__mult,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ).

% dvd_mult
tff(fact_3505_dvd__mult,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ).

% dvd_mult
tff(fact_3506_dvd__mult2,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ).

% dvd_mult2
tff(fact_3507_dvd__mult2,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(times_times_real(B),C)) ) ).

% dvd_mult2
tff(fact_3508_dvd__mult2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(times_times_rat(B),C)) ) ).

% dvd_mult2
tff(fact_3509_dvd__mult2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ).

% dvd_mult2
tff(fact_3510_dvd__mult2,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ).

% dvd_mult2
tff(fact_3511_dvd__mult__left,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ).

% dvd_mult_left
tff(fact_3512_dvd__mult__left,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(times_times_real(A),B)),C)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),C) ) ).

% dvd_mult_left
tff(fact_3513_dvd__mult__left,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(A),B)),C)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C) ) ).

% dvd_mult_left
tff(fact_3514_dvd__mult__left,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ).

% dvd_mult_left
tff(fact_3515_dvd__mult__left,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ).

% dvd_mult_left
tff(fact_3516_dvd__triv__left,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) ).

% dvd_triv_left
tff(fact_3517_dvd__triv__left,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(times_times_real(A),B)) ).

% dvd_triv_left
tff(fact_3518_dvd__triv__left,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(times_times_rat(A),B)) ).

% dvd_triv_left
tff(fact_3519_dvd__triv__left,axiom,
    ! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) ).

% dvd_triv_left
tff(fact_3520_dvd__triv__left,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ).

% dvd_triv_left
tff(fact_3521_mult__dvd__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),D)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D)) ) ) ).

% mult_dvd_mono
tff(fact_3522_mult__dvd__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,C),D)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),D)) ) ) ).

% mult_dvd_mono
tff(fact_3523_mult__dvd__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C),D)
       => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D)) ) ) ).

% mult_dvd_mono
tff(fact_3524_mult__dvd__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),D)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ).

% mult_dvd_mono
tff(fact_3525_mult__dvd__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),D)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ).

% mult_dvd_mono
tff(fact_3526_dvd__mult__right,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C) ) ).

% dvd_mult_right
tff(fact_3527_dvd__mult__right,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(times_times_real(A),B)),C)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),C) ) ).

% dvd_mult_right
tff(fact_3528_dvd__mult__right,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(A),B)),C)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),C) ) ).

% dvd_mult_right
tff(fact_3529_dvd__mult__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ).

% dvd_mult_right
tff(fact_3530_dvd__mult__right,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ).

% dvd_mult_right
tff(fact_3531_dvd__triv__right,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)) ).

% dvd_triv_right
tff(fact_3532_dvd__triv__right,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(times_times_real(B),A)) ).

% dvd_triv_right
tff(fact_3533_dvd__triv__right,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(times_times_rat(B),A)) ).

% dvd_triv_right
tff(fact_3534_dvd__triv__right,axiom,
    ! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)) ).

% dvd_triv_right
tff(fact_3535_dvd__triv__right,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),A)) ).

% dvd_triv_right
tff(fact_3536_division__decomp,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))
     => ? [B6: nat,C5: nat] :
          ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B6),C5) )
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B6),B)
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C5),C) ) ) ).

% division_decomp
tff(fact_3537_division__decomp,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
     => ? [B6: int,C5: int] :
          ( ( A = aa_int_int(aa_int_fun_int_int(times_times_int,B6),C5) )
          & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B6),B)
          & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C5),C) ) ) ).

% division_decomp
tff(fact_3538_dvd__productE,axiom,
    ! [P2: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,P2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B))
     => ~ ! [X4: nat,Y4: nat] :
            ( ( P2 = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X4),Y4) )
           => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,X4),A)
             => ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Y4),B) ) ) ) ).

% dvd_productE
tff(fact_3539_dvd__productE,axiom,
    ! [P2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,P2),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))
     => ~ ! [X4: int,Y4: int] :
            ( ( P2 = aa_int_int(aa_int_fun_int_int(times_times_int,X4),Y4) )
           => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X4),A)
             => ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Y4),B) ) ) ) ).

% dvd_productE
tff(fact_3540_one__dvd,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,one_one_Code_integer),A) ).

% one_dvd
tff(fact_3541_one__dvd,axiom,
    ! [A: complex] : aa_complex_o(dvd_dvd_complex(one_one_complex),A) ).

% one_dvd
tff(fact_3542_one__dvd,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,one_one_real),A) ).

% one_dvd
tff(fact_3543_one__dvd,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,one_one_rat),A) ).

% one_dvd
tff(fact_3544_one__dvd,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,one_one_nat),A) ).

% one_dvd
tff(fact_3545_one__dvd,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,one_one_int),A) ).

% one_dvd
tff(fact_3546_unit__imp__dvd,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A) ) ).

% unit_imp_dvd
tff(fact_3547_unit__imp__dvd,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A) ) ).

% unit_imp_dvd
tff(fact_3548_unit__imp__dvd,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A) ) ).

% unit_imp_dvd
tff(fact_3549_dvd__unit__imp__unit,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer) ) ) ).

% dvd_unit_imp_unit
tff(fact_3550_dvd__unit__imp__unit,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat) ) ) ).

% dvd_unit_imp_unit
tff(fact_3551_dvd__unit__imp__unit,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int) ) ) ).

% dvd_unit_imp_unit
tff(fact_3552_dvd__add,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(B,C)) ) ) ).

% dvd_add
tff(fact_3553_dvd__add,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),C)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(plus_plus_real(B),C)) ) ) ).

% dvd_add
tff(fact_3554_dvd__add,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C)
       => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(plus_plus_rat(B),C)) ) ) ).

% dvd_add
tff(fact_3555_dvd__add,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ) ) ).

% dvd_add
tff(fact_3556_dvd__add,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ) ) ).

% dvd_add
tff(fact_3557_dvd__add__left__iff,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(B,C))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ) ).

% dvd_add_left_iff
tff(fact_3558_dvd__add__left__iff,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),C)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(plus_plus_real(B),C))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ) ).

% dvd_add_left_iff
tff(fact_3559_dvd__add__left__iff,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(plus_plus_rat(B),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ) ).

% dvd_add_left_iff
tff(fact_3560_dvd__add__left__iff,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ) ).

% dvd_add_left_iff
tff(fact_3561_dvd__add__left__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ) ).

% dvd_add_left_iff
tff(fact_3562_dvd__add__right__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(B,C))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% dvd_add_right_iff
tff(fact_3563_dvd__add__right__iff,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(plus_plus_real(B),C))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),C) ) ) ).

% dvd_add_right_iff
tff(fact_3564_dvd__add__right__iff,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(plus_plus_rat(B),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C) ) ) ).

% dvd_add_right_iff
tff(fact_3565_dvd__add__right__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% dvd_add_right_iff
tff(fact_3566_dvd__add__right__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% dvd_add_right_iff
tff(fact_3567_dvd__div__eq__iff,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
       => ( ( divide6298287555418463151nteger(A,C) = divide6298287555418463151nteger(B,C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_3568_dvd__div__eq__iff,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( aa_complex_o(dvd_dvd_complex(C),A)
     => ( aa_complex_o(dvd_dvd_complex(C),B)
       => ( ( aa_complex_complex(divide1717551699836669952omplex(A),C) = aa_complex_complex(divide1717551699836669952omplex(B),C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_3569_dvd__div__eq__iff,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,C),A)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,C),B)
       => ( ( aa_real_real(divide_divide_real(A),C) = aa_real_real(divide_divide_real(B),C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_3570_dvd__div__eq__iff,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C),B)
       => ( ( aa_rat_rat(divide_divide_rat(A),C) = aa_rat_rat(divide_divide_rat(B),C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_3571_dvd__div__eq__iff,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_3572_dvd__div__eq__iff,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
       => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_3573_dvd__div__eq__cancel,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( divide6298287555418463151nteger(A,C) = divide6298287555418463151nteger(B,C) )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_3574_dvd__div__eq__cancel,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A),C) = aa_complex_complex(divide1717551699836669952omplex(B),C) )
     => ( aa_complex_o(dvd_dvd_complex(C),A)
       => ( aa_complex_o(dvd_dvd_complex(C),B)
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_3575_dvd__div__eq__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( ( aa_real_real(divide_divide_real(A),C) = aa_real_real(divide_divide_real(B),C) )
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,C),A)
       => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,C),B)
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_3576_dvd__div__eq__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( aa_rat_rat(divide_divide_rat(A),C) = aa_rat_rat(divide_divide_rat(B),C) )
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C),B)
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_3577_dvd__div__eq__cancel,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C) )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_3578_dvd__div__eq__cancel,axiom,
    ! [A: int,C: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C) )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_3579_div__div__div__same,axiom,
    ! [D: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
       => ( divide6298287555418463151nteger(divide6298287555418463151nteger(A,D),divide6298287555418463151nteger(B,D)) = divide6298287555418463151nteger(A,B) ) ) ) ).

% div_div_div_same
tff(fact_3580_div__div__div__same,axiom,
    ! [D: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),D)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),D)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ) ).

% div_div_div_same
tff(fact_3581_div__div__div__same,axiom,
    ! [D: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),D)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),D)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ) ).

% div_div_div_same
tff(fact_3582_dvd__power__same,axiom,
    ! [X: code_integer,Y2: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,X),Y2)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(X),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(Y2),Nb)) ) ).

% dvd_power_same
tff(fact_3583_dvd__power__same,axiom,
    ! [X: nat,Y2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,X),Y2)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y2),Nb)) ) ).

% dvd_power_same
tff(fact_3584_dvd__power__same,axiom,
    ! [X: real,Y2: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,X),Y2)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),Nb)) ) ).

% dvd_power_same
tff(fact_3585_dvd__power__same,axiom,
    ! [X: int,Y2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X),Y2)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X),Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y2),Nb)) ) ).

% dvd_power_same
tff(fact_3586_dvd__power__same,axiom,
    ! [X: complex,Y2: complex,Nb: nat] :
      ( aa_complex_o(dvd_dvd_complex(X),Y2)
     => aa_complex_o(dvd_dvd_complex(aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,X),Nb)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Y2),Nb)) ) ).

% dvd_power_same
tff(fact_3587_mod__mod__cancel,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C) ) ) ).

% mod_mod_cancel
tff(fact_3588_mod__mod__cancel,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C) ) ) ).

% mod_mod_cancel
tff(fact_3589_mod__mod__cancel,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
     => ( modulo364778990260209775nteger(modulo364778990260209775nteger(A,B),C) = modulo364778990260209775nteger(A,C) ) ) ).

% mod_mod_cancel
tff(fact_3590_dvd__mod,axiom,
    ! [K2: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K2),Ma)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K2),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),Nb)) ) ) ).

% dvd_mod
tff(fact_3591_dvd__mod,axiom,
    ! [K2: int,Ma: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K2),Ma)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K2),Nb)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K2),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Ma),Nb)) ) ) ).

% dvd_mod
tff(fact_3592_dvd__mod,axiom,
    ! [K2: code_integer,Ma: code_integer,Nb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,K2),Ma)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,K2),Nb)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,K2),modulo364778990260209775nteger(Ma,Nb)) ) ) ).

% dvd_mod
tff(fact_3593_dvd__mod__iff,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A) ) ) ).

% dvd_mod_iff
tff(fact_3594_dvd__mod__iff,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A) ) ) ).

% dvd_mod_iff
tff(fact_3595_dvd__mod__iff,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),modulo364778990260209775nteger(A,B))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A) ) ) ).

% dvd_mod_iff
tff(fact_3596_dvd__mod__imp__dvd,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A) ) ) ).

% dvd_mod_imp_dvd
tff(fact_3597_dvd__mod__imp__dvd,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A) ) ) ).

% dvd_mod_imp_dvd
tff(fact_3598_dvd__mod__imp__dvd,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),modulo364778990260209775nteger(A,B))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A) ) ) ).

% dvd_mod_imp_dvd
tff(fact_3599_signed__take__bit__mult,axiom,
    ! [Nb: nat,K2: int,L: int] : aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(Nb),K2)),aa_int_int(bit_ri631733984087533419it_int(Nb),L))) = aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(aa_int_fun_int_int(times_times_int,K2),L)) ).

% signed_take_bit_mult
tff(fact_3600_signed__take__bit__add,axiom,
    ! [Nb: nat,K2: int,L: int] : aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(bit_ri631733984087533419it_int(Nb),K2)),aa_int_int(bit_ri631733984087533419it_int(Nb),L))) = aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),L)) ).

% signed_take_bit_add
tff(fact_3601_subset__divisors__dvd,axiom,
    ! [A: complex,B: complex] :
      ( ord_le211207098394363844omplex(collect_complex(aTP_Lamm_bf(A)),collect_complex(aTP_Lamm_bf(B)))
    <=> aa_complex_o(dvd_dvd_complex(A),B) ) ).

% subset_divisors_dvd
tff(fact_3602_subset__divisors__dvd,axiom,
    ! [A: nat,B: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bg,A))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bg,B)))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% subset_divisors_dvd
tff(fact_3603_subset__divisors__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ord_le7084787975880047091nteger(collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_bh,A)),collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_bh,B)))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B) ) ).

% subset_divisors_dvd
tff(fact_3604_subset__divisors__dvd,axiom,
    ! [A: int,B: int] :
      ( ord_less_eq_set_int(collect_int(aa_int_fun_int_o(aTP_Lamm_bi,A)),collect_int(aa_int_fun_int_o(aTP_Lamm_bi,B)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% subset_divisors_dvd
tff(fact_3605_strict__subset__divisors__dvd,axiom,
    ! [A: complex,B: complex] :
      ( ord_less_set_complex(collect_complex(aTP_Lamm_bf(A)),collect_complex(aTP_Lamm_bf(B)))
    <=> ( aa_complex_o(dvd_dvd_complex(A),B)
        & ~ aa_complex_o(dvd_dvd_complex(B),A) ) ) ).

% strict_subset_divisors_dvd
tff(fact_3606_strict__subset__divisors__dvd,axiom,
    ! [A: nat,B: nat] :
      ( aa_set_nat_o(ord_less_set_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bg,A))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bg,B)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
        & ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A) ) ) ).

% strict_subset_divisors_dvd
tff(fact_3607_strict__subset__divisors__dvd,axiom,
    ! [A: int,B: int] :
      ( ord_less_set_int(collect_int(aa_int_fun_int_o(aTP_Lamm_bi,A)),collect_int(aa_int_fun_int_o(aTP_Lamm_bi,B)))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
        & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A) ) ) ).

% strict_subset_divisors_dvd
tff(fact_3608_strict__subset__divisors__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ord_le1307284697595431911nteger(collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_bh,A)),collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_bh,B)))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
        & ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A) ) ) ).

% strict_subset_divisors_dvd
tff(fact_3609_even__signed__take__bit__iff,axiom,
    ! [Ma: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_ri6519982836138164636nteger(Ma,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A) ) ).

% even_signed_take_bit_iff
tff(fact_3610_even__signed__take__bit__iff,axiom,
    ! [Ma: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri631733984087533419it_int(Ma),A))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) ) ).

% even_signed_take_bit_iff
tff(fact_3611_mod__int__unique,axiom,
    ! [K2: int,L: int,Q: int,R: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(K2,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q),R))
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),L) = R ) ) ).

% mod_int_unique
tff(fact_3612_not__is__unit__0,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,zero_z3403309356797280102nteger),one_one_Code_integer) ).

% not_is_unit_0
tff(fact_3613_not__is__unit__0,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),one_one_nat) ).

% not_is_unit_0
tff(fact_3614_not__is__unit__0,axiom,
    ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,zero_zero_int),one_one_int) ).

% not_is_unit_0
tff(fact_3615_minf_I10_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z: code_integer] :
    ! [X2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X2),Z)
     => ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X2,S))
      <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X2,S)) ) ) ).

% minf(10)
tff(fact_3616_minf_I10_J,axiom,
    ! [D: real,S: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Z)
     => ( ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(plus_plus_real(X2),S))
      <=> ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(plus_plus_real(X2),S)) ) ) ).

% minf(10)
tff(fact_3617_minf_I10_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z: rat] :
    ! [X2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Z)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X2),S))
      <=> ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X2),S)) ) ) ).

% minf(10)
tff(fact_3618_minf_I10_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Z)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),S))
      <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),S)) ) ) ).

% minf(10)
tff(fact_3619_minf_I10_J,axiom,
    ! [D: int,S: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Z)
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),S))
      <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),S)) ) ) ).

% minf(10)
tff(fact_3620_minf_I9_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z: code_integer] :
    ! [X2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X2),Z)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X2,S))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X2,S)) ) ) ).

% minf(9)
tff(fact_3621_minf_I9_J,axiom,
    ! [D: real,S: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X2),Z)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(plus_plus_real(X2),S))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(plus_plus_real(X2),S)) ) ) ).

% minf(9)
tff(fact_3622_minf_I9_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z: rat] :
    ! [X2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),Z)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X2),S))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X2),S)) ) ) ).

% minf(9)
tff(fact_3623_minf_I9_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Z)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),S))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),S)) ) ) ).

% minf(9)
tff(fact_3624_minf_I9_J,axiom,
    ! [D: int,S: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Z)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),S))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),S)) ) ) ).

% minf(9)
tff(fact_3625_pinf_I10_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z: code_integer] :
    ! [X2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z),X2)
     => ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X2,S))
      <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X2,S)) ) ) ).

% pinf(10)
tff(fact_3626_pinf_I10_J,axiom,
    ! [D: real,S: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),X2)
     => ( ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(plus_plus_real(X2),S))
      <=> ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(plus_plus_real(X2),S)) ) ) ).

% pinf(10)
tff(fact_3627_pinf_I10_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z: rat] :
    ! [X2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),X2)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X2),S))
      <=> ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X2),S)) ) ) ).

% pinf(10)
tff(fact_3628_pinf_I10_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),X2)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),S))
      <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),S)) ) ) ).

% pinf(10)
tff(fact_3629_pinf_I10_J,axiom,
    ! [D: int,S: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),X2)
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),S))
      <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),S)) ) ) ).

% pinf(10)
tff(fact_3630_pinf_I9_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z: code_integer] :
    ! [X2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z),X2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X2,S))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X2,S)) ) ) ).

% pinf(9)
tff(fact_3631_pinf_I9_J,axiom,
    ! [D: real,S: real] :
    ? [Z: real] :
    ! [X2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),X2)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(plus_plus_real(X2),S))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(plus_plus_real(X2),S)) ) ) ).

% pinf(9)
tff(fact_3632_pinf_I9_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z: rat] :
    ! [X2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),X2)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X2),S))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X2),S)) ) ) ).

% pinf(9)
tff(fact_3633_pinf_I9_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z: nat] :
    ! [X2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),X2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),S))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),S)) ) ) ).

% pinf(9)
tff(fact_3634_pinf_I9_J,axiom,
    ! [D: int,S: int] :
    ? [Z: int] :
    ! [X2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),X2)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),S))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),S)) ) ) ).

% pinf(9)
tff(fact_3635_dvd__div__eq__0__iff,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
     => ( ( divide6298287555418463151nteger(A,B) = zero_z3403309356797280102nteger )
      <=> ( A = zero_z3403309356797280102nteger ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_3636_dvd__div__eq__0__iff,axiom,
    ! [B: complex,A: complex] :
      ( aa_complex_o(dvd_dvd_complex(B),A)
     => ( ( aa_complex_complex(divide1717551699836669952omplex(A),B) = zero_zero_complex )
      <=> ( A = zero_zero_complex ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_3637_dvd__div__eq__0__iff,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A)
     => ( ( aa_real_real(divide_divide_real(A),B) = zero_zero_real )
      <=> ( A = zero_zero_real ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_3638_dvd__div__eq__0__iff,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A)
     => ( ( aa_rat_rat(divide_divide_rat(A),B) = zero_zero_rat )
      <=> ( A = zero_zero_rat ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_3639_dvd__div__eq__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = zero_zero_nat )
      <=> ( A = zero_zero_nat ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_3640_dvd__div__eq__0__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = zero_zero_int )
      <=> ( A = zero_zero_int ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_3641_unit__mult__right__cancel,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A) )
      <=> ( B = C ) ) ) ).

% unit_mult_right_cancel
tff(fact_3642_unit__mult__right__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) )
      <=> ( B = C ) ) ) ).

% unit_mult_right_cancel
tff(fact_3643_unit__mult__right__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,B),A) = aa_int_int(aa_int_fun_int_int(times_times_int,C),A) )
      <=> ( B = C ) ) ) ).

% unit_mult_right_cancel
tff(fact_3644_unit__mult__left__cancel,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C) )
      <=> ( B = C ) ) ) ).

% unit_mult_left_cancel
tff(fact_3645_unit__mult__left__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C) )
      <=> ( B = C ) ) ) ).

% unit_mult_left_cancel
tff(fact_3646_unit__mult__left__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) = aa_int_int(aa_int_fun_int_int(times_times_int,A),C) )
      <=> ( B = C ) ) ) ).

% unit_mult_left_cancel
tff(fact_3647_mult__unit__dvd__iff_H,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C)
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),C) ) ) ).

% mult_unit_dvd_iff'
tff(fact_3648_mult__unit__dvd__iff_H,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ) ).

% mult_unit_dvd_iff'
tff(fact_3649_mult__unit__dvd__iff_H,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ) ).

% mult_unit_dvd_iff'
tff(fact_3650_dvd__mult__unit__iff_H,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% dvd_mult_unit_iff'
tff(fact_3651_dvd__mult__unit__iff_H,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% dvd_mult_unit_iff'
tff(fact_3652_dvd__mult__unit__iff_H,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% dvd_mult_unit_iff'
tff(fact_3653_mult__unit__dvd__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C)
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% mult_unit_dvd_iff
tff(fact_3654_mult__unit__dvd__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% mult_unit_dvd_iff
tff(fact_3655_mult__unit__dvd__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% mult_unit_dvd_iff
tff(fact_3656_dvd__mult__unit__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% dvd_mult_unit_iff
tff(fact_3657_dvd__mult__unit__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% dvd_mult_unit_iff
tff(fact_3658_dvd__mult__unit__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% dvd_mult_unit_iff
tff(fact_3659_is__unit__mult__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),one_one_Code_integer)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer) ) ) ).

% is_unit_mult_iff
tff(fact_3660_is__unit__mult__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),one_one_nat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
        & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat) ) ) ).

% is_unit_mult_iff
tff(fact_3661_is__unit__mult__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),one_one_int)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
        & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int) ) ) ).

% is_unit_mult_iff
tff(fact_3662_dvd__div__mult,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(B,C)),A) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A),C) ) ) ).

% dvd_div_mult
tff(fact_3663_dvd__div__mult,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)),A) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),C) ) ) ).

% dvd_div_mult
tff(fact_3664_dvd__div__mult,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)),A) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),C) ) ) ).

% dvd_div_mult
tff(fact_3665_div__mult__swap,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),divide6298287555418463151nteger(B,C)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),C) ) ) ).

% div_mult_swap
tff(fact_3666_div__mult__swap,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ) ) ).

% div_mult_swap
tff(fact_3667_div__mult__swap,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) ) ) ).

% div_mult_swap
tff(fact_3668_div__div__eq__right,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
       => ( divide6298287555418463151nteger(A,divide6298287555418463151nteger(B,C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),C) ) ) ) ).

% div_div_eq_right
tff(fact_3669_div__div__eq__right,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ) ).

% div_div_eq_right
tff(fact_3670_div__div__eq__right,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ) ).

% div_div_eq_right
tff(fact_3671_dvd__div__mult2__eq,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)),A)
     => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = divide6298287555418463151nteger(divide6298287555418463151nteger(A,B),C) ) ) ).

% dvd_div_mult2_eq
tff(fact_3672_dvd__div__mult2__eq,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ).

% dvd_div_mult2_eq
tff(fact_3673_dvd__div__mult2__eq,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),C)),A)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ).

% dvd_div_mult2_eq
tff(fact_3674_dvd__mult__imp__div,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),divide6298287555418463151nteger(B,C)) ) ).

% dvd_mult_imp_div
tff(fact_3675_dvd__mult__imp__div,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),B)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) ) ).

% dvd_mult_imp_div
tff(fact_3676_dvd__mult__imp__div,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),B)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) ) ).

% dvd_mult_imp_div
tff(fact_3677_div__mult__div__if__dvd,axiom,
    ! [B: code_integer,A: code_integer,D: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),C)
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),divide6298287555418463151nteger(C,D)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_3678_div__mult__div__if__dvd,axiom,
    ! [B: nat,A: nat,D: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),C)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),D)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_3679_div__mult__div__if__dvd,axiom,
    ! [B: int,A: int,D: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),C)
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),aa_int_int(aa_int_fun_int_int(divide_divide_int,C),D)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_3680_unit__div__cancel,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( ( divide6298287555418463151nteger(B,A) = divide6298287555418463151nteger(C,A) )
      <=> ( B = C ) ) ) ).

% unit_div_cancel
tff(fact_3681_unit__div__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),A) )
      <=> ( B = C ) ) ) ).

% unit_div_cancel
tff(fact_3682_unit__div__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A) = aa_int_int(aa_int_fun_int_int(divide_divide_int,C),A) )
      <=> ( B = C ) ) ) ).

% unit_div_cancel
tff(fact_3683_div__unit__dvd__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,divide6298287555418463151nteger(A,B)),C)
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% div_unit_dvd_iff
tff(fact_3684_div__unit__dvd__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% div_unit_dvd_iff
tff(fact_3685_div__unit__dvd__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% div_unit_dvd_iff
tff(fact_3686_dvd__div__unit__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),divide6298287555418463151nteger(C,B))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% dvd_div_unit_iff
tff(fact_3687_dvd__div__unit__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),B))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% dvd_div_unit_iff
tff(fact_3688_dvd__div__unit__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,C),B))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% dvd_div_unit_iff
tff(fact_3689_div__plus__div__distrib__dvd__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A)
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,B),C) = plus_p5714425477246183910nteger(divide6298287555418463151nteger(A,C),divide6298287555418463151nteger(B,C)) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_3690_div__plus__div__distrib__dvd__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_3691_div__plus__div__distrib__dvd__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_3692_div__plus__div__distrib__dvd__right,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,B),C) = plus_p5714425477246183910nteger(divide6298287555418463151nteger(A,C),divide6298287555418463151nteger(B,C)) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_3693_div__plus__div__distrib__dvd__right,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_3694_div__plus__div__distrib__dvd__right,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_3695_div__power,axiom,
    ! [B: code_integer,A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(divide6298287555418463151nteger(A,B)),Nb) = divide6298287555418463151nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb),aa_nat_Code_integer(power_8256067586552552935nteger(B),Nb)) ) ) ).

% div_power
tff(fact_3696_div__power,axiom,
    ! [B: nat,A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)) ) ) ).

% div_power
tff(fact_3697_div__power,axiom,
    ! [B: int,A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),Nb) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Nb)) ) ) ).

% div_power
tff(fact_3698_mod__eq__0__iff__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A) ) ).

% mod_eq_0_iff_dvd
tff(fact_3699_mod__eq__0__iff__dvd,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int )
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A) ) ).

% mod_eq_0_iff_dvd
tff(fact_3700_mod__eq__0__iff__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger(A,B) = zero_z3403309356797280102nteger )
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A) ) ).

% mod_eq_0_iff_dvd
tff(fact_3701_dvd__eq__mod__eq__0,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),A) = zero_zero_nat ) ) ).

% dvd_eq_mod_eq_0
tff(fact_3702_dvd__eq__mod__eq__0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),A) = zero_zero_int ) ) ).

% dvd_eq_mod_eq_0
tff(fact_3703_dvd__eq__mod__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
    <=> ( modulo364778990260209775nteger(B,A) = zero_z3403309356797280102nteger ) ) ).

% dvd_eq_mod_eq_0
tff(fact_3704_mod__0__imp__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A) ) ).

% mod_0_imp_dvd
tff(fact_3705_mod__0__imp__dvd,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A) ) ).

% mod_0_imp_dvd
tff(fact_3706_mod__0__imp__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger(A,B) = zero_z3403309356797280102nteger )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A) ) ).

% mod_0_imp_dvd
tff(fact_3707_le__imp__power__dvd,axiom,
    ! [Ma: nat,Nb: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),Ma)),aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)) ) ).

% le_imp_power_dvd
tff(fact_3708_le__imp__power__dvd,axiom,
    ! [Ma: nat,Nb: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ).

% le_imp_power_dvd
tff(fact_3709_le__imp__power__dvd,axiom,
    ! [Ma: nat,Nb: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Ma)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ).

% le_imp_power_dvd
tff(fact_3710_le__imp__power__dvd,axiom,
    ! [Ma: nat,Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Ma)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)) ) ).

% le_imp_power_dvd
tff(fact_3711_le__imp__power__dvd,axiom,
    ! [Ma: nat,Nb: nat,A: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => aa_complex_o(dvd_dvd_complex(aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),Ma)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),Nb)) ) ).

% le_imp_power_dvd
tff(fact_3712_power__le__dvd,axiom,
    ! [A: code_integer,Nb: nat,B: code_integer,Ma: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),Ma)),B) ) ) ).

% power_le_dvd
tff(fact_3713_power__le__dvd,axiom,
    ! [A: nat,Nb: nat,B: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Ma)),B) ) ) ).

% power_le_dvd
tff(fact_3714_power__le__dvd,axiom,
    ! [A: real,Nb: nat,B: real,Ma: nat] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Ma)),B) ) ) ).

% power_le_dvd
tff(fact_3715_power__le__dvd,axiom,
    ! [A: int,Nb: nat,B: int,Ma: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Ma)),B) ) ) ).

% power_le_dvd
tff(fact_3716_power__le__dvd,axiom,
    ! [A: complex,Nb: nat,B: complex,Ma: nat] :
      ( aa_complex_o(dvd_dvd_complex(aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),Nb)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
       => aa_complex_o(dvd_dvd_complex(aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),Ma)),B) ) ) ).

% power_le_dvd
tff(fact_3717_dvd__power__le,axiom,
    ! [X: code_integer,Y2: code_integer,Nb: nat,Ma: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,X),Y2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(X),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(Y2),Ma)) ) ) ).

% dvd_power_le
tff(fact_3718_dvd__power__le,axiom,
    ! [X: nat,Y2: nat,Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,X),Y2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y2),Ma)) ) ) ).

% dvd_power_le
tff(fact_3719_dvd__power__le,axiom,
    ! [X: real,Y2: real,Nb: nat,Ma: nat] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,X),Y2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),Ma)) ) ) ).

% dvd_power_le
tff(fact_3720_dvd__power__le,axiom,
    ! [X: int,Y2: int,Nb: nat,Ma: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X),Y2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X),Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y2),Ma)) ) ) ).

% dvd_power_le
tff(fact_3721_dvd__power__le,axiom,
    ! [X: complex,Y2: complex,Nb: nat,Ma: nat] :
      ( aa_complex_o(dvd_dvd_complex(X),Y2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma)
       => aa_complex_o(dvd_dvd_complex(aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,X),Nb)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Y2),Ma)) ) ) ).

% dvd_power_le
tff(fact_3722_nat__dvd__not__less,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ma)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
       => ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),Ma) ) ) ).

% nat_dvd_not_less
tff(fact_3723_dvd__pos__nat,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ma),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ma) ) ) ).

% dvd_pos_nat
tff(fact_3724_zdvd__antisym__nonneg,axiom,
    ! [Ma: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ma)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Nb)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Ma),Nb)
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Nb),Ma)
           => ( Ma = Nb ) ) ) ) ) ).

% zdvd_antisym_nonneg
tff(fact_3725_zdvd__mono,axiom,
    ! [K2: int,Ma: int,T: int] :
      ( ( K2 != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Ma),T)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,K2),Ma)),aa_int_int(aa_int_fun_int_int(times_times_int,K2),T)) ) ) ).

% zdvd_mono
tff(fact_3726_zdvd__mult__cancel,axiom,
    ! [K2: int,Ma: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,K2),Ma)),aa_int_int(aa_int_fun_int_int(times_times_int,K2),Nb))
     => ( ( K2 != zero_zero_int )
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Ma),Nb) ) ) ).

% zdvd_mult_cancel
tff(fact_3727_bezout__add__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D2: nat,X4: nat,Y4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),A)
      & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),B)
      & ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y4)),D2) )
        | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y4)),D2) ) ) ) ).

% bezout_add_nat
tff(fact_3728_bezout__lemma__nat,axiom,
    ! [D: nat,A: nat,B: nat,X: nat,Y2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),B)
       => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y2)),D) )
            | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y2)),D) ) )
         => ? [X4: nat,Y4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),A)
              & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
              & ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),Y4)),D) )
                | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),X4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y4)),D) ) ) ) ) ) ) ).

% bezout_lemma_nat
tff(fact_3729_zdvd__reduce,axiom,
    ! [K2: int,Nb: int,Ma: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K2),aa_int_int(aa_int_fun_int_int(plus_plus_int,Nb),aa_int_int(aa_int_fun_int_int(times_times_int,K2),Ma)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K2),Nb) ) ).

% zdvd_reduce
tff(fact_3730_zdvd__period,axiom,
    ! [A: int,D: int,X: int,T: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),D)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),T))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,C),D))),T)) ) ) ).

% zdvd_period
tff(fact_3731_unit__dvdE,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ~ ( ( A != zero_z3403309356797280102nteger )
         => ! [C3: code_integer] : B != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C3) ) ) ).

% unit_dvdE
tff(fact_3732_unit__dvdE,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ~ ( ( A != zero_zero_nat )
         => ! [C3: nat] : B != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C3) ) ) ).

% unit_dvdE
tff(fact_3733_unit__dvdE,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ~ ( ( A != zero_zero_int )
         => ! [C3: int] : B != aa_int_int(aa_int_fun_int_int(times_times_int,A),C3) ) ) ).

% unit_dvdE
tff(fact_3734_unity__coeff__ex,axiom,
    ! [P: fun_Code_integer_o,L: code_integer] :
      ( ? [X3: code_integer] : aa_Code_integer_o(P,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,L),X3))
    <=> ? [X3: code_integer] :
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,L),plus_p5714425477246183910nteger(X3,zero_z3403309356797280102nteger))
          & aa_Code_integer_o(P,X3) ) ) ).

% unity_coeff_ex
tff(fact_3735_unity__coeff__ex,axiom,
    ! [P: fun_complex_o,L: complex] :
      ( ? [X3: complex] : aa_complex_o(P,aa_complex_complex(times_times_complex(L),X3))
    <=> ? [X3: complex] :
          ( aa_complex_o(dvd_dvd_complex(L),aa_complex_complex(plus_plus_complex(X3),zero_zero_complex))
          & aa_complex_o(P,X3) ) ) ).

% unity_coeff_ex
tff(fact_3736_unity__coeff__ex,axiom,
    ! [P: fun_real_o,L: real] :
      ( ? [X3: real] : aa_real_o(P,aa_real_real(times_times_real(L),X3))
    <=> ? [X3: real] :
          ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,L),aa_real_real(plus_plus_real(X3),zero_zero_real))
          & aa_real_o(P,X3) ) ) ).

% unity_coeff_ex
tff(fact_3737_unity__coeff__ex,axiom,
    ! [P: fun_rat_o,L: rat] :
      ( ? [X3: rat] : aa_rat_o(P,aa_rat_rat(times_times_rat(L),X3))
    <=> ? [X3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,L),aa_rat_rat(plus_plus_rat(X3),zero_zero_rat))
          & aa_rat_o(P,X3) ) ) ).

% unity_coeff_ex
tff(fact_3738_unity__coeff__ex,axiom,
    ! [P: fun_nat_o,L: nat] :
      ( ? [X3: nat] : aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,L),X3))
    <=> ? [X3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,L),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),zero_zero_nat))
          & aa_nat_o(P,X3) ) ) ).

% unity_coeff_ex
tff(fact_3739_unity__coeff__ex,axiom,
    ! [P: fun_int_o,L: int] :
      ( ? [X3: int] : aa_int_o(P,aa_int_int(aa_int_fun_int_int(times_times_int,L),X3))
    <=> ? [X3: int] :
          ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),zero_zero_int))
          & aa_int_o(P,X3) ) ) ).

% unity_coeff_ex
tff(fact_3740_dvd__div__eq__mult,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
       => ( ( divide6298287555418463151nteger(B,A) = C )
        <=> ( B = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_3741_dvd__div__eq__mult,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) = C )
        <=> ( B = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_3742_dvd__div__eq__mult,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
       => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A) = C )
        <=> ( B = aa_int_int(aa_int_fun_int_int(times_times_int,C),A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_3743_div__dvd__iff__mult,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,divide6298287555418463151nteger(A,B)),C)
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) ) ) ) ).

% div_dvd_iff_mult
tff(fact_3744_div__dvd__iff__mult,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C)
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ) ).

% div_dvd_iff_mult
tff(fact_3745_div__dvd__iff__mult,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C)
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ) ).

% div_dvd_iff_mult
tff(fact_3746_dvd__div__iff__mult,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),divide6298287555418463151nteger(B,C))
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),B) ) ) ) ).

% dvd_div_iff_mult
tff(fact_3747_dvd__div__iff__mult,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( C != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C))
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),B) ) ) ) ).

% dvd_div_iff_mult
tff(fact_3748_dvd__div__iff__mult,axiom,
    ! [C: int,B: int,A: int] :
      ( ( C != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C))
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),B) ) ) ) ).

% dvd_div_iff_mult
tff(fact_3749_dvd__div__div__eq__mult,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( C != zero_z3403309356797280102nteger )
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),D)
           => ( ( divide6298287555418463151nteger(B,A) = divide6298287555418463151nteger(D,C) )
            <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),D) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_3750_dvd__div__div__eq__mult,axiom,
    ! [A: nat,C: nat,B: nat,D: nat] :
      ( ( A != zero_zero_nat )
     => ( ( C != zero_zero_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),D)
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,D),C) )
            <=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),D) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_3751_dvd__div__div__eq__mult,axiom,
    ! [A: int,C: int,B: int,D: int] :
      ( ( A != zero_zero_int )
     => ( ( C != zero_zero_int )
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),D)
           => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A) = aa_int_int(aa_int_fun_int_int(divide_divide_int,D),C) )
            <=> ( aa_int_int(aa_int_fun_int_int(times_times_int,B),C) = aa_int_int(aa_int_fun_int_int(times_times_int,A),D) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_3752_unit__div__eq__0__iff,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( ( divide6298287555418463151nteger(A,B) = zero_z3403309356797280102nteger )
      <=> ( A = zero_z3403309356797280102nteger ) ) ) ).

% unit_div_eq_0_iff
tff(fact_3753_unit__div__eq__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = zero_zero_nat )
      <=> ( A = zero_zero_nat ) ) ) ).

% unit_div_eq_0_iff
tff(fact_3754_unit__div__eq__0__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = zero_zero_int )
      <=> ( A = zero_zero_int ) ) ) ).

% unit_div_eq_0_iff
tff(fact_3755_even__numeral,axiom,
    ! [Nb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),numera6620942414471956472nteger(aa_num_num(bit0,Nb))) ).

% even_numeral
tff(fact_3756_even__numeral,axiom,
    ! [Nb: num] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb))) ).

% even_numeral
tff(fact_3757_even__numeral,axiom,
    ! [Nb: num] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))) ).

% even_numeral
tff(fact_3758_unit__eq__div1,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( ( divide6298287555418463151nteger(A,B) = C )
      <=> ( A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B) ) ) ) ).

% unit_eq_div1
tff(fact_3759_unit__eq__div1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) = C )
      <=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B) ) ) ) ).

% unit_eq_div1
tff(fact_3760_unit__eq__div1,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = C )
      <=> ( A = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) ) ) ) ).

% unit_eq_div1
tff(fact_3761_unit__eq__div2,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( ( A = divide6298287555418463151nteger(C,B) )
      <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) = C ) ) ) ).

% unit_eq_div2
tff(fact_3762_unit__eq__div2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),B) )
      <=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = C ) ) ) ).

% unit_eq_div2
tff(fact_3763_unit__eq__div2,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( ( A = aa_int_int(aa_int_fun_int_int(divide_divide_int,C),B) )
      <=> ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) = C ) ) ) ).

% unit_eq_div2
tff(fact_3764_div__mult__unit2,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
       => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = divide6298287555418463151nteger(divide6298287555418463151nteger(A,B),C) ) ) ) ).

% div_mult_unit2
tff(fact_3765_div__mult__unit2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ) ).

% div_mult_unit2
tff(fact_3766_div__mult__unit2,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ) ).

% div_mult_unit2
tff(fact_3767_unit__div__commute,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),C) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C),B) ) ) ).

% unit_div_commute
tff(fact_3768_unit__div__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),B) ) ) ).

% unit_div_commute
tff(fact_3769_unit__div__commute,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),B) ) ) ).

% unit_div_commute
tff(fact_3770_unit__div__mult__swap,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),one_one_Code_integer)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),divide6298287555418463151nteger(B,C)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),C) ) ) ).

% unit_div_mult_swap
tff(fact_3771_unit__div__mult__swap,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ) ) ).

% unit_div_mult_swap
tff(fact_3772_unit__div__mult__swap,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) ) ) ).

% unit_div_mult_swap
tff(fact_3773_is__unit__div__mult2__eq,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),one_one_Code_integer)
       => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = divide6298287555418463151nteger(divide6298287555418463151nteger(A,B),C) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_3774_is__unit__div__mult2__eq,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),one_one_nat)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_3775_is__unit__div__mult2__eq,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),one_one_int)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_3776_unit__imp__mod__eq__0,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = zero_zero_nat ) ) ).

% unit_imp_mod_eq_0
tff(fact_3777_unit__imp__mod__eq__0,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int ) ) ).

% unit_imp_mod_eq_0
tff(fact_3778_unit__imp__mod__eq__0,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
     => ( modulo364778990260209775nteger(A,B) = zero_z3403309356797280102nteger ) ) ).

% unit_imp_mod_eq_0
tff(fact_3779_is__unit__power__iff,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)),one_one_Code_integer)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
        | ( Nb = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_3780_is__unit__power__iff,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),one_one_nat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
        | ( Nb = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_3781_is__unit__power__iff,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)),one_one_int)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
        | ( Nb = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_3782_eucl__rel__int__dividesI,axiom,
    ! [L: int,K2: int,Q: int] :
      ( ( L != zero_zero_int )
     => ( ( K2 = aa_int_int(aa_int_fun_int_int(times_times_int,Q),L) )
       => aa_Pro4592068669816069595_int_o(eucl_rel_int(K2,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q),zero_zero_int)) ) ) ).

% eucl_rel_int_dividesI
tff(fact_3783_dvd__imp__le,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K2),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb) ) ) ).

% dvd_imp_le
tff(fact_3784_dvd__mult__cancel,axiom,
    ! [K2: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ma),Nb) ) ) ).

% dvd_mult_cancel
tff(fact_3785_nat__mult__dvd__cancel1,axiom,
    ! [K2: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ma),Nb) ) ) ).

% nat_mult_dvd_cancel1
tff(fact_3786_bezout__add__strong__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [D2: nat,X4: nat,Y4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),A)
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),B)
          & ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y4)),D2) ) ) ) ).

% bezout_add_strong_nat
tff(fact_3787_zdvd__imp__le,axiom,
    ! [Z2: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Z2),Nb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z2),Nb) ) ) ).

% zdvd_imp_le
tff(fact_3788_mod__greater__zero__iff__not__dvd,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),Nb))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),Ma) ) ).

% mod_greater_zero_iff_not_dvd
tff(fact_3789_eucl__rel__int,axiom,
    ! [K2: int,L: int] : aa_Pro4592068669816069595_int_o(eucl_rel_int(K2,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),L)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),L))) ).

% eucl_rel_int
tff(fact_3790_even__zero,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_z3403309356797280102nteger) ).

% even_zero
tff(fact_3791_even__zero,axiom,
    aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),zero_zero_nat) ).

% even_zero
tff(fact_3792_even__zero,axiom,
    aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),zero_zero_int) ).

% even_zero
tff(fact_3793_is__unitE,axiom,
    ! [A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ~ ( ( A != zero_z3403309356797280102nteger )
         => ! [B2: code_integer] :
              ( ( B2 != zero_z3403309356797280102nteger )
             => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),one_one_Code_integer)
               => ( ( divide6298287555418463151nteger(one_one_Code_integer,A) = B2 )
                 => ( ( divide6298287555418463151nteger(one_one_Code_integer,B2) = A )
                   => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B2) = one_one_Code_integer )
                     => ( divide6298287555418463151nteger(C,A) != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B2) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_3794_is__unitE,axiom,
    ! [A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ~ ( ( A != zero_zero_nat )
         => ! [B2: nat] :
              ( ( B2 != zero_zero_nat )
             => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),one_one_nat)
               => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A) = B2 )
                 => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B2) = A )
                   => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B2) = one_one_nat )
                     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),A) != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B2) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_3795_is__unitE,axiom,
    ! [A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ~ ( ( A != zero_zero_int )
         => ! [B2: int] :
              ( ( B2 != zero_zero_int )
             => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),one_one_int)
               => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),A) = B2 )
                 => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),B2) = A )
                   => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B2) = one_one_int )
                     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,C),A) != aa_int_int(aa_int_fun_int_int(times_times_int,C),B2) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_3796_is__unit__div__mult__cancel__left,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
       => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) = divide6298287555418463151nteger(one_one_Code_integer,B) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_3797_is__unit__div__mult__cancel__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_3798_is__unit__div__mult__cancel__left,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),B) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_3799_is__unit__div__mult__cancel__right,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),one_one_Code_integer)
       => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)) = divide6298287555418463151nteger(one_one_Code_integer,B) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_3800_is__unit__div__mult__cancel__right,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_3801_is__unit__div__mult__cancel__right,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),A)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),B) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_3802_evenE,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
     => ~ ! [B2: code_integer] : A != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B2) ) ).

% evenE
tff(fact_3803_evenE,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ~ ! [B2: nat] : A != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2) ) ).

% evenE
tff(fact_3804_evenE,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ~ ! [B2: int] : A != aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2) ) ).

% evenE
tff(fact_3805_odd__one,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),one_one_Code_integer) ).

% odd_one
tff(fact_3806_odd__one,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),one_one_nat) ).

% odd_one
tff(fact_3807_odd__one,axiom,
    ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),one_one_int) ).

% odd_one
tff(fact_3808_odd__even__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
     => ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,B)) ) ) ).

% odd_even_add
tff(fact_3809_odd__even__add,axiom,
    ! [A: nat,B: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% odd_even_add
tff(fact_3810_odd__even__add,axiom,
    ! [A: int,B: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% odd_even_add
tff(fact_3811_bit__eq__rec,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = B )
    <=> ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),B) )
        & ( divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = divide6298287555418463151nteger(B,numera6620942414471956472nteger(aa_num_num(bit0,one))) ) ) ) ).

% bit_eq_rec
tff(fact_3812_bit__eq__rec,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B) )
        & ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ) ).

% bit_eq_rec
tff(fact_3813_bit__eq__rec,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
    <=> ( ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B) )
        & ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ) ).

% bit_eq_rec
tff(fact_3814_dvd__power__iff,axiom,
    ! [X: code_integer,Ma: nat,Nb: nat] :
      ( ( X != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(X),Ma)),aa_nat_Code_integer(power_8256067586552552935nteger(X),Nb))
      <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,X),one_one_Code_integer)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ) ) ).

% dvd_power_iff
tff(fact_3815_dvd__power__iff,axiom,
    ! [X: nat,Ma: nat,Nb: nat] :
      ( ( X != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),Nb))
      <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,X),one_one_nat)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ) ) ).

% dvd_power_iff
tff(fact_3816_dvd__power__iff,axiom,
    ! [X: int,Ma: nat,Nb: nat] :
      ( ( X != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X),Ma)),aa_nat_int(aa_int_fun_nat_int(power_power_int,X),Nb))
      <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X),one_one_int)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ) ) ).

% dvd_power_iff
tff(fact_3817_dvd__power,axiom,
    ! [Nb: nat,X: code_integer] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( X = one_one_Code_integer ) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,X),aa_nat_Code_integer(power_8256067586552552935nteger(X),Nb)) ) ).

% dvd_power
tff(fact_3818_dvd__power,axiom,
    ! [Nb: nat,X: rat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( X = one_one_rat ) )
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,X),aa_nat_rat(power_power_rat(X),Nb)) ) ).

% dvd_power
tff(fact_3819_dvd__power,axiom,
    ! [Nb: nat,X: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( X = one_one_nat ) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),Nb)) ) ).

% dvd_power
tff(fact_3820_dvd__power,axiom,
    ! [Nb: nat,X: real] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( X = one_one_real ) )
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,X),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Nb)) ) ).

% dvd_power
tff(fact_3821_dvd__power,axiom,
    ! [Nb: nat,X: int] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( X = one_one_int ) )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X),aa_nat_int(aa_int_fun_nat_int(power_power_int,X),Nb)) ) ).

% dvd_power
tff(fact_3822_dvd__power,axiom,
    ! [Nb: nat,X: complex] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( X = one_one_complex ) )
     => aa_complex_o(dvd_dvd_complex(X),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,X),Nb)) ) ).

% dvd_power
tff(fact_3823_even__even__mod__4__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))))) ) ).

% even_even_mod_4_iff
tff(fact_3824_dvd__mult__cancel1,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ma)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Nb)),Ma)
      <=> ( Nb = one_one_nat ) ) ) ).

% dvd_mult_cancel1
tff(fact_3825_dvd__mult__cancel2,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ma)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Ma)),Ma)
      <=> ( Nb = one_one_nat ) ) ) ).

% dvd_mult_cancel2
tff(fact_3826_power__dvd__imp__le,axiom,
    ! [I2: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I2),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I2),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),I2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ) ).

% power_dvd_imp_le
tff(fact_3827_mod__int__pos__iff,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),L))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K2)
        | ( ( L = zero_zero_int )
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2) )
        | aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L) ) ) ).

% mod_int_pos_iff
tff(fact_3828_even__two__times__div__two,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))) = A ) ) ).

% even_two_times_div_two
tff(fact_3829_even__two__times__div__two,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = A ) ) ).

% even_two_times_div_two
tff(fact_3830_even__two__times__div__two,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) = A ) ) ).

% even_two_times_div_two
tff(fact_3831_even__iff__mod__2__eq__zero,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_3832_even__iff__mod__2__eq__zero,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_3833_even__iff__mod__2__eq__zero,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
    <=> ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_3834_odd__iff__mod__2__eq__one,axiom,
    ! [A: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_3835_odd__iff__mod__2__eq__one,axiom,
    ! [A: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_3836_odd__iff__mod__2__eq__one,axiom,
    ! [A: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
    <=> ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = one_one_Code_integer ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_3837_power__mono__odd,axiom,
    ! [Nb: nat,A: real,B: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)) ) ) ).

% power_mono_odd
tff(fact_3838_power__mono__odd,axiom,
    ! [Nb: nat,A: rat,B: rat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb)) ) ) ).

% power_mono_odd
tff(fact_3839_power__mono__odd,axiom,
    ! [Nb: nat,A: int,B: int] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Nb)) ) ) ).

% power_mono_odd
tff(fact_3840_odd__pos,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% odd_pos
tff(fact_3841_dvd__power__iff__le,axiom,
    ! [K2: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K2),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K2),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ) ).

% dvd_power_iff_le
tff(fact_3842_signed__take__bit__int__less__exp,axiom,
    ! [Nb: nat,K2: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_ri631733984087533419it_int(Nb),K2)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% signed_take_bit_int_less_exp
tff(fact_3843_even__unset__bit__iff,axiom,
    ! [Ma: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se8260200283734997820nteger(Ma,A))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
        | ( Ma = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_3844_even__unset__bit__iff,axiom,
    ! [Ma: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Ma),A))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        | ( Ma = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_3845_even__unset__bit__iff,axiom,
    ! [Ma: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,Ma),A))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        | ( Ma = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_3846_even__set__bit__iff,axiom,
    ! [Ma: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se2793503036327961859nteger(Ma,A))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
        & ( Ma != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_3847_even__set__bit__iff,axiom,
    ! [Ma: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,Ma),A))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        & ( Ma != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_3848_even__set__bit__iff,axiom,
    ! [Ma: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,Ma),A))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        & ( Ma != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_3849_even__flip__bit__iff,axiom,
    ! [Ma: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se1345352211410354436nteger(Ma,A))
    <=> ~ ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
        <=> ( Ma = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_3850_even__flip__bit__iff,axiom,
    ! [Ma: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se2159334234014336723it_int(Ma),A))
    <=> ~ ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        <=> ( Ma = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_3851_even__flip__bit__iff,axiom,
    ! [Ma: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,Ma),A))
    <=> ~ ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        <=> ( Ma = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_3852_oddE,axiom,
    ! [A: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
     => ~ ! [B2: code_integer] : A != plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B2),one_one_Code_integer) ) ).

% oddE
tff(fact_3853_oddE,axiom,
    ! [A: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ~ ! [B2: nat] : A != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2)),one_one_nat) ) ).

% oddE
tff(fact_3854_oddE,axiom,
    ! [A: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ~ ! [B2: int] : A != aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2)),one_one_int) ) ).

% oddE
tff(fact_3855_mod2__eq__if,axiom,
    ! [A: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A),zero_zero_nat,one_one_nat) ).

% mod2_eq_if
tff(fact_3856_mod2__eq__if,axiom,
    ! [A: int] :
      aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = $ite(aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A),zero_zero_int,one_one_int) ).

% mod2_eq_if
tff(fact_3857_mod2__eq__if,axiom,
    ! [A: code_integer] :
      modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A),zero_z3403309356797280102nteger,one_one_Code_integer) ).

% mod2_eq_if
tff(fact_3858_parity__cases,axiom,
    ! [A: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != zero_zero_nat ) )
     => ~ ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
         => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != one_one_nat ) ) ) ).

% parity_cases
tff(fact_3859_parity__cases,axiom,
    ! [A: int] :
      ( ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != zero_zero_int ) )
     => ~ ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != one_one_int ) ) ) ).

% parity_cases
tff(fact_3860_parity__cases,axiom,
    ! [A: code_integer] :
      ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
       => ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) != zero_z3403309356797280102nteger ) )
     => ~ ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
         => ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) != one_one_Code_integer ) ) ) ).

% parity_cases
tff(fact_3861_zero__le__power__eq,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ) ) ).

% zero_le_power_eq
tff(fact_3862_zero__le__power__eq,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ) ) ).

% zero_le_power_eq
tff(fact_3863_zero__le__power__eq,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ) ).

% zero_le_power_eq
tff(fact_3864_zero__le__odd__power,axiom,
    ! [Nb: nat,A: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ) ).

% zero_le_odd_power
tff(fact_3865_zero__le__odd__power,axiom,
    ! [Nb: nat,A: rat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Nb))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ) ).

% zero_le_odd_power
tff(fact_3866_zero__le__odd__power,axiom,
    ! [Nb: nat,A: int] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ).

% zero_le_odd_power
tff(fact_3867_zero__le__even__power,axiom,
    ! [Nb: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ).

% zero_le_even_power
tff(fact_3868_zero__le__even__power,axiom,
    ! [Nb: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Nb)) ) ).

% zero_le_even_power
tff(fact_3869_zero__le__even__power,axiom,
    ! [Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)) ) ).

% zero_le_even_power
tff(fact_3870_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),aa_int_int(bit_ri631733984087533419it_int(Nb),K2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K2),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ) ).

% signed_take_bit_int_greater_eq_self_iff
tff(fact_3871_signed__take__bit__int__less__self__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_ri631733984087533419it_int(Nb),K2)),K2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),K2) ) ).

% signed_take_bit_int_less_self_iff
tff(fact_3872_zero__less__power__eq,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))
    <=> ( ( Nb = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & ( A != zero_zero_real ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ) ) ).

% zero_less_power_eq
tff(fact_3873_zero__less__power__eq,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Nb))
    <=> ( ( Nb = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & ( A != zero_zero_rat ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ) ) ).

% zero_less_power_eq
tff(fact_3874_zero__less__power__eq,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb))
    <=> ( ( Nb = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & ( A != zero_zero_int ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ) ) ).

% zero_less_power_eq
tff(fact_3875_eucl__rel__int__iff,axiom,
    ! [K2: int,L: int,Q: int,R: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(K2,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q),R))
    <=> ( ( K2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,L),Q)),R) )
        & $ite(
            aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L),
            ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R)
            & aa_int_o(aa_int_fun_int_o(ord_less_int,R),L) ),
            $ite(
              aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int),
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),R)
              & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R),zero_zero_int) ),
              Q = zero_zero_int ) ) ) ) ).

% eucl_rel_int_iff
tff(fact_3876_power__le__zero__eq,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),zero_zero_real)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
            & ( A = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_3877_power__le__zero__eq,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Nb)),zero_zero_rat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_3878_power__le__zero__eq,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)),zero_zero_int)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_3879_option_Osize__gen_I1_J,axiom,
    ! [X: fun_Pr2602117396471556196at_nat] : size_o8335143837870341156at_nat(X,none_P5556105721700978146at_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size_gen(1)
tff(fact_3880_option_Osize__gen_I1_J,axiom,
    ! [X: fun_num_nat] : size_option_num(X,none_num) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size_gen(1)
tff(fact_3881_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ~ aa_nat_o(vEBT_V5719532721284313246member(X),Xa)
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),produc738532404422230701BT_nat(X,Xa))
       => ( ! [A3: $o,B2: $o] :
              ( ( X = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B2)) )
             => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),produc738532404422230701BT_nat(aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B2)),Xa))
               => $ite(
                    Xa = zero_zero_nat,
                    (A3),
                    $ite(Xa = one_one_nat,(B2),$false) ) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,zero_zero_nat,Uv2),Uw) )
               => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,zero_zero_nat,Uv2),Uw),Xa)) )
           => ~ ! [Uy: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy,aa_nat_nat(suc,V2),TreeList),S2) )
                 => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy,aa_nat_nat(suc,V2),TreeList),S2),Xa))
                   => $let(
                        pos: nat,
                        pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_V5719532721284313246member(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
tff(fact_3882_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( aa_nat_o(vEBT_V5719532721284313246member(X),Xa)
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),produc738532404422230701BT_nat(X,Xa))
       => ( ! [A3: $o,B2: $o] :
              ( ( X = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B2)) )
             => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),produc738532404422230701BT_nat(aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B2)),Xa))
               => ~ $ite(
                      Xa = zero_zero_nat,
                      (A3),
                      $ite(Xa = one_one_nat,(B2),$false) ) ) )
         => ~ ! [Uy: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S2: vEBT_VEBT] :
                ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy,aa_nat_nat(suc,V2),TreeList),S2) )
               => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy,aa_nat_nat(suc,V2),TreeList),S2),Xa))
                 => ~ $let(
                        pos: nat,
                        pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_V5719532721284313246member(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
tff(fact_3883_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y2: $o] :
      ( ( aa_nat_o(vEBT_V5719532721284313246member(X),Xa)
      <=> (Y2) )
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),produc738532404422230701BT_nat(X,Xa))
       => ( ! [A3: $o,B2: $o] :
              ( ( X = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B2)) )
             => ( ( (Y2)
                <=> $ite(
                      Xa = zero_zero_nat,
                      (A3),
                      $ite(Xa = one_one_nat,(B2),$false) ) )
               => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),produc738532404422230701BT_nat(aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B2)),Xa)) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,zero_zero_nat,Uv2),Uw) )
               => ( ~ (Y2)
                 => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,zero_zero_nat,Uv2),Uw),Xa)) ) )
           => ~ ! [Uy: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy,aa_nat_nat(suc,V2),TreeList),S2) )
                 => ( ( (Y2)
                    <=> $let(
                          pos: nat,
                          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_V5719532721284313246member(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList),pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) )
                   => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy,aa_nat_nat(suc,V2),TreeList),S2),Xa)) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
tff(fact_3884_div2__even__ext__nat,axiom,
    ! [X: nat,Y2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X)
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Y2) )
       => ( X = Y2 ) ) ) ).

% div2_even_ext_nat
tff(fact_3885_vebt__buildup_Oelims,axiom,
    ! [X: nat,Y2: vEBT_VEBT] :
      ( ( aa_nat_VEBT_VEBT(vEBT_vebt_buildup,X) = Y2 )
     => ( ( ( X = zero_zero_nat )
         => ( Y2 != aa_o_VEBT_VEBT(vEBT_Leaf($false),$false) ) )
       => ( ( ( X = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y2 != aa_o_VEBT_VEBT(vEBT_Leaf($false),$false) ) )
         => ~ ! [Va: nat] :
                ( ( X = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
               => ( Y2 != $ite(
                      aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))),
                      $let(
                        half: nat,
                        half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half),aa_nat_VEBT_VEBT(vEBT_vebt_buildup,half))),aa_nat_VEBT_VEBT(vEBT_vebt_buildup,half)) ),
                      $let(
                        half: nat,
                        half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half)),aa_nat_VEBT_VEBT(vEBT_vebt_buildup,half))),aa_nat_VEBT_VEBT(vEBT_vebt_buildup,aa_nat_nat(suc,half))) ) ) ) ) ) ) ) ).

% vebt_buildup.elims
tff(fact_3886_neg__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q: int,R: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
     => ( aa_Pro4592068669816069595_int_o(eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int),B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q),R))
       => aa_Pro4592068669816069595_int_o(eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),R)),one_one_int))) ) ) ).

% neg_eucl_rel_int_mult_2
tff(fact_3887_flip__bit__0,axiom,
    ! [A: code_integer] : bit_se1345352211410354436nteger(zero_zero_nat,A) = plus_p5714425477246183910nteger(zero_n356916108424825756nteger(aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ).

% flip_bit_0
tff(fact_3888_flip__bit__0,axiom,
    ! [A: int] : aa_int_int(bit_se2159334234014336723it_int(zero_zero_nat),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_o_int(zero_n2684676970156552555ol_int,aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ).

% flip_bit_0
tff(fact_3889_flip__bit__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,zero_zero_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% flip_bit_0
tff(fact_3890_add__scale__eq__noteq,axiom,
    ! [R: complex,A: complex,B: complex,C: complex,D: complex] :
      ( ( R != zero_zero_complex )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_complex_complex(plus_plus_complex(A),aa_complex_complex(times_times_complex(R),C)) != aa_complex_complex(plus_plus_complex(B),aa_complex_complex(times_times_complex(R),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_3891_add__scale__eq__noteq,axiom,
    ! [R: real,A: real,B: real,C: real,D: real] :
      ( ( R != zero_zero_real )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_real_real(plus_plus_real(A),aa_real_real(times_times_real(R),C)) != aa_real_real(plus_plus_real(B),aa_real_real(times_times_real(R),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_3892_add__scale__eq__noteq,axiom,
    ! [R: rat,A: rat,B: rat,C: rat,D: rat] :
      ( ( R != zero_zero_rat )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_rat_rat(plus_plus_rat(A),aa_rat_rat(times_times_rat(R),C)) != aa_rat_rat(plus_plus_rat(B),aa_rat_rat(times_times_rat(R),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_3893_add__scale__eq__noteq,axiom,
    ! [R: nat,A: nat,B: nat,C: nat,D: nat] :
      ( ( R != zero_zero_nat )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R),C)) != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_3894_add__scale__eq__noteq,axiom,
    ! [R: int,A: int,B: int,C: int,D: int] :
      ( ( R != zero_zero_int )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,R),C)) != aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(times_times_int,R),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_3895_intind,axiom,
    ! [I2: nat,Nb: nat,P: fun_nat_o,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Nb)
     => ( aa_nat_o(P,X)
       => aa_nat_o(P,aa_nat_nat(nth_nat(replicate_nat(Nb,X)),I2)) ) ) ).

% intind
tff(fact_3896_intind,axiom,
    ! [I2: nat,Nb: nat,P: fun_int_o,X: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Nb)
     => ( aa_int_o(P,X)
       => aa_int_o(P,aa_nat_int(nth_int(replicate_int(Nb,X)),I2)) ) ) ).

% intind
tff(fact_3897_intind,axiom,
    ! [I2: nat,Nb: nat,P: fun_VEBT_VEBT_o,X: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Nb)
     => ( aa_VEBT_VEBT_o(P,X)
       => aa_VEBT_VEBT_o(P,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(replicate_VEBT_VEBT(Nb,X)),I2)) ) ) ).

% intind
tff(fact_3898_diff__self,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(A),A) = zero_zero_complex ).

% diff_self
tff(fact_3899_diff__self,axiom,
    ! [A: real] : aa_real_real(minus_minus_real(A),A) = zero_zero_real ).

% diff_self
tff(fact_3900_diff__self,axiom,
    ! [A: rat] : aa_rat_rat(minus_minus_rat(A),A) = zero_zero_rat ).

% diff_self
tff(fact_3901_diff__self,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),A) = zero_zero_int ).

% diff_self
tff(fact_3902_diff__0__right,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(A),zero_zero_complex) = A ).

% diff_0_right
tff(fact_3903_diff__0__right,axiom,
    ! [A: real] : aa_real_real(minus_minus_real(A),zero_zero_real) = A ).

% diff_0_right
tff(fact_3904_diff__0__right,axiom,
    ! [A: rat] : aa_rat_rat(minus_minus_rat(A),zero_zero_rat) = A ).

% diff_0_right
tff(fact_3905_diff__0__right,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),zero_zero_int) = A ).

% diff_0_right
tff(fact_3906_zero__diff,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,zero_zero_nat),A) = zero_zero_nat ).

% zero_diff
tff(fact_3907_diff__zero,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(A),zero_zero_complex) = A ).

% diff_zero
tff(fact_3908_diff__zero,axiom,
    ! [A: real] : aa_real_real(minus_minus_real(A),zero_zero_real) = A ).

% diff_zero
tff(fact_3909_diff__zero,axiom,
    ! [A: rat] : aa_rat_rat(minus_minus_rat(A),zero_zero_rat) = A ).

% diff_zero
tff(fact_3910_diff__zero,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),zero_zero_nat) = A ).

% diff_zero
tff(fact_3911_diff__zero,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),zero_zero_int) = A ).

% diff_zero
tff(fact_3912_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(A),A) = zero_zero_complex ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_3913_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: real] : aa_real_real(minus_minus_real(A),A) = zero_zero_real ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_3914_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: rat] : aa_rat_rat(minus_minus_rat(A),A) = zero_zero_rat ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_3915_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),A) = zero_zero_nat ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_3916_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),A) = zero_zero_int ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_3917_add__diff__cancel__right_H,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(A),B)),B) = A ).

% add_diff_cancel_right'
tff(fact_3918_add__diff__cancel__right_H,axiom,
    ! [A: real,B: real] : aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(A),B)),B) = A ).

% add_diff_cancel_right'
tff(fact_3919_add__diff__cancel__right_H,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),B)),B) = A ).

% add_diff_cancel_right'
tff(fact_3920_add__diff__cancel__right_H,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B) = A ).

% add_diff_cancel_right'
tff(fact_3921_add__diff__cancel__right_H,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B) = A ).

% add_diff_cancel_right'
tff(fact_3922_add__diff__cancel__right,axiom,
    ! [A: complex,C: complex,B: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(A),C)),aa_complex_complex(plus_plus_complex(B),C)) = aa_complex_complex(minus_minus_complex(A),B) ).

% add_diff_cancel_right
tff(fact_3923_add__diff__cancel__right,axiom,
    ! [A: real,C: real,B: real] : aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(A),C)),aa_real_real(plus_plus_real(B),C)) = aa_real_real(minus_minus_real(A),B) ).

% add_diff_cancel_right
tff(fact_3924_add__diff__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),C)) = aa_rat_rat(minus_minus_rat(A),B) ).

% add_diff_cancel_right
tff(fact_3925_add__diff__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ).

% add_diff_cancel_right
tff(fact_3926_add__diff__cancel__right,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) ).

% add_diff_cancel_right
tff(fact_3927_add__diff__cancel__left_H,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(A),B)),A) = B ).

% add_diff_cancel_left'
tff(fact_3928_add__diff__cancel__left_H,axiom,
    ! [A: real,B: real] : aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(A),B)),A) = B ).

% add_diff_cancel_left'
tff(fact_3929_add__diff__cancel__left_H,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),B)),A) = B ).

% add_diff_cancel_left'
tff(fact_3930_add__diff__cancel__left_H,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),A) = B ).

% add_diff_cancel_left'
tff(fact_3931_add__diff__cancel__left_H,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),A) = B ).

% add_diff_cancel_left'
tff(fact_3932_add__diff__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(C),A)),aa_complex_complex(plus_plus_complex(C),B)) = aa_complex_complex(minus_minus_complex(A),B) ).

% add_diff_cancel_left
tff(fact_3933_add__diff__cancel__left,axiom,
    ! [C: real,A: real,B: real] : aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(C),A)),aa_real_real(plus_plus_real(C),B)) = aa_real_real(minus_minus_real(A),B) ).

% add_diff_cancel_left
tff(fact_3934_add__diff__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(C),A)),aa_rat_rat(plus_plus_rat(C),B)) = aa_rat_rat(minus_minus_rat(A),B) ).

% add_diff_cancel_left
tff(fact_3935_add__diff__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ).

% add_diff_cancel_left
tff(fact_3936_add__diff__cancel__left,axiom,
    ! [C: int,A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) ).

% add_diff_cancel_left
tff(fact_3937_diff__add__cancel,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(minus_minus_complex(A),B)),B) = A ).

% diff_add_cancel
tff(fact_3938_diff__add__cancel,axiom,
    ! [A: real,B: real] : aa_real_real(plus_plus_real(aa_real_real(minus_minus_real(A),B)),B) = A ).

% diff_add_cancel
tff(fact_3939_diff__add__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(A),B)),B) = A ).

% diff_add_cancel
tff(fact_3940_diff__add__cancel,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),B) = A ).

% diff_add_cancel
tff(fact_3941_add__diff__cancel,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(A),B)),B) = A ).

% add_diff_cancel
tff(fact_3942_add__diff__cancel,axiom,
    ! [A: real,B: real] : aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(A),B)),B) = A ).

% add_diff_cancel
tff(fact_3943_add__diff__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),B)),B) = A ).

% add_diff_cancel
tff(fact_3944_add__diff__cancel,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B) = A ).

% add_diff_cancel
tff(fact_3945_minus__mod__self2,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% minus_mod_self2
tff(fact_3946_minus__mod__self2,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(minus_8373710615458151222nteger(A,B),B) = modulo364778990260209775nteger(A,B) ).

% minus_mod_self2
tff(fact_3947_of__bool__less__eq__iff,axiom,
    ! [P: $o,Q2: $o] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_n2052037380579107095ol_rat((P))),zero_n2052037380579107095ol_rat((Q2)))
    <=> ( (P)
       => (Q2) ) ) ).

% of_bool_less_eq_iff
tff(fact_3948_of__bool__less__eq__iff,axiom,
    ! [P: $o,Q2: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(P))),aa_o_nat(zero_n2687167440665602831ol_nat,(Q2)))
    <=> ( (P)
       => (Q2) ) ) ).

% of_bool_less_eq_iff
tff(fact_3949_of__bool__less__eq__iff,axiom,
    ! [P: $o,Q2: $o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_o_int(zero_n2684676970156552555ol_int,(P))),aa_o_int(zero_n2684676970156552555ol_int,(Q2)))
    <=> ( (P)
       => (Q2) ) ) ).

% of_bool_less_eq_iff
tff(fact_3950_of__bool__less__eq__iff,axiom,
    ! [P: $o,Q2: $o] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_n356916108424825756nteger((P))),zero_n356916108424825756nteger((Q2)))
    <=> ( (P)
       => (Q2) ) ) ).

% of_bool_less_eq_iff
tff(fact_3951_of__bool__less__iff,axiom,
    ! [P: $o,Q2: $o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_n3304061248610475627l_real((P))),zero_n3304061248610475627l_real((Q2)))
    <=> ( ~ (P)
        & (Q2) ) ) ).

% of_bool_less_iff
tff(fact_3952_of__bool__less__iff,axiom,
    ! [P: $o,Q2: $o] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_n2052037380579107095ol_rat((P))),zero_n2052037380579107095ol_rat((Q2)))
    <=> ( ~ (P)
        & (Q2) ) ) ).

% of_bool_less_iff
tff(fact_3953_of__bool__less__iff,axiom,
    ! [P: $o,Q2: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(P))),aa_o_nat(zero_n2687167440665602831ol_nat,(Q2)))
    <=> ( ~ (P)
        & (Q2) ) ) ).

% of_bool_less_iff
tff(fact_3954_of__bool__less__iff,axiom,
    ! [P: $o,Q2: $o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_o_int(zero_n2684676970156552555ol_int,(P))),aa_o_int(zero_n2684676970156552555ol_int,(Q2)))
    <=> ( ~ (P)
        & (Q2) ) ) ).

% of_bool_less_iff
tff(fact_3955_of__bool__less__iff,axiom,
    ! [P: $o,Q2: $o] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_n356916108424825756nteger((P))),zero_n356916108424825756nteger((Q2)))
    <=> ( ~ (P)
        & (Q2) ) ) ).

% of_bool_less_iff
tff(fact_3956_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( zero_n1201886186963655149omplex((P)) = one_one_complex )
    <=> (P) ) ).

% of_bool_eq_1_iff
tff(fact_3957_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( zero_n3304061248610475627l_real((P)) = one_one_real )
    <=> (P) ) ).

% of_bool_eq_1_iff
tff(fact_3958_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( zero_n2052037380579107095ol_rat((P)) = one_one_rat )
    <=> (P) ) ).

% of_bool_eq_1_iff
tff(fact_3959_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( aa_o_nat(zero_n2687167440665602831ol_nat,(P)) = one_one_nat )
    <=> (P) ) ).

% of_bool_eq_1_iff
tff(fact_3960_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( aa_o_int(zero_n2684676970156552555ol_int,(P)) = one_one_int )
    <=> (P) ) ).

% of_bool_eq_1_iff
tff(fact_3961_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( zero_n356916108424825756nteger((P)) = one_one_Code_integer )
    <=> (P) ) ).

% of_bool_eq_1_iff
tff(fact_3962_of__bool__eq_I2_J,axiom,
    zero_n1201886186963655149omplex($true) = one_one_complex ).

% of_bool_eq(2)
tff(fact_3963_of__bool__eq_I2_J,axiom,
    zero_n3304061248610475627l_real($true) = one_one_real ).

% of_bool_eq(2)
tff(fact_3964_of__bool__eq_I2_J,axiom,
    zero_n2052037380579107095ol_rat($true) = one_one_rat ).

% of_bool_eq(2)
tff(fact_3965_of__bool__eq_I2_J,axiom,
    aa_o_nat(zero_n2687167440665602831ol_nat,$true) = one_one_nat ).

% of_bool_eq(2)
tff(fact_3966_of__bool__eq_I2_J,axiom,
    aa_o_int(zero_n2684676970156552555ol_int,$true) = one_one_int ).

% of_bool_eq(2)
tff(fact_3967_of__bool__eq_I2_J,axiom,
    zero_n356916108424825756nteger($true) = one_one_Code_integer ).

% of_bool_eq(2)
tff(fact_3968_replicate__eq__replicate,axiom,
    ! [Ma: nat,X: vEBT_VEBT,Nb: nat,Y2: vEBT_VEBT] :
      ( ( replicate_VEBT_VEBT(Ma,X) = replicate_VEBT_VEBT(Nb,Y2) )
    <=> ( ( Ma = Nb )
        & ( ( Ma != zero_zero_nat )
         => ( X = Y2 ) ) ) ) ).

% replicate_eq_replicate
tff(fact_3969_length__replicate,axiom,
    ! [Nb: nat,X: vEBT_VEBT] : size_s6755466524823107622T_VEBT(replicate_VEBT_VEBT(Nb,X)) = Nb ).

% length_replicate
tff(fact_3970_length__replicate,axiom,
    ! [Nb: nat,X: $o] : size_size_list_o(replicate_o(Nb,(X))) = Nb ).

% length_replicate
tff(fact_3971_length__replicate,axiom,
    ! [Nb: nat,X: nat] : size_size_list_nat(replicate_nat(Nb,X)) = Nb ).

% length_replicate
tff(fact_3972_length__replicate,axiom,
    ! [Nb: nat,X: int] : size_size_list_int(replicate_int(Nb,X)) = Nb ).

% length_replicate
tff(fact_3973_map__replicate,axiom,
    ! [F: fun_nat_nat,Nb: nat,X: nat] : map_nat_nat(F,replicate_nat(Nb,X)) = replicate_nat(Nb,aa_nat_nat(F,X)) ).

% map_replicate
tff(fact_3974_map__replicate,axiom,
    ! [F: fun_VE3165209603783583043T_VEBT,Nb: nat,X: vEBT_VEBT] : map_VE8901447254227204932T_VEBT(F,replicate_VEBT_VEBT(Nb,X)) = replicate_VEBT_VEBT(Nb,aa_VEB5832953974749077948T_VEBT(F,X)) ).

% map_replicate
tff(fact_3975_diff__ge__0__iff__ge,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(minus_minus_real(A),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ).

% diff_ge_0_iff_ge
tff(fact_3976_diff__ge__0__iff__ge,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(minus_minus_rat(A),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ).

% diff_ge_0_iff_ge
tff(fact_3977_diff__ge__0__iff__ge,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ).

% diff_ge_0_iff_ge
tff(fact_3978_diff__gt__0__iff__gt,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(minus_minus_real(A),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ).

% diff_gt_0_iff_gt
tff(fact_3979_diff__gt__0__iff__gt,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(minus_minus_rat(A),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ).

% diff_gt_0_iff_gt
tff(fact_3980_diff__gt__0__iff__gt,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ).

% diff_gt_0_iff_gt
tff(fact_3981_le__add__diff__inverse2,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_real(plus_plus_real(aa_real_real(minus_minus_real(A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_3982_le__add__diff__inverse2,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_3983_le__add__diff__inverse2,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_3984_le__add__diff__inverse2,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_3985_le__add__diff__inverse,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_real(plus_plus_real(B),aa_real_real(minus_minus_real(A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_3986_le__add__diff__inverse,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_rat(plus_plus_rat(B),aa_rat_rat(minus_minus_rat(A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_3987_le__add__diff__inverse,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_3988_le__add__diff__inverse,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_3989_diff__numeral__special_I9_J,axiom,
    aa_complex_complex(minus_minus_complex(one_one_complex),one_one_complex) = zero_zero_complex ).

% diff_numeral_special(9)
tff(fact_3990_diff__numeral__special_I9_J,axiom,
    aa_real_real(minus_minus_real(one_one_real),one_one_real) = zero_zero_real ).

% diff_numeral_special(9)
tff(fact_3991_diff__numeral__special_I9_J,axiom,
    aa_rat_rat(minus_minus_rat(one_one_rat),one_one_rat) = zero_zero_rat ).

% diff_numeral_special(9)
tff(fact_3992_diff__numeral__special_I9_J,axiom,
    aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),one_one_int) = zero_zero_int ).

% diff_numeral_special(9)
tff(fact_3993_diff__add__zero,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) = zero_zero_nat ).

% diff_add_zero
tff(fact_3994_right__diff__distrib__numeral,axiom,
    ! [V: num,B: complex,C: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),aa_complex_complex(minus_minus_complex(B),C)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),B)),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),C)) ).

% right_diff_distrib_numeral
tff(fact_3995_right__diff__distrib__numeral,axiom,
    ! [V: num,B: real,C: real] : aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,V)),aa_real_real(minus_minus_real(B),C)) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,V)),B)),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,V)),C)) ).

% right_diff_distrib_numeral
tff(fact_3996_right__diff__distrib__numeral,axiom,
    ! [V: num,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(minus_minus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),B)),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),C)) ).

% right_diff_distrib_numeral
tff(fact_3997_right__diff__distrib__numeral,axiom,
    ! [V: num,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),B)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),C)) ).

% right_diff_distrib_numeral
tff(fact_3998_left__diff__distrib__numeral,axiom,
    ! [A: complex,B: complex,V: num] : aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(A),B)),numera6690914467698888265omplex(V)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(V))),aa_complex_complex(times_times_complex(B),numera6690914467698888265omplex(V))) ).

% left_diff_distrib_numeral
tff(fact_3999_left__diff__distrib__numeral,axiom,
    ! [A: real,B: real,V: num] : aa_real_real(times_times_real(aa_real_real(minus_minus_real(A),B)),aa_num_real(numeral_numeral_real,V)) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(A),aa_num_real(numeral_numeral_real,V))),aa_real_real(times_times_real(B),aa_num_real(numeral_numeral_real,V))) ).

% left_diff_distrib_numeral
tff(fact_4000_left__diff__distrib__numeral,axiom,
    ! [A: rat,B: rat,V: num] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A),B)),aa_num_rat(numeral_numeral_rat,V)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(times_times_rat(B),aa_num_rat(numeral_numeral_rat,V))) ).

% left_diff_distrib_numeral
tff(fact_4001_left__diff__distrib__numeral,axiom,
    ! [A: int,B: int,V: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),aa_num_int(numeral_numeral_int,V)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_num_int(numeral_numeral_int,V))),aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_num_int(numeral_numeral_int,V))) ).

% left_diff_distrib_numeral
tff(fact_4002_div__diff,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B)
       => ( divide6298287555418463151nteger(minus_8373710615458151222nteger(A,B),C) = minus_8373710615458151222nteger(divide6298287555418463151nteger(A,C),divide6298287555418463151nteger(B,C)) ) ) ) ).

% div_diff
tff(fact_4003_div__diff,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) ) ) ) ).

% div_diff
tff(fact_4004_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),zero_n3304061248610475627l_real((P)))
    <=> (P) ) ).

% zero_less_of_bool_iff
tff(fact_4005_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),zero_n2052037380579107095ol_rat((P)))
    <=> (P) ) ).

% zero_less_of_bool_iff
tff(fact_4006_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_o_nat(zero_n2687167440665602831ol_nat,(P)))
    <=> (P) ) ).

% zero_less_of_bool_iff
tff(fact_4007_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_o_int(zero_n2684676970156552555ol_int,(P)))
    <=> (P) ) ).

% zero_less_of_bool_iff
tff(fact_4008_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),zero_n356916108424825756nteger((P)))
    <=> (P) ) ).

% zero_less_of_bool_iff
tff(fact_4009_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_n3304061248610475627l_real((P))),one_one_real)
    <=> ~ (P) ) ).

% of_bool_less_one_iff
tff(fact_4010_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_n2052037380579107095ol_rat((P))),one_one_rat)
    <=> ~ (P) ) ).

% of_bool_less_one_iff
tff(fact_4011_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(P))),one_one_nat)
    <=> ~ (P) ) ).

% of_bool_less_one_iff
tff(fact_4012_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_o_int(zero_n2684676970156552555ol_int,(P))),one_one_int)
    <=> ~ (P) ) ).

% of_bool_less_one_iff
tff(fact_4013_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_n356916108424825756nteger((P))),one_one_Code_integer)
    <=> ~ (P) ) ).

% of_bool_less_one_iff
tff(fact_4014_of__bool__not__iff,axiom,
    ! [P: $o] : zero_n1201886186963655149omplex(~ (P)) = aa_complex_complex(minus_minus_complex(one_one_complex),zero_n1201886186963655149omplex((P))) ).

% of_bool_not_iff
tff(fact_4015_of__bool__not__iff,axiom,
    ! [P: $o] : zero_n3304061248610475627l_real(~ (P)) = aa_real_real(minus_minus_real(one_one_real),zero_n3304061248610475627l_real((P))) ).

% of_bool_not_iff
tff(fact_4016_of__bool__not__iff,axiom,
    ! [P: $o] : zero_n2052037380579107095ol_rat(~ (P)) = aa_rat_rat(minus_minus_rat(one_one_rat),zero_n2052037380579107095ol_rat((P))) ).

% of_bool_not_iff
tff(fact_4017_of__bool__not__iff,axiom,
    ! [P: $o] : aa_o_int(zero_n2684676970156552555ol_int,~ (P)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_o_int(zero_n2684676970156552555ol_int,(P))) ).

% of_bool_not_iff
tff(fact_4018_of__bool__not__iff,axiom,
    ! [P: $o] : zero_n356916108424825756nteger(~ (P)) = minus_8373710615458151222nteger(one_one_Code_integer,zero_n356916108424825756nteger((P))) ).

% of_bool_not_iff
tff(fact_4019_Suc__0__mod__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = aa_o_nat(zero_n2687167440665602831ol_nat,Nb != aa_nat_nat(suc,zero_zero_nat)) ).

% Suc_0_mod_eq
tff(fact_4020_in__set__replicate,axiom,
    ! [X: complex,Nb: nat,Y2: complex] :
      ( member_complex(X,set_complex2(replicate_complex(Nb,Y2)))
    <=> ( ( X = Y2 )
        & ( Nb != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_4021_in__set__replicate,axiom,
    ! [X: real,Nb: nat,Y2: real] :
      ( member_real(X,set_real2(replicate_real(Nb,Y2)))
    <=> ( ( X = Y2 )
        & ( Nb != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_4022_in__set__replicate,axiom,
    ! [X: set_nat,Nb: nat,Y2: set_nat] :
      ( member_set_nat(X,set_set_nat2(replicate_set_nat(Nb,Y2)))
    <=> ( ( X = Y2 )
        & ( Nb != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_4023_in__set__replicate,axiom,
    ! [X: nat,Nb: nat,Y2: nat] :
      ( aa_set_nat_o(member_nat(X),set_nat2(replicate_nat(Nb,Y2)))
    <=> ( ( X = Y2 )
        & ( Nb != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_4024_in__set__replicate,axiom,
    ! [X: int,Nb: nat,Y2: int] :
      ( member_int(X,set_int2(replicate_int(Nb,Y2)))
    <=> ( ( X = Y2 )
        & ( Nb != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_4025_in__set__replicate,axiom,
    ! [X: vEBT_VEBT,Nb: nat,Y2: vEBT_VEBT] :
      ( member_VEBT_VEBT(X,set_VEBT_VEBT2(replicate_VEBT_VEBT(Nb,Y2)))
    <=> ( ( X = Y2 )
        & ( Nb != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_4026_Bex__set__replicate,axiom,
    ! [Nb: nat,A: nat,P: fun_nat_o] :
      ( ? [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),set_nat2(replicate_nat(Nb,A)))
          & aa_nat_o(P,X3) )
    <=> ( aa_nat_o(P,A)
        & ( Nb != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_4027_Bex__set__replicate,axiom,
    ! [Nb: nat,A: int,P: fun_int_o] :
      ( ? [X3: int] :
          ( member_int(X3,set_int2(replicate_int(Nb,A)))
          & aa_int_o(P,X3) )
    <=> ( aa_int_o(P,A)
        & ( Nb != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_4028_Bex__set__replicate,axiom,
    ! [Nb: nat,A: vEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( ? [X3: vEBT_VEBT] :
          ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(replicate_VEBT_VEBT(Nb,A)))
          & aa_VEBT_VEBT_o(P,X3) )
    <=> ( aa_VEBT_VEBT_o(P,A)
        & ( Nb != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_4029_Ball__set__replicate,axiom,
    ! [Nb: nat,A: nat,P: fun_nat_o] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),set_nat2(replicate_nat(Nb,A)))
         => aa_nat_o(P,X3) )
    <=> ( aa_nat_o(P,A)
        | ( Nb = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_4030_Ball__set__replicate,axiom,
    ! [Nb: nat,A: int,P: fun_int_o] :
      ( ! [X3: int] :
          ( member_int(X3,set_int2(replicate_int(Nb,A)))
         => aa_int_o(P,X3) )
    <=> ( aa_int_o(P,A)
        | ( Nb = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_4031_Ball__set__replicate,axiom,
    ! [Nb: nat,A: vEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( ! [X3: vEBT_VEBT] :
          ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(replicate_VEBT_VEBT(Nb,A)))
         => aa_VEBT_VEBT_o(P,X3) )
    <=> ( aa_VEBT_VEBT_o(P,A)
        | ( Nb = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_4032_nth__replicate,axiom,
    ! [I2: nat,Nb: nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Nb)
     => ( aa_nat_nat(nth_nat(replicate_nat(Nb,X)),I2) = X ) ) ).

% nth_replicate
tff(fact_4033_nth__replicate,axiom,
    ! [I2: nat,Nb: nat,X: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Nb)
     => ( aa_nat_int(nth_int(replicate_int(Nb,X)),I2) = X ) ) ).

% nth_replicate
tff(fact_4034_nth__replicate,axiom,
    ! [I2: nat,Nb: nat,X: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Nb)
     => ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(replicate_VEBT_VEBT(Nb,X)),I2) = X ) ) ).

% nth_replicate
tff(fact_4035_zle__diff1__eq,axiom,
    ! [W: int,Z2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),aa_int_int(aa_int_fun_int_int(minus_minus_int,Z2),one_one_int))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z2) ) ).

% zle_diff1_eq
tff(fact_4036_odd__of__bool__self,axiom,
    ! [P2: $o] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_o_nat(zero_n2687167440665602831ol_nat,(P2)))
    <=> (P2) ) ).

% odd_of_bool_self
tff(fact_4037_odd__of__bool__self,axiom,
    ! [P2: $o] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_o_int(zero_n2684676970156552555ol_int,(P2)))
    <=> (P2) ) ).

% odd_of_bool_self
tff(fact_4038_odd__of__bool__self,axiom,
    ! [P2: $o] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_n356916108424825756nteger((P2)))
    <=> (P2) ) ).

% odd_of_bool_self
tff(fact_4039_even__diff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),minus_8373710615458151222nteger(A,B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,B)) ) ).

% even_diff
tff(fact_4040_even__diff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ).

% even_diff
tff(fact_4041_of__bool__half__eq__0,axiom,
    ! [B: $o] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(B))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% of_bool_half_eq_0
tff(fact_4042_of__bool__half__eq__0,axiom,
    ! [B: $o] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_o_int(zero_n2684676970156552555ol_int,(B))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ).

% of_bool_half_eq_0
tff(fact_4043_of__bool__half__eq__0,axiom,
    ! [B: $o] : divide6298287555418463151nteger(zero_n356916108424825756nteger((B)),numera6620942414471956472nteger(aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ).

% of_bool_half_eq_0
tff(fact_4044_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb),one_one_Code_integer))
    <=> ( Nb = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_4045_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),one_one_nat))
    <=> ( Nb = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_4046_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),one_one_int))
    <=> ( Nb = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_4047_one__div__2__pow__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_o_nat(zero_n2687167440665602831ol_nat,Nb = zero_zero_nat) ).

% one_div_2_pow_eq
tff(fact_4048_one__div__2__pow__eq,axiom,
    ! [Nb: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_o_int(zero_n2684676970156552555ol_int,Nb = zero_zero_nat) ).

% one_div_2_pow_eq
tff(fact_4049_one__div__2__pow__eq,axiom,
    ! [Nb: nat] : divide6298287555418463151nteger(one_one_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)) = zero_n356916108424825756nteger(Nb = zero_zero_nat) ).

% one_div_2_pow_eq
tff(fact_4050_bits__1__div__exp,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_o_nat(zero_n2687167440665602831ol_nat,Nb = zero_zero_nat) ).

% bits_1_div_exp
tff(fact_4051_bits__1__div__exp,axiom,
    ! [Nb: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_o_int(zero_n2684676970156552555ol_int,Nb = zero_zero_nat) ).

% bits_1_div_exp
tff(fact_4052_bits__1__div__exp,axiom,
    ! [Nb: nat] : divide6298287555418463151nteger(one_one_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)) = zero_n356916108424825756nteger(Nb = zero_zero_nat) ).

% bits_1_div_exp
tff(fact_4053_one__mod__2__pow__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).

% one_mod_2_pow_eq
tff(fact_4054_one__mod__2__pow__eq,axiom,
    ! [Nb: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,one_one_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).

% one_mod_2_pow_eq
tff(fact_4055_one__mod__2__pow__eq,axiom,
    ! [Nb: nat] : modulo364778990260209775nteger(one_one_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)) = zero_n356916108424825756nteger(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).

% one_mod_2_pow_eq
tff(fact_4056_signed__take__bit__diff,axiom,
    ! [Nb: nat,K2: int,L: int] : aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(bit_ri631733984087533419it_int(Nb),K2)),aa_int_int(bit_ri631733984087533419it_int(Nb),L))) = aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(aa_int_fun_int_int(minus_minus_int,K2),L)) ).

% signed_take_bit_diff
tff(fact_4057_dvd__antisym,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ma),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),Ma)
       => ( Ma = Nb ) ) ) ).

% dvd_antisym
tff(fact_4058_diff__eq__diff__eq,axiom,
    ! [A: complex,B: complex,C: complex,D: complex] :
      ( ( aa_complex_complex(minus_minus_complex(A),B) = aa_complex_complex(minus_minus_complex(C),D) )
     => ( ( A = B )
      <=> ( C = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_4059_diff__eq__diff__eq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( aa_real_real(minus_minus_real(A),B) = aa_real_real(minus_minus_real(C),D) )
     => ( ( A = B )
      <=> ( C = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_4060_diff__eq__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(minus_minus_rat(C),D) )
     => ( ( A = B )
      <=> ( C = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_4061_diff__eq__diff__eq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D) )
     => ( ( A = B )
      <=> ( C = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_4062_diff__right__commute,axiom,
    ! [A: complex,C: complex,B: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(minus_minus_complex(A),C)),B) = aa_complex_complex(minus_minus_complex(aa_complex_complex(minus_minus_complex(A),B)),C) ).

% diff_right_commute
tff(fact_4063_diff__right__commute,axiom,
    ! [A: real,C: real,B: real] : aa_real_real(minus_minus_real(aa_real_real(minus_minus_real(A),C)),B) = aa_real_real(minus_minus_real(aa_real_real(minus_minus_real(A),B)),C) ).

% diff_right_commute
tff(fact_4064_diff__right__commute,axiom,
    ! [A: rat,C: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A),C)),B) = aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A),B)),C) ).

% diff_right_commute
tff(fact_4065_diff__right__commute,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),C)),B) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)),C) ).

% diff_right_commute
tff(fact_4066_diff__right__commute,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) ).

% diff_right_commute
tff(fact_4067_of__bool__conj,axiom,
    ! [P: $o,Q2: $o] :
      zero_n3304061248610475627l_real(( (P)
        & (Q2) )) = aa_real_real(times_times_real(zero_n3304061248610475627l_real((P))),zero_n3304061248610475627l_real((Q2))) ).

% of_bool_conj
tff(fact_4068_of__bool__conj,axiom,
    ! [P: $o,Q2: $o] :
      zero_n2052037380579107095ol_rat(( (P)
        & (Q2) )) = aa_rat_rat(times_times_rat(zero_n2052037380579107095ol_rat((P))),zero_n2052037380579107095ol_rat((Q2))) ).

% of_bool_conj
tff(fact_4069_of__bool__conj,axiom,
    ! [P: $o,Q2: $o] :
      aa_o_nat(zero_n2687167440665602831ol_nat,
        ( (P)
        & (Q2) )) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(P))),aa_o_nat(zero_n2687167440665602831ol_nat,(Q2))) ).

% of_bool_conj
tff(fact_4070_of__bool__conj,axiom,
    ! [P: $o,Q2: $o] :
      aa_o_int(zero_n2684676970156552555ol_int,
        ( (P)
        & (Q2) )) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,(P))),aa_o_int(zero_n2684676970156552555ol_int,(Q2))) ).

% of_bool_conj
tff(fact_4071_of__bool__conj,axiom,
    ! [P: $o,Q2: $o] :
      zero_n356916108424825756nteger(( (P)
        & (Q2) )) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_n356916108424825756nteger((P))),zero_n356916108424825756nteger((Q2))) ).

% of_bool_conj
tff(fact_4072_diff__eq__diff__less__eq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( aa_real_real(minus_minus_real(A),B) = aa_real_real(minus_minus_real(C),D) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D) ) ) ).

% diff_eq_diff_less_eq
tff(fact_4073_diff__eq__diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(minus_minus_rat(C),D) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D) ) ) ).

% diff_eq_diff_less_eq
tff(fact_4074_diff__eq__diff__less__eq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D) ) ) ).

% diff_eq_diff_less_eq
tff(fact_4075_diff__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(A),C)),aa_real_real(minus_minus_real(B),C)) ) ).

% diff_right_mono
tff(fact_4076_diff__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),C)),aa_rat_rat(minus_minus_rat(B),C)) ) ).

% diff_right_mono
tff(fact_4077_diff__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) ) ).

% diff_right_mono
tff(fact_4078_diff__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(C),A)),aa_real_real(minus_minus_real(C),B)) ) ).

% diff_left_mono
tff(fact_4079_diff__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(C),A)),aa_rat_rat(minus_minus_rat(C),B)) ) ).

% diff_left_mono
tff(fact_4080_diff__left__mono,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,C),A)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B)) ) ).

% diff_left_mono
tff(fact_4081_diff__mono,axiom,
    ! [A: real,B: real,D: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,D),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(A),C)),aa_real_real(minus_minus_real(B),D)) ) ) ).

% diff_mono
tff(fact_4082_diff__mono,axiom,
    ! [A: rat,B: rat,D: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,D),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),C)),aa_rat_rat(minus_minus_rat(B),D)) ) ) ).

% diff_mono
tff(fact_4083_diff__mono,axiom,
    ! [A: int,B: int,D: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,D),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D)) ) ) ).

% diff_mono
tff(fact_4084_eq__iff__diff__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( A = B )
    <=> ( aa_complex_complex(minus_minus_complex(A),B) = zero_zero_complex ) ) ).

% eq_iff_diff_eq_0
tff(fact_4085_eq__iff__diff__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( A = B )
    <=> ( aa_real_real(minus_minus_real(A),B) = zero_zero_real ) ) ).

% eq_iff_diff_eq_0
tff(fact_4086_eq__iff__diff__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
    <=> ( aa_rat_rat(minus_minus_rat(A),B) = zero_zero_rat ) ) ).

% eq_iff_diff_eq_0
tff(fact_4087_eq__iff__diff__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
    <=> ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = zero_zero_int ) ) ).

% eq_iff_diff_eq_0
tff(fact_4088_diff__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(A),C)),aa_real_real(minus_minus_real(B),C)) ) ).

% diff_strict_right_mono
tff(fact_4089_diff__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(A),C)),aa_rat_rat(minus_minus_rat(B),C)) ) ).

% diff_strict_right_mono
tff(fact_4090_diff__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) ) ).

% diff_strict_right_mono
tff(fact_4091_diff__strict__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(C),A)),aa_real_real(minus_minus_real(C),B)) ) ).

% diff_strict_left_mono
tff(fact_4092_diff__strict__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(C),A)),aa_rat_rat(minus_minus_rat(C),B)) ) ).

% diff_strict_left_mono
tff(fact_4093_diff__strict__left__mono,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,C),A)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B)) ) ).

% diff_strict_left_mono
tff(fact_4094_diff__eq__diff__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( aa_real_real(minus_minus_real(A),B) = aa_real_real(minus_minus_real(C),D) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,C),D) ) ) ).

% diff_eq_diff_less
tff(fact_4095_diff__eq__diff__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(minus_minus_rat(C),D) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D) ) ) ).

% diff_eq_diff_less
tff(fact_4096_diff__eq__diff__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,C),D) ) ) ).

% diff_eq_diff_less
tff(fact_4097_diff__strict__mono,axiom,
    ! [A: real,B: real,D: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,D),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(A),C)),aa_real_real(minus_minus_real(B),D)) ) ) ).

% diff_strict_mono
tff(fact_4098_diff__strict__mono,axiom,
    ! [A: rat,B: rat,D: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,D),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(A),C)),aa_rat_rat(minus_minus_rat(B),D)) ) ) ).

% diff_strict_mono
tff(fact_4099_diff__strict__mono,axiom,
    ! [A: int,B: int,D: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,D),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D)) ) ) ).

% diff_strict_mono
tff(fact_4100_right__diff__distrib_H,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(times_times_complex(A),aa_complex_complex(minus_minus_complex(B),C)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(A),B)),aa_complex_complex(times_times_complex(A),C)) ).

% right_diff_distrib'
tff(fact_4101_right__diff__distrib_H,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(times_times_real(A),aa_real_real(minus_minus_real(B),C)) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(A),B)),aa_real_real(times_times_real(A),C)) ).

% right_diff_distrib'
tff(fact_4102_right__diff__distrib_H,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),aa_rat_rat(minus_minus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A),B)),aa_rat_rat(times_times_rat(A),C)) ).

% right_diff_distrib'
tff(fact_4103_right__diff__distrib_H,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)) ).

% right_diff_distrib'
tff(fact_4104_right__diff__distrib_H,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ).

% right_diff_distrib'
tff(fact_4105_left__diff__distrib_H,axiom,
    ! [B: complex,C: complex,A: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(B),C)),A) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(B),A)),aa_complex_complex(times_times_complex(C),A)) ).

% left_diff_distrib'
tff(fact_4106_left__diff__distrib_H,axiom,
    ! [B: real,C: real,A: real] : aa_real_real(times_times_real(aa_real_real(minus_minus_real(B),C)),A) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(B),A)),aa_real_real(times_times_real(C),A)) ).

% left_diff_distrib'
tff(fact_4107_left__diff__distrib_H,axiom,
    ! [B: rat,C: rat,A: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(B),C)),A) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(B),A)),aa_rat_rat(times_times_rat(C),A)) ).

% left_diff_distrib'
tff(fact_4108_left__diff__distrib_H,axiom,
    ! [B: nat,C: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),C)),A) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)) ).

% left_diff_distrib'
tff(fact_4109_left__diff__distrib_H,axiom,
    ! [B: int,C: int,A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)),A) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),A)) ).

% left_diff_distrib'
tff(fact_4110_right__diff__distrib,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(times_times_complex(A),aa_complex_complex(minus_minus_complex(B),C)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(A),B)),aa_complex_complex(times_times_complex(A),C)) ).

% right_diff_distrib
tff(fact_4111_right__diff__distrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(times_times_real(A),aa_real_real(minus_minus_real(B),C)) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(A),B)),aa_real_real(times_times_real(A),C)) ).

% right_diff_distrib
tff(fact_4112_right__diff__distrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),aa_rat_rat(minus_minus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A),B)),aa_rat_rat(times_times_rat(A),C)) ).

% right_diff_distrib
tff(fact_4113_right__diff__distrib,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ).

% right_diff_distrib
tff(fact_4114_left__diff__distrib,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(A),B)),C) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(A),C)),aa_complex_complex(times_times_complex(B),C)) ).

% left_diff_distrib
tff(fact_4115_left__diff__distrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(times_times_real(aa_real_real(minus_minus_real(A),B)),C) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),C)) ).

% left_diff_distrib
tff(fact_4116_left__diff__distrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A),B)),C) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) ).

% left_diff_distrib
tff(fact_4117_left__diff__distrib,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% left_diff_distrib
tff(fact_4118_inf__period_I2_J,axiom,
    ! [P: fun_complex_o,D3: complex,Q2: fun_complex_o] :
      ( ! [X4: complex,K: complex] :
          ( aa_complex_o(P,X4)
        <=> aa_complex_o(P,aa_complex_complex(minus_minus_complex(X4),aa_complex_complex(times_times_complex(K),D3))) )
     => ( ! [X4: complex,K: complex] :
            ( aa_complex_o(Q2,X4)
          <=> aa_complex_o(Q2,aa_complex_complex(minus_minus_complex(X4),aa_complex_complex(times_times_complex(K),D3))) )
       => ! [X2: complex,K4: complex] :
            ( ( aa_complex_o(P,X2)
              | aa_complex_o(Q2,X2) )
          <=> ( aa_complex_o(P,aa_complex_complex(minus_minus_complex(X2),aa_complex_complex(times_times_complex(K4),D3)))
              | aa_complex_o(Q2,aa_complex_complex(minus_minus_complex(X2),aa_complex_complex(times_times_complex(K4),D3))) ) ) ) ) ).

% inf_period(2)
tff(fact_4119_inf__period_I2_J,axiom,
    ! [P: fun_real_o,D3: real,Q2: fun_real_o] :
      ( ! [X4: real,K: real] :
          ( aa_real_o(P,X4)
        <=> aa_real_o(P,aa_real_real(minus_minus_real(X4),aa_real_real(times_times_real(K),D3))) )
     => ( ! [X4: real,K: real] :
            ( aa_real_o(Q2,X4)
          <=> aa_real_o(Q2,aa_real_real(minus_minus_real(X4),aa_real_real(times_times_real(K),D3))) )
       => ! [X2: real,K4: real] :
            ( ( aa_real_o(P,X2)
              | aa_real_o(Q2,X2) )
          <=> ( aa_real_o(P,aa_real_real(minus_minus_real(X2),aa_real_real(times_times_real(K4),D3)))
              | aa_real_o(Q2,aa_real_real(minus_minus_real(X2),aa_real_real(times_times_real(K4),D3))) ) ) ) ) ).

% inf_period(2)
tff(fact_4120_inf__period_I2_J,axiom,
    ! [P: fun_rat_o,D3: rat,Q2: fun_rat_o] :
      ( ! [X4: rat,K: rat] :
          ( aa_rat_o(P,X4)
        <=> aa_rat_o(P,aa_rat_rat(minus_minus_rat(X4),aa_rat_rat(times_times_rat(K),D3))) )
     => ( ! [X4: rat,K: rat] :
            ( aa_rat_o(Q2,X4)
          <=> aa_rat_o(Q2,aa_rat_rat(minus_minus_rat(X4),aa_rat_rat(times_times_rat(K),D3))) )
       => ! [X2: rat,K4: rat] :
            ( ( aa_rat_o(P,X2)
              | aa_rat_o(Q2,X2) )
          <=> ( aa_rat_o(P,aa_rat_rat(minus_minus_rat(X2),aa_rat_rat(times_times_rat(K4),D3)))
              | aa_rat_o(Q2,aa_rat_rat(minus_minus_rat(X2),aa_rat_rat(times_times_rat(K4),D3))) ) ) ) ) ).

% inf_period(2)
tff(fact_4121_inf__period_I2_J,axiom,
    ! [P: fun_int_o,D3: int,Q2: fun_int_o] :
      ( ! [X4: int,K: int] :
          ( aa_int_o(P,X4)
        <=> aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K),D3))) )
     => ( ! [X4: int,K: int] :
            ( aa_int_o(Q2,X4)
          <=> aa_int_o(Q2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K),D3))) )
       => ! [X2: int,K4: int] :
            ( ( aa_int_o(P,X2)
              | aa_int_o(Q2,X2) )
          <=> ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D3)))
              | aa_int_o(Q2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D3))) ) ) ) ) ).

% inf_period(2)
tff(fact_4122_inf__period_I1_J,axiom,
    ! [P: fun_complex_o,D3: complex,Q2: fun_complex_o] :
      ( ! [X4: complex,K: complex] :
          ( aa_complex_o(P,X4)
        <=> aa_complex_o(P,aa_complex_complex(minus_minus_complex(X4),aa_complex_complex(times_times_complex(K),D3))) )
     => ( ! [X4: complex,K: complex] :
            ( aa_complex_o(Q2,X4)
          <=> aa_complex_o(Q2,aa_complex_complex(minus_minus_complex(X4),aa_complex_complex(times_times_complex(K),D3))) )
       => ! [X2: complex,K4: complex] :
            ( ( aa_complex_o(P,X2)
              & aa_complex_o(Q2,X2) )
          <=> ( aa_complex_o(P,aa_complex_complex(minus_minus_complex(X2),aa_complex_complex(times_times_complex(K4),D3)))
              & aa_complex_o(Q2,aa_complex_complex(minus_minus_complex(X2),aa_complex_complex(times_times_complex(K4),D3))) ) ) ) ) ).

% inf_period(1)
tff(fact_4123_inf__period_I1_J,axiom,
    ! [P: fun_real_o,D3: real,Q2: fun_real_o] :
      ( ! [X4: real,K: real] :
          ( aa_real_o(P,X4)
        <=> aa_real_o(P,aa_real_real(minus_minus_real(X4),aa_real_real(times_times_real(K),D3))) )
     => ( ! [X4: real,K: real] :
            ( aa_real_o(Q2,X4)
          <=> aa_real_o(Q2,aa_real_real(minus_minus_real(X4),aa_real_real(times_times_real(K),D3))) )
       => ! [X2: real,K4: real] :
            ( ( aa_real_o(P,X2)
              & aa_real_o(Q2,X2) )
          <=> ( aa_real_o(P,aa_real_real(minus_minus_real(X2),aa_real_real(times_times_real(K4),D3)))
              & aa_real_o(Q2,aa_real_real(minus_minus_real(X2),aa_real_real(times_times_real(K4),D3))) ) ) ) ) ).

% inf_period(1)
tff(fact_4124_inf__period_I1_J,axiom,
    ! [P: fun_rat_o,D3: rat,Q2: fun_rat_o] :
      ( ! [X4: rat,K: rat] :
          ( aa_rat_o(P,X4)
        <=> aa_rat_o(P,aa_rat_rat(minus_minus_rat(X4),aa_rat_rat(times_times_rat(K),D3))) )
     => ( ! [X4: rat,K: rat] :
            ( aa_rat_o(Q2,X4)
          <=> aa_rat_o(Q2,aa_rat_rat(minus_minus_rat(X4),aa_rat_rat(times_times_rat(K),D3))) )
       => ! [X2: rat,K4: rat] :
            ( ( aa_rat_o(P,X2)
              & aa_rat_o(Q2,X2) )
          <=> ( aa_rat_o(P,aa_rat_rat(minus_minus_rat(X2),aa_rat_rat(times_times_rat(K4),D3)))
              & aa_rat_o(Q2,aa_rat_rat(minus_minus_rat(X2),aa_rat_rat(times_times_rat(K4),D3))) ) ) ) ) ).

% inf_period(1)
tff(fact_4125_inf__period_I1_J,axiom,
    ! [P: fun_int_o,D3: int,Q2: fun_int_o] :
      ( ! [X4: int,K: int] :
          ( aa_int_o(P,X4)
        <=> aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K),D3))) )
     => ( ! [X4: int,K: int] :
            ( aa_int_o(Q2,X4)
          <=> aa_int_o(Q2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K),D3))) )
       => ! [X2: int,K4: int] :
            ( ( aa_int_o(P,X2)
              & aa_int_o(Q2,X2) )
          <=> ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D3)))
              & aa_int_o(Q2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D3))) ) ) ) ) ).

% inf_period(1)
tff(fact_4126_diff__diff__eq,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(minus_minus_complex(A),B)),C) = aa_complex_complex(minus_minus_complex(A),aa_complex_complex(plus_plus_complex(B),C)) ).

% diff_diff_eq
tff(fact_4127_diff__diff__eq,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(minus_minus_real(aa_real_real(minus_minus_real(A),B)),C) = aa_real_real(minus_minus_real(A),aa_real_real(plus_plus_real(B),C)) ).

% diff_diff_eq
tff(fact_4128_diff__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A),B)),C) = aa_rat_rat(minus_minus_rat(A),aa_rat_rat(plus_plus_rat(B),C)) ).

% diff_diff_eq
tff(fact_4129_diff__diff__eq,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ).

% diff_diff_eq
tff(fact_4130_diff__diff__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ).

% diff_diff_eq
tff(fact_4131_add__implies__diff,axiom,
    ! [C: complex,B: complex,A: complex] :
      ( ( aa_complex_complex(plus_plus_complex(C),B) = A )
     => ( C = aa_complex_complex(minus_minus_complex(A),B) ) ) ).

% add_implies_diff
tff(fact_4132_add__implies__diff,axiom,
    ! [C: real,B: real,A: real] :
      ( ( aa_real_real(plus_plus_real(C),B) = A )
     => ( C = aa_real_real(minus_minus_real(A),B) ) ) ).

% add_implies_diff
tff(fact_4133_add__implies__diff,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( aa_rat_rat(plus_plus_rat(C),B) = A )
     => ( C = aa_rat_rat(minus_minus_rat(A),B) ) ) ).

% add_implies_diff
tff(fact_4134_add__implies__diff,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B) = A )
     => ( C = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ) ) ).

% add_implies_diff
tff(fact_4135_add__implies__diff,axiom,
    ! [C: int,B: int,A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B) = A )
     => ( C = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) ) ) ).

% add_implies_diff
tff(fact_4136_diff__add__eq__diff__diff__swap,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(minus_minus_complex(A),aa_complex_complex(plus_plus_complex(B),C)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(minus_minus_complex(A),C)),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_4137_diff__add__eq__diff__diff__swap,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(minus_minus_real(A),aa_real_real(plus_plus_real(B),C)) = aa_real_real(minus_minus_real(aa_real_real(minus_minus_real(A),C)),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_4138_diff__add__eq__diff__diff__swap,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(minus_minus_rat(A),aa_rat_rat(plus_plus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A),C)),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_4139_diff__add__eq__diff__diff__swap,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_4140_diff__add__eq,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(minus_minus_complex(A),B)),C) = aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(A),C)),B) ).

% diff_add_eq
tff(fact_4141_diff__add__eq,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(plus_plus_real(aa_real_real(minus_minus_real(A),B)),C) = aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(A),C)),B) ).

% diff_add_eq
tff(fact_4142_diff__add__eq,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(A),B)),C) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),C)),B) ).

% diff_add_eq
tff(fact_4143_diff__add__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B) ).

% diff_add_eq
tff(fact_4144_diff__diff__eq2,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(minus_minus_complex(A),aa_complex_complex(minus_minus_complex(B),C)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(A),C)),B) ).

% diff_diff_eq2
tff(fact_4145_diff__diff__eq2,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(minus_minus_real(A),aa_real_real(minus_minus_real(B),C)) = aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(A),C)),B) ).

% diff_diff_eq2
tff(fact_4146_diff__diff__eq2,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(minus_minus_rat(A),aa_rat_rat(minus_minus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),C)),B) ).

% diff_diff_eq2
tff(fact_4147_diff__diff__eq2,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B) ).

% diff_diff_eq2
tff(fact_4148_add__diff__eq,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(plus_plus_complex(A),aa_complex_complex(minus_minus_complex(B),C)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(A),B)),C) ).

% add_diff_eq
tff(fact_4149_add__diff__eq,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(plus_plus_real(A),aa_real_real(minus_minus_real(B),C)) = aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(A),B)),C) ).

% add_diff_eq
tff(fact_4150_add__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(plus_plus_rat(A),aa_rat_rat(minus_minus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),B)),C) ).

% add_diff_eq
tff(fact_4151_add__diff__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ).

% add_diff_eq
tff(fact_4152_eq__diff__eq,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( ( A = aa_complex_complex(minus_minus_complex(C),B) )
    <=> ( aa_complex_complex(plus_plus_complex(A),B) = C ) ) ).

% eq_diff_eq
tff(fact_4153_eq__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( ( A = aa_real_real(minus_minus_real(C),B) )
    <=> ( aa_real_real(plus_plus_real(A),B) = C ) ) ).

% eq_diff_eq
tff(fact_4154_eq__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( A = aa_rat_rat(minus_minus_rat(C),B) )
    <=> ( aa_rat_rat(plus_plus_rat(A),B) = C ) ) ).

% eq_diff_eq
tff(fact_4155_eq__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = C ) ) ).

% eq_diff_eq
tff(fact_4156_diff__eq__eq,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( aa_complex_complex(minus_minus_complex(A),B) = C )
    <=> ( A = aa_complex_complex(plus_plus_complex(C),B) ) ) ).

% diff_eq_eq
tff(fact_4157_diff__eq__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( aa_real_real(minus_minus_real(A),B) = C )
    <=> ( A = aa_real_real(plus_plus_real(C),B) ) ) ).

% diff_eq_eq
tff(fact_4158_diff__eq__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(minus_minus_rat(A),B) = C )
    <=> ( A = aa_rat_rat(plus_plus_rat(C),B) ) ) ).

% diff_eq_eq
tff(fact_4159_diff__eq__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = C )
    <=> ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B) ) ) ).

% diff_eq_eq
tff(fact_4160_group__cancel_Osub1,axiom,
    ! [A2: complex,K2: complex,A: complex,B: complex] :
      ( ( A2 = aa_complex_complex(plus_plus_complex(K2),A) )
     => ( aa_complex_complex(minus_minus_complex(A2),B) = aa_complex_complex(plus_plus_complex(K2),aa_complex_complex(minus_minus_complex(A),B)) ) ) ).

% group_cancel.sub1
tff(fact_4161_group__cancel_Osub1,axiom,
    ! [A2: real,K2: real,A: real,B: real] :
      ( ( A2 = aa_real_real(plus_plus_real(K2),A) )
     => ( aa_real_real(minus_minus_real(A2),B) = aa_real_real(plus_plus_real(K2),aa_real_real(minus_minus_real(A),B)) ) ) ).

% group_cancel.sub1
tff(fact_4162_group__cancel_Osub1,axiom,
    ! [A2: rat,K2: rat,A: rat,B: rat] :
      ( ( A2 = aa_rat_rat(plus_plus_rat(K2),A) )
     => ( aa_rat_rat(minus_minus_rat(A2),B) = aa_rat_rat(plus_plus_rat(K2),aa_rat_rat(minus_minus_rat(A),B)) ) ) ).

% group_cancel.sub1
tff(fact_4163_group__cancel_Osub1,axiom,
    ! [A2: int,K2: int,A: int,B: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),A) )
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) ) ) ).

% group_cancel.sub1
tff(fact_4164_add__diff__add,axiom,
    ! [A: complex,C: complex,B: complex,D: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(A),C)),aa_complex_complex(plus_plus_complex(B),D)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(minus_minus_complex(A),B)),aa_complex_complex(minus_minus_complex(C),D)) ).

% add_diff_add
tff(fact_4165_add__diff__add,axiom,
    ! [A: real,C: real,B: real,D: real] : aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(A),C)),aa_real_real(plus_plus_real(B),D)) = aa_real_real(plus_plus_real(aa_real_real(minus_minus_real(A),B)),aa_real_real(minus_minus_real(C),D)) ).

% add_diff_add
tff(fact_4166_add__diff__add,axiom,
    ! [A: rat,C: rat,B: rat,D: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),D)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(A),B)),aa_rat_rat(minus_minus_rat(C),D)) ).

% add_diff_add
tff(fact_4167_add__diff__add,axiom,
    ! [A: int,C: int,B: int,D: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D)) ).

% add_diff_add
tff(fact_4168_diff__divide__distrib,axiom,
    ! [A: complex,B: complex,C: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(A),B)),C) = aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(A),C)),aa_complex_complex(divide1717551699836669952omplex(B),C)) ).

% diff_divide_distrib
tff(fact_4169_diff__divide__distrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(A),B)),C) = aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(A),C)),aa_real_real(divide_divide_real(B),C)) ).

% diff_divide_distrib
tff(fact_4170_diff__divide__distrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(A),B)),C) = aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C)) ).

% diff_divide_distrib
tff(fact_4171_dvd__diff__commute,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),minus_8373710615458151222nteger(C,B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),minus_8373710615458151222nteger(B,C)) ) ).

% dvd_diff_commute
tff(fact_4172_dvd__diff__commute,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) ) ).

% dvd_diff_commute
tff(fact_4173_mod__diff__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) ).

% mod_diff_eq
tff(fact_4174_mod__diff__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(minus_8373710615458151222nteger(modulo364778990260209775nteger(A,C),modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(minus_8373710615458151222nteger(A,B),C) ).

% mod_diff_eq
tff(fact_4175_mod__diff__cong,axiom,
    ! [A: int,C: int,A4: int,B: int,B4: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A4),C) )
     => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B4),C) )
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A4),B4)),C) ) ) ) ).

% mod_diff_cong
tff(fact_4176_mod__diff__cong,axiom,
    ! [A: code_integer,C: code_integer,A4: code_integer,B: code_integer,B4: code_integer] :
      ( ( modulo364778990260209775nteger(A,C) = modulo364778990260209775nteger(A4,C) )
     => ( ( modulo364778990260209775nteger(B,C) = modulo364778990260209775nteger(B4,C) )
       => ( modulo364778990260209775nteger(minus_8373710615458151222nteger(A,B),C) = modulo364778990260209775nteger(minus_8373710615458151222nteger(A4,B4),C) ) ) ) ).

% mod_diff_cong
tff(fact_4177_mod__diff__left__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C)),B)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) ).

% mod_diff_left_eq
tff(fact_4178_mod__diff__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(minus_8373710615458151222nteger(modulo364778990260209775nteger(A,C),B),C) = modulo364778990260209775nteger(minus_8373710615458151222nteger(A,B),C) ).

% mod_diff_left_eq
tff(fact_4179_mod__diff__right__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) ).

% mod_diff_right_eq
tff(fact_4180_mod__diff__right__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : modulo364778990260209775nteger(minus_8373710615458151222nteger(A,modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(minus_8373710615458151222nteger(A,B),C) ).

% mod_diff_right_eq
tff(fact_4181_int__distrib_I4_J,axiom,
    ! [W: int,Z1: int,Z22: int] : aa_int_int(aa_int_fun_int_int(times_times_int,W),aa_int_int(aa_int_fun_int_int(minus_minus_int,Z1),Z22)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z1)),aa_int_int(aa_int_fun_int_int(times_times_int,W),Z22)) ).

% int_distrib(4)
tff(fact_4182_int__distrib_I3_J,axiom,
    ! [Z1: int,Z22: int,W: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z1),Z22)),W) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z1),W)),aa_int_int(aa_int_fun_int_int(times_times_int,Z22),W)) ).

% int_distrib(3)
tff(fact_4183_zero__less__eq__of__bool,axiom,
    ! [P: $o] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),zero_n3304061248610475627l_real((P))) ).

% zero_less_eq_of_bool
tff(fact_4184_zero__less__eq__of__bool,axiom,
    ! [P: $o] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),zero_n2052037380579107095ol_rat((P))) ).

% zero_less_eq_of_bool
tff(fact_4185_zero__less__eq__of__bool,axiom,
    ! [P: $o] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_o_nat(zero_n2687167440665602831ol_nat,(P))) ).

% zero_less_eq_of_bool
tff(fact_4186_zero__less__eq__of__bool,axiom,
    ! [P: $o] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_o_int(zero_n2684676970156552555ol_int,(P))) ).

% zero_less_eq_of_bool
tff(fact_4187_zero__less__eq__of__bool,axiom,
    ! [P: $o] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),zero_n356916108424825756nteger((P))) ).

% zero_less_eq_of_bool
tff(fact_4188_of__bool__less__eq__one,axiom,
    ! [P: $o] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_n3304061248610475627l_real((P))),one_one_real) ).

% of_bool_less_eq_one
tff(fact_4189_of__bool__less__eq__one,axiom,
    ! [P: $o] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_n2052037380579107095ol_rat((P))),one_one_rat) ).

% of_bool_less_eq_one
tff(fact_4190_of__bool__less__eq__one,axiom,
    ! [P: $o] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(P))),one_one_nat) ).

% of_bool_less_eq_one
tff(fact_4191_of__bool__less__eq__one,axiom,
    ! [P: $o] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_o_int(zero_n2684676970156552555ol_int,(P))),one_one_int) ).

% of_bool_less_eq_one
tff(fact_4192_of__bool__less__eq__one,axiom,
    ! [P: $o] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_n356916108424825756nteger((P))),one_one_Code_integer) ).

% of_bool_less_eq_one
tff(fact_4193_of__bool__def,axiom,
    ! [P2: $o] :
      zero_n1201886186963655149omplex((P2)) = $ite((P2),one_one_complex,zero_zero_complex) ).

% of_bool_def
tff(fact_4194_of__bool__def,axiom,
    ! [P2: $o] :
      zero_n3304061248610475627l_real((P2)) = $ite((P2),one_one_real,zero_zero_real) ).

% of_bool_def
tff(fact_4195_of__bool__def,axiom,
    ! [P2: $o] :
      zero_n2052037380579107095ol_rat((P2)) = $ite((P2),one_one_rat,zero_zero_rat) ).

% of_bool_def
tff(fact_4196_of__bool__def,axiom,
    ! [P2: $o] :
      aa_o_nat(zero_n2687167440665602831ol_nat,(P2)) = $ite((P2),one_one_nat,zero_zero_nat) ).

% of_bool_def
tff(fact_4197_of__bool__def,axiom,
    ! [P2: $o] :
      aa_o_int(zero_n2684676970156552555ol_int,(P2)) = $ite((P2),one_one_int,zero_zero_int) ).

% of_bool_def
tff(fact_4198_of__bool__def,axiom,
    ! [P2: $o] :
      zero_n356916108424825756nteger((P2)) = $ite((P2),one_one_Code_integer,zero_z3403309356797280102nteger) ).

% of_bool_def
tff(fact_4199_split__of__bool,axiom,
    ! [P: fun_complex_o,P2: $o] :
      ( aa_complex_o(P,zero_n1201886186963655149omplex((P2)))
    <=> ( ( (P2)
         => aa_complex_o(P,one_one_complex) )
        & ( ~ (P2)
         => aa_complex_o(P,zero_zero_complex) ) ) ) ).

% split_of_bool
tff(fact_4200_split__of__bool,axiom,
    ! [P: fun_real_o,P2: $o] :
      ( aa_real_o(P,zero_n3304061248610475627l_real((P2)))
    <=> ( ( (P2)
         => aa_real_o(P,one_one_real) )
        & ( ~ (P2)
         => aa_real_o(P,zero_zero_real) ) ) ) ).

% split_of_bool
tff(fact_4201_split__of__bool,axiom,
    ! [P: fun_rat_o,P2: $o] :
      ( aa_rat_o(P,zero_n2052037380579107095ol_rat((P2)))
    <=> ( ( (P2)
         => aa_rat_o(P,one_one_rat) )
        & ( ~ (P2)
         => aa_rat_o(P,zero_zero_rat) ) ) ) ).

% split_of_bool
tff(fact_4202_split__of__bool,axiom,
    ! [P: fun_nat_o,P2: $o] :
      ( aa_nat_o(P,aa_o_nat(zero_n2687167440665602831ol_nat,(P2)))
    <=> ( ( (P2)
         => aa_nat_o(P,one_one_nat) )
        & ( ~ (P2)
         => aa_nat_o(P,zero_zero_nat) ) ) ) ).

% split_of_bool
tff(fact_4203_split__of__bool,axiom,
    ! [P: fun_int_o,P2: $o] :
      ( aa_int_o(P,aa_o_int(zero_n2684676970156552555ol_int,(P2)))
    <=> ( ( (P2)
         => aa_int_o(P,one_one_int) )
        & ( ~ (P2)
         => aa_int_o(P,zero_zero_int) ) ) ) ).

% split_of_bool
tff(fact_4204_split__of__bool,axiom,
    ! [P: fun_Code_integer_o,P2: $o] :
      ( aa_Code_integer_o(P,zero_n356916108424825756nteger((P2)))
    <=> ( ( (P2)
         => aa_Code_integer_o(P,one_one_Code_integer) )
        & ( ~ (P2)
         => aa_Code_integer_o(P,zero_z3403309356797280102nteger) ) ) ) ).

% split_of_bool
tff(fact_4205_split__of__bool__asm,axiom,
    ! [P: fun_complex_o,P2: $o] :
      ( aa_complex_o(P,zero_n1201886186963655149omplex((P2)))
    <=> ~ ( ( (P2)
            & ~ aa_complex_o(P,one_one_complex) )
          | ( ~ (P2)
            & ~ aa_complex_o(P,zero_zero_complex) ) ) ) ).

% split_of_bool_asm
tff(fact_4206_split__of__bool__asm,axiom,
    ! [P: fun_real_o,P2: $o] :
      ( aa_real_o(P,zero_n3304061248610475627l_real((P2)))
    <=> ~ ( ( (P2)
            & ~ aa_real_o(P,one_one_real) )
          | ( ~ (P2)
            & ~ aa_real_o(P,zero_zero_real) ) ) ) ).

% split_of_bool_asm
tff(fact_4207_split__of__bool__asm,axiom,
    ! [P: fun_rat_o,P2: $o] :
      ( aa_rat_o(P,zero_n2052037380579107095ol_rat((P2)))
    <=> ~ ( ( (P2)
            & ~ aa_rat_o(P,one_one_rat) )
          | ( ~ (P2)
            & ~ aa_rat_o(P,zero_zero_rat) ) ) ) ).

% split_of_bool_asm
tff(fact_4208_split__of__bool__asm,axiom,
    ! [P: fun_nat_o,P2: $o] :
      ( aa_nat_o(P,aa_o_nat(zero_n2687167440665602831ol_nat,(P2)))
    <=> ~ ( ( (P2)
            & ~ aa_nat_o(P,one_one_nat) )
          | ( ~ (P2)
            & ~ aa_nat_o(P,zero_zero_nat) ) ) ) ).

% split_of_bool_asm
tff(fact_4209_split__of__bool__asm,axiom,
    ! [P: fun_int_o,P2: $o] :
      ( aa_int_o(P,aa_o_int(zero_n2684676970156552555ol_int,(P2)))
    <=> ~ ( ( (P2)
            & ~ aa_int_o(P,one_one_int) )
          | ( ~ (P2)
            & ~ aa_int_o(P,zero_zero_int) ) ) ) ).

% split_of_bool_asm
tff(fact_4210_split__of__bool__asm,axiom,
    ! [P: fun_Code_integer_o,P2: $o] :
      ( aa_Code_integer_o(P,zero_n356916108424825756nteger((P2)))
    <=> ~ ( ( (P2)
            & ~ aa_Code_integer_o(P,one_one_Code_integer) )
          | ( ~ (P2)
            & ~ aa_Code_integer_o(P,zero_z3403309356797280102nteger) ) ) ) ).

% split_of_bool_asm
tff(fact_4211_le__iff__diff__le__0,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(A),B)),zero_zero_real) ) ).

% le_iff_diff_le_0
tff(fact_4212_le__iff__diff__le__0,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),B)),zero_zero_rat) ) ).

% le_iff_diff_le_0
tff(fact_4213_le__iff__diff__le__0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),zero_zero_int) ) ).

% le_iff_diff_le_0
tff(fact_4214_less__iff__diff__less__0,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(A),B)),zero_zero_real) ) ).

% less_iff_diff_less_0
tff(fact_4215_less__iff__diff__less__0,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(A),B)),zero_zero_rat) ) ).

% less_iff_diff_less_0
tff(fact_4216_less__iff__diff__less__0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),zero_zero_int) ) ).

% less_iff_diff_less_0
tff(fact_4217_add__le__add__imp__diff__le,axiom,
    ! [I2: real,K2: real,Nb: real,J: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(I2),K2)),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Nb),aa_real_real(plus_plus_real(J),K2))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(I2),K2)),Nb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Nb),aa_real_real(plus_plus_real(J),K2))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(Nb),K2)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_4218_add__le__add__imp__diff__le,axiom,
    ! [I2: rat,K2: rat,Nb: rat,J: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I2),K2)),Nb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Nb),aa_rat_rat(plus_plus_rat(J),K2))
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I2),K2)),Nb)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Nb),aa_rat_rat(plus_plus_rat(J),K2))
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(Nb),K2)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_4219_add__le__add__imp__diff__le,axiom,
    ! [I2: nat,K2: nat,Nb: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K2))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),Nb)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K2))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_4220_add__le__add__imp__diff__le,axiom,
    ! [I2: int,K2: int,Nb: int,J: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),K2)),Nb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),K2))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),K2)),Nb)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),K2))
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Nb),K2)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_4221_add__le__imp__le__diff,axiom,
    ! [I2: real,K2: real,Nb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(I2),K2)),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I2),aa_real_real(minus_minus_real(Nb),K2)) ) ).

% add_le_imp_le_diff
tff(fact_4222_add__le__imp__le__diff,axiom,
    ! [I2: rat,K2: rat,Nb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I2),K2)),Nb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I2),aa_rat_rat(minus_minus_rat(Nb),K2)) ) ).

% add_le_imp_le_diff
tff(fact_4223_add__le__imp__le__diff,axiom,
    ! [I2: nat,K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)) ) ).

% add_le_imp_le_diff
tff(fact_4224_add__le__imp__le__diff,axiom,
    ! [I2: int,K2: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),K2)),Nb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),aa_int_int(aa_int_fun_int_int(minus_minus_int,Nb),K2)) ) ).

% add_le_imp_le_diff
tff(fact_4225_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A) = C )
        <=> ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
tff(fact_4226_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) = B ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
tff(fact_4227_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),B) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
tff(fact_4228_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)),C) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_4229_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)),C) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)),A) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
tff(fact_4230_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_4231_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)),A) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
tff(fact_4232_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),B) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
tff(fact_4233_le__add__diff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)),A)) ) ).

% le_add_diff
tff(fact_4234_diff__add,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)),A) = B ) ) ).

% diff_add
tff(fact_4235_le__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(minus_minus_real(C),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(A),B)),C) ) ).

% le_diff_eq
tff(fact_4236_le__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(minus_minus_rat(C),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),B)),C) ) ).

% le_diff_eq
tff(fact_4237_le__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ) ).

% le_diff_eq
tff(fact_4238_diff__le__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(A),B)),C)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(plus_plus_real(C),B)) ) ).

% diff_le_eq
tff(fact_4239_diff__le__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),B)),C)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(plus_plus_rat(C),B)) ) ).

% diff_le_eq
tff(fact_4240_diff__le__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)) ) ).

% diff_le_eq
tff(fact_4241_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: real,B: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_real(plus_plus_real(B),aa_real_real(minus_minus_real(A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_4242_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: rat,B: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_rat(plus_plus_rat(B),aa_rat_rat(minus_minus_rat(A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_4243_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_4244_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: int,B: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_4245_less__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(minus_minus_real(C),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(A),B)),C) ) ).

% less_diff_eq
tff(fact_4246_less__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(minus_minus_rat(C),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),B)),C) ) ).

% less_diff_eq
tff(fact_4247_less__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ) ).

% less_diff_eq
tff(fact_4248_diff__less__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(A),B)),C)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(plus_plus_real(C),B)) ) ).

% diff_less_eq
tff(fact_4249_diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(A),B)),C)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(plus_plus_rat(C),B)) ) ).

% diff_less_eq
tff(fact_4250_diff__less__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)) ) ).

% diff_less_eq
tff(fact_4251_square__diff__square__factored,axiom,
    ! [X: complex,Y2: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(X),X)),aa_complex_complex(times_times_complex(Y2),Y2)) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(X),Y2)),aa_complex_complex(minus_minus_complex(X),Y2)) ).

% square_diff_square_factored
tff(fact_4252_square__diff__square__factored,axiom,
    ! [X: real,Y2: real] : aa_real_real(minus_minus_real(aa_real_real(times_times_real(X),X)),aa_real_real(times_times_real(Y2),Y2)) = aa_real_real(times_times_real(aa_real_real(plus_plus_real(X),Y2)),aa_real_real(minus_minus_real(X),Y2)) ).

% square_diff_square_factored
tff(fact_4253_square__diff__square__factored,axiom,
    ! [X: rat,Y2: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(X),X)),aa_rat_rat(times_times_rat(Y2),Y2)) = aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(X),Y2)),aa_rat_rat(minus_minus_rat(X),Y2)) ).

% square_diff_square_factored
tff(fact_4254_square__diff__square__factored,axiom,
    ! [X: int,Y2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y2),Y2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y2)) ).

% square_diff_square_factored
tff(fact_4255_eq__add__iff2,axiom,
    ! [A: complex,E: complex,C: complex,B: complex,D: complex] :
      ( ( aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),E)),C) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(B),E)),D) )
    <=> ( C = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(B),A)),E)),D) ) ) ).

% eq_add_iff2
tff(fact_4256_eq__add__iff2,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( ( aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),E)),C) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(B),E)),D) )
    <=> ( C = aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(minus_minus_real(B),A)),E)),D) ) ) ).

% eq_add_iff2
tff(fact_4257_eq__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E)),C) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E)),D) )
    <=> ( C = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(B),A)),E)),D) ) ) ).

% eq_add_iff2
tff(fact_4258_eq__add__iff2,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D) )
    <=> ( C = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),E)),D) ) ) ).

% eq_add_iff2
tff(fact_4259_eq__add__iff1,axiom,
    ! [A: complex,E: complex,C: complex,B: complex,D: complex] :
      ( ( aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),E)),C) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(B),E)),D) )
    <=> ( aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(A),B)),E)),C) = D ) ) ).

% eq_add_iff1
tff(fact_4260_eq__add__iff1,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( ( aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),E)),C) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(B),E)),D) )
    <=> ( aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(minus_minus_real(A),B)),E)),C) = D ) ) ).

% eq_add_iff1
tff(fact_4261_eq__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E)),C) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E)),D) )
    <=> ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A),B)),E)),C) = D ) ) ).

% eq_add_iff1
tff(fact_4262_eq__add__iff1,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),E)),C) = D ) ) ).

% eq_add_iff1
tff(fact_4263_mult__diff__mult,axiom,
    ! [X: complex,Y2: complex,A: complex,B: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(X),Y2)),aa_complex_complex(times_times_complex(A),B)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(X),aa_complex_complex(minus_minus_complex(Y2),B))),aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(X),A)),B)) ).

% mult_diff_mult
tff(fact_4264_mult__diff__mult,axiom,
    ! [X: real,Y2: real,A: real,B: real] : aa_real_real(minus_minus_real(aa_real_real(times_times_real(X),Y2)),aa_real_real(times_times_real(A),B)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(X),aa_real_real(minus_minus_real(Y2),B))),aa_real_real(times_times_real(aa_real_real(minus_minus_real(X),A)),B)) ).

% mult_diff_mult
tff(fact_4265_mult__diff__mult,axiom,
    ! [X: rat,Y2: rat,A: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(X),Y2)),aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(X),aa_rat_rat(minus_minus_rat(Y2),B))),aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(X),A)),B)) ).

% mult_diff_mult
tff(fact_4266_mult__diff__mult,axiom,
    ! [X: int,Y2: int,A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),Y2)),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),aa_int_int(aa_int_fun_int_int(minus_minus_int,Y2),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),A)),B)) ).

% mult_diff_mult
tff(fact_4267_replicate__eqI,axiom,
    ! [Xs: list_complex,Nb: nat,X: complex] :
      ( ( size_s3451745648224563538omplex(Xs) = Nb )
     => ( ! [Y4: complex] :
            ( member_complex(Y4,set_complex2(Xs))
           => ( Y4 = X ) )
       => ( Xs = replicate_complex(Nb,X) ) ) ) ).

% replicate_eqI
tff(fact_4268_replicate__eqI,axiom,
    ! [Xs: list_real,Nb: nat,X: real] :
      ( ( size_size_list_real(Xs) = Nb )
     => ( ! [Y4: real] :
            ( member_real(Y4,set_real2(Xs))
           => ( Y4 = X ) )
       => ( Xs = replicate_real(Nb,X) ) ) ) ).

% replicate_eqI
tff(fact_4269_replicate__eqI,axiom,
    ! [Xs: list_set_nat,Nb: nat,X: set_nat] :
      ( ( size_s3254054031482475050et_nat(Xs) = Nb )
     => ( ! [Y4: set_nat] :
            ( member_set_nat(Y4,set_set_nat2(Xs))
           => ( Y4 = X ) )
       => ( Xs = replicate_set_nat(Nb,X) ) ) ) ).

% replicate_eqI
tff(fact_4270_replicate__eqI,axiom,
    ! [Xs: list_VEBT_VEBT,Nb: nat,X: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = Nb )
     => ( ! [Y4: vEBT_VEBT] :
            ( member_VEBT_VEBT(Y4,set_VEBT_VEBT2(Xs))
           => ( Y4 = X ) )
       => ( Xs = replicate_VEBT_VEBT(Nb,X) ) ) ) ).

% replicate_eqI
tff(fact_4271_replicate__eqI,axiom,
    ! [Xs: list_o,Nb: nat,X: $o] :
      ( ( size_size_list_o(Xs) = Nb )
     => ( ! [Y4: $o] :
            ( member_o((Y4),set_o2(Xs))
           => ( (Y4)
            <=> (X) ) )
       => ( Xs = replicate_o(Nb,(X)) ) ) ) ).

% replicate_eqI
tff(fact_4272_replicate__eqI,axiom,
    ! [Xs: list_nat,Nb: nat,X: nat] :
      ( ( size_size_list_nat(Xs) = Nb )
     => ( ! [Y4: nat] :
            ( aa_set_nat_o(member_nat(Y4),set_nat2(Xs))
           => ( Y4 = X ) )
       => ( Xs = replicate_nat(Nb,X) ) ) ) ).

% replicate_eqI
tff(fact_4273_replicate__eqI,axiom,
    ! [Xs: list_int,Nb: nat,X: int] :
      ( ( size_size_list_int(Xs) = Nb )
     => ( ! [Y4: int] :
            ( member_int(Y4,set_int2(Xs))
           => ( Y4 = X ) )
       => ( Xs = replicate_int(Nb,X) ) ) ) ).

% replicate_eqI
tff(fact_4274_replicate__length__same,axiom,
    ! [Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ! [X4: vEBT_VEBT] :
          ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(Xs))
         => ( X4 = X ) )
     => ( replicate_VEBT_VEBT(size_s6755466524823107622T_VEBT(Xs),X) = Xs ) ) ).

% replicate_length_same
tff(fact_4275_replicate__length__same,axiom,
    ! [Xs: list_o,X: $o] :
      ( ! [X4: $o] :
          ( member_o((X4),set_o2(Xs))
         => ( (X4)
          <=> (X) ) )
     => ( replicate_o(size_size_list_o(Xs),(X)) = Xs ) ) ).

% replicate_length_same
tff(fact_4276_replicate__length__same,axiom,
    ! [Xs: list_nat,X: nat] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),set_nat2(Xs))
         => ( X4 = X ) )
     => ( replicate_nat(size_size_list_nat(Xs),X) = Xs ) ) ).

% replicate_length_same
tff(fact_4277_replicate__length__same,axiom,
    ! [Xs: list_int,X: int] :
      ( ! [X4: int] :
          ( member_int(X4,set_int2(Xs))
         => ( X4 = X ) )
     => ( replicate_int(size_size_list_int(Xs),X) = Xs ) ) ).

% replicate_length_same
tff(fact_4278_mod__eq__dvd__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C) )
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) ) ).

% mod_eq_dvd_iff
tff(fact_4279_mod__eq__dvd__iff,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger(A,C) = modulo364778990260209775nteger(B,C) )
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),minus_8373710615458151222nteger(A,B)) ) ).

% mod_eq_dvd_iff
tff(fact_4280_dvd__minus__mod,axiom,
    ! [B: nat,A: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))) ).

% dvd_minus_mod
tff(fact_4281_dvd__minus__mod,axiom,
    ! [B: int,A: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))) ).

% dvd_minus_mod
tff(fact_4282_dvd__minus__mod,axiom,
    ! [B: code_integer,A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),minus_8373710615458151222nteger(A,modulo364778990260209775nteger(A,B))) ).

% dvd_minus_mod
tff(fact_4283_int__le__induct,axiom,
    ! [I2: int,K2: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),K2)
     => ( aa_int_o(P,K2)
       => ( ! [I3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I3),K2)
             => ( aa_int_o(P,I3)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I3),one_one_int)) ) )
         => aa_int_o(P,I2) ) ) ) ).

% int_le_induct
tff(fact_4284_int__less__induct,axiom,
    ! [I2: int,K2: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,I2),K2)
     => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,K2),one_one_int))
       => ( ! [I3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,I3),K2)
             => ( aa_int_o(P,I3)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I3),one_one_int)) ) )
         => aa_int_o(P,I2) ) ) ) ).

% int_less_induct
tff(fact_4285_map__replicate__const,axiom,
    ! [K2: vEBT_VEBT,Lst: list_VEBT_VEBT] : map_VE8901447254227204932T_VEBT(aTP_Lamm_bj(K2),Lst) = replicate_VEBT_VEBT(size_s6755466524823107622T_VEBT(Lst),K2) ).

% map_replicate_const
tff(fact_4286_map__replicate__const,axiom,
    ! [K2: vEBT_VEBT,Lst: list_o] : map_o_VEBT_VEBT(aTP_Lamm_bk(K2),Lst) = replicate_VEBT_VEBT(size_size_list_o(Lst),K2) ).

% map_replicate_const
tff(fact_4287_map__replicate__const,axiom,
    ! [K2: nat,Lst: list_nat] : map_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_bl,K2),Lst) = replicate_nat(size_size_list_nat(Lst),K2) ).

% map_replicate_const
tff(fact_4288_map__replicate__const,axiom,
    ! [K2: vEBT_VEBT,Lst: list_nat] : map_nat_VEBT_VEBT(aTP_Lamm_bm(K2),Lst) = replicate_VEBT_VEBT(size_size_list_nat(Lst),K2) ).

% map_replicate_const
tff(fact_4289_map__replicate__const,axiom,
    ! [K2: vEBT_VEBT,Lst: list_int] : map_int_VEBT_VEBT(aTP_Lamm_bn(K2),Lst) = replicate_VEBT_VEBT(size_size_list_int(Lst),K2) ).

% map_replicate_const
tff(fact_4290_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),E)),C)),aa_real_real(plus_plus_real(aa_real_real(times_times_real(B),E)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(minus_minus_real(B),A)),E)),D)) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_4291_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E)),C)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E)),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(B),A)),E)),D)) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_4292_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),E)),D)) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_4293_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),E)),C)),aa_real_real(plus_plus_real(aa_real_real(times_times_real(B),E)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(minus_minus_real(A),B)),E)),C)),D) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_4294_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E)),C)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E)),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A),B)),E)),C)),D) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_4295_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),E)),C)),D) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_4296_less__add__iff2,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),E)),C)),aa_real_real(plus_plus_real(aa_real_real(times_times_real(B),E)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(minus_minus_real(B),A)),E)),D)) ) ).

% less_add_iff2
tff(fact_4297_less__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E)),C)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E)),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(B),A)),E)),D)) ) ).

% less_add_iff2
tff(fact_4298_less__add__iff2,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),E)),D)) ) ).

% less_add_iff2
tff(fact_4299_less__add__iff1,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),E)),C)),aa_real_real(plus_plus_real(aa_real_real(times_times_real(B),E)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(minus_minus_real(A),B)),E)),C)),D) ) ).

% less_add_iff1
tff(fact_4300_less__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E)),C)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E)),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A),B)),E)),C)),D) ) ).

% less_add_iff1
tff(fact_4301_less__add__iff1,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E)),D))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),E)),C)),D) ) ).

% less_add_iff1
tff(fact_4302_divide__diff__eq__iff,axiom,
    ! [Z2: complex,X: complex,Y2: complex] :
      ( ( Z2 != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(X),Z2)),Y2) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(X),aa_complex_complex(times_times_complex(Y2),Z2))),Z2) ) ) ).

% divide_diff_eq_iff
tff(fact_4303_divide__diff__eq__iff,axiom,
    ! [Z2: real,X: real,Y2: real] :
      ( ( Z2 != zero_zero_real )
     => ( aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(X),Z2)),Y2) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(X),aa_real_real(times_times_real(Y2),Z2))),Z2) ) ) ).

% divide_diff_eq_iff
tff(fact_4304_divide__diff__eq__iff,axiom,
    ! [Z2: rat,X: rat,Y2: rat] :
      ( ( Z2 != zero_zero_rat )
     => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(X),Z2)),Y2) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(X),aa_rat_rat(times_times_rat(Y2),Z2))),Z2) ) ) ).

% divide_diff_eq_iff
tff(fact_4305_diff__divide__eq__iff,axiom,
    ! [Z2: complex,X: complex,Y2: complex] :
      ( ( Z2 != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(X),aa_complex_complex(divide1717551699836669952omplex(Y2),Z2)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(X),Z2)),Y2)),Z2) ) ) ).

% diff_divide_eq_iff
tff(fact_4306_diff__divide__eq__iff,axiom,
    ! [Z2: real,X: real,Y2: real] :
      ( ( Z2 != zero_zero_real )
     => ( aa_real_real(minus_minus_real(X),aa_real_real(divide_divide_real(Y2),Z2)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(times_times_real(X),Z2)),Y2)),Z2) ) ) ).

% diff_divide_eq_iff
tff(fact_4307_diff__divide__eq__iff,axiom,
    ! [Z2: rat,X: rat,Y2: rat] :
      ( ( Z2 != zero_zero_rat )
     => ( aa_rat_rat(minus_minus_rat(X),aa_rat_rat(divide_divide_rat(Y2),Z2)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(X),Z2)),Y2)),Z2) ) ) ).

% diff_divide_eq_iff
tff(fact_4308_diff__frac__eq,axiom,
    ! [Y2: complex,Z2: complex,X: complex,W: complex] :
      ( ( Y2 != zero_zero_complex )
     => ( ( Z2 != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(X),Y2)),aa_complex_complex(divide1717551699836669952omplex(W),Z2)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(X),Z2)),aa_complex_complex(times_times_complex(W),Y2))),aa_complex_complex(times_times_complex(Y2),Z2)) ) ) ) ).

% diff_frac_eq
tff(fact_4309_diff__frac__eq,axiom,
    ! [Y2: real,Z2: real,X: real,W: real] :
      ( ( Y2 != zero_zero_real )
     => ( ( Z2 != zero_zero_real )
       => ( aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(X),Y2)),aa_real_real(divide_divide_real(W),Z2)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(times_times_real(X),Z2)),aa_real_real(times_times_real(W),Y2))),aa_real_real(times_times_real(Y2),Z2)) ) ) ) ).

% diff_frac_eq
tff(fact_4310_diff__frac__eq,axiom,
    ! [Y2: rat,Z2: rat,X: rat,W: rat] :
      ( ( Y2 != zero_zero_rat )
     => ( ( Z2 != zero_zero_rat )
       => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(X),Y2)),aa_rat_rat(divide_divide_rat(W),Z2)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(X),Z2)),aa_rat_rat(times_times_rat(W),Y2))),aa_rat_rat(times_times_rat(Y2),Z2)) ) ) ) ).

% diff_frac_eq
tff(fact_4311_add__divide__eq__if__simps_I4_J,axiom,
    ! [A: complex,B: complex,Z2: complex] :
      aa_complex_complex(minus_minus_complex(A),aa_complex_complex(divide1717551699836669952omplex(B),Z2)) = $ite(Z2 = zero_zero_complex,A,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(A),Z2)),B)),Z2)) ).

% add_divide_eq_if_simps(4)
tff(fact_4312_add__divide__eq__if__simps_I4_J,axiom,
    ! [A: real,B: real,Z2: real] :
      aa_real_real(minus_minus_real(A),aa_real_real(divide_divide_real(B),Z2)) = $ite(Z2 = zero_zero_real,A,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(times_times_real(A),Z2)),B)),Z2)) ).

% add_divide_eq_if_simps(4)
tff(fact_4313_add__divide__eq__if__simps_I4_J,axiom,
    ! [A: rat,B: rat,Z2: rat] :
      aa_rat_rat(minus_minus_rat(A),aa_rat_rat(divide_divide_rat(B),Z2)) = $ite(Z2 = zero_zero_rat,A,aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A),Z2)),B)),Z2)) ).

% add_divide_eq_if_simps(4)
tff(fact_4314_square__diff__one__factored,axiom,
    ! [X: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(X),X)),one_one_complex) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(X),one_one_complex)),aa_complex_complex(minus_minus_complex(X),one_one_complex)) ).

% square_diff_one_factored
tff(fact_4315_square__diff__one__factored,axiom,
    ! [X: real] : aa_real_real(minus_minus_real(aa_real_real(times_times_real(X),X)),one_one_real) = aa_real_real(times_times_real(aa_real_real(plus_plus_real(X),one_one_real)),aa_real_real(minus_minus_real(X),one_one_real)) ).

% square_diff_one_factored
tff(fact_4316_square__diff__one__factored,axiom,
    ! [X: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(X),X)),one_one_rat) = aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(X),one_one_rat)),aa_rat_rat(minus_minus_rat(X),one_one_rat)) ).

% square_diff_one_factored
tff(fact_4317_square__diff__one__factored,axiom,
    ! [X: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),one_one_int) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,X),one_one_int)) ).

% square_diff_one_factored
tff(fact_4318_inf__period_I3_J,axiom,
    ! [D: code_integer,D3: code_integer,T: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),D3)
     => ! [X2: code_integer,K4: code_integer] :
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X2,T))
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(minus_8373710615458151222nteger(X2,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K4),D3)),T)) ) ) ).

% inf_period(3)
tff(fact_4319_inf__period_I3_J,axiom,
    ! [D: complex,D3: complex,T: complex] :
      ( aa_complex_o(dvd_dvd_complex(D),D3)
     => ! [X2: complex,K4: complex] :
          ( aa_complex_o(dvd_dvd_complex(D),aa_complex_complex(plus_plus_complex(X2),T))
        <=> aa_complex_o(dvd_dvd_complex(D),aa_complex_complex(plus_plus_complex(aa_complex_complex(minus_minus_complex(X2),aa_complex_complex(times_times_complex(K4),D3))),T)) ) ) ).

% inf_period(3)
tff(fact_4320_inf__period_I3_J,axiom,
    ! [D: real,D3: real,T: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),D3)
     => ! [X2: real,K4: real] :
          ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(plus_plus_real(X2),T))
        <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(plus_plus_real(aa_real_real(minus_minus_real(X2),aa_real_real(times_times_real(K4),D3))),T)) ) ) ).

% inf_period(3)
tff(fact_4321_inf__period_I3_J,axiom,
    ! [D: rat,D3: rat,T: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),D3)
     => ! [X2: rat,K4: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X2),T))
        <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(X2),aa_rat_rat(times_times_rat(K4),D3))),T)) ) ) ).

% inf_period(3)
tff(fact_4322_inf__period_I3_J,axiom,
    ! [D: int,D3: int,T: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D3)
     => ! [X2: int,K4: int] :
          ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),T))
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D3))),T)) ) ) ).

% inf_period(3)
tff(fact_4323_inf__period_I4_J,axiom,
    ! [D: code_integer,D3: code_integer,T: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),D3)
     => ! [X2: code_integer,K4: code_integer] :
          ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X2,T))
        <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(minus_8373710615458151222nteger(X2,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K4),D3)),T)) ) ) ).

% inf_period(4)
tff(fact_4324_inf__period_I4_J,axiom,
    ! [D: complex,D3: complex,T: complex] :
      ( aa_complex_o(dvd_dvd_complex(D),D3)
     => ! [X2: complex,K4: complex] :
          ( ~ aa_complex_o(dvd_dvd_complex(D),aa_complex_complex(plus_plus_complex(X2),T))
        <=> ~ aa_complex_o(dvd_dvd_complex(D),aa_complex_complex(plus_plus_complex(aa_complex_complex(minus_minus_complex(X2),aa_complex_complex(times_times_complex(K4),D3))),T)) ) ) ).

% inf_period(4)
tff(fact_4325_inf__period_I4_J,axiom,
    ! [D: real,D3: real,T: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),D3)
     => ! [X2: real,K4: real] :
          ( ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(plus_plus_real(X2),T))
        <=> ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(plus_plus_real(aa_real_real(minus_minus_real(X2),aa_real_real(times_times_real(K4),D3))),T)) ) ) ).

% inf_period(4)
tff(fact_4326_inf__period_I4_J,axiom,
    ! [D: rat,D3: rat,T: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),D3)
     => ! [X2: rat,K4: rat] :
          ( ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X2),T))
        <=> ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(X2),aa_rat_rat(times_times_rat(K4),D3))),T)) ) ) ).

% inf_period(4)
tff(fact_4327_inf__period_I4_J,axiom,
    ! [D: int,D3: int,T: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D3)
     => ! [X2: int,K4: int] :
          ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),T))
        <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D3))),T)) ) ) ).

% inf_period(4)
tff(fact_4328_minus__mult__div__eq__mod,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% minus_mult_div_eq_mod
tff(fact_4329_minus__mult__div__eq__mod,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% minus_mult_div_eq_mod
tff(fact_4330_minus__mult__div__eq__mod,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(A,B))) = modulo364778990260209775nteger(A,B) ).

% minus_mult_div_eq_mod
tff(fact_4331_minus__mod__eq__mult__div,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ).

% minus_mod_eq_mult_div
tff(fact_4332_minus__mod__eq__mult__div,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ).

% minus_mod_eq_mult_div
tff(fact_4333_minus__mod__eq__mult__div,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,modulo364778990260209775nteger(A,B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(A,B)) ).

% minus_mod_eq_mult_div
tff(fact_4334_minus__mod__eq__div__mult,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B) ).

% minus_mod_eq_div_mult
tff(fact_4335_minus__mod__eq__div__mult,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B) ).

% minus_mod_eq_div_mult
tff(fact_4336_minus__mod__eq__div__mult,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,modulo364778990260209775nteger(A,B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),B) ).

% minus_mod_eq_div_mult
tff(fact_4337_minus__div__mult__eq__mod,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% minus_div_mult_eq_mod
tff(fact_4338_minus__div__mult__eq__mod,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% minus_div_mult_eq_mod
tff(fact_4339_minus__div__mult__eq__mod,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),B)) = modulo364778990260209775nteger(A,B) ).

% minus_div_mult_eq_mod
tff(fact_4340_plusinfinity,axiom,
    ! [D: int,P3: fun_int_o,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ! [X4: int,K: int] :
            ( aa_int_o(P3,X4)
          <=> aa_int_o(P3,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K),D))) )
       => ( ? [Z3: int] :
            ! [X4: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X4)
             => ( aa_int_o(P,X4)
              <=> aa_int_o(P3,X4) ) )
         => ( ? [X_1: int] : aa_int_o(P3,X_1)
           => ? [X_13: int] : aa_int_o(P,X_13) ) ) ) ) ).

% plusinfinity
tff(fact_4341_minusinfinity,axiom,
    ! [D: int,P1: fun_int_o,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ! [X4: int,K: int] :
            ( aa_int_o(P1,X4)
          <=> aa_int_o(P1,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K),D))) )
       => ( ? [Z3: int] :
            ! [X4: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Z3)
             => ( aa_int_o(P,X4)
              <=> aa_int_o(P1,X4) ) )
         => ( ? [X_1: int] : aa_int_o(P1,X_1)
           => ? [X_13: int] : aa_int_o(P,X_13) ) ) ) ) ).

% minusinfinity
tff(fact_4342_int__induct,axiom,
    ! [P: fun_int_o,K2: int,I2: int] :
      ( aa_int_o(P,K2)
     => ( ! [I3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),I3)
           => ( aa_int_o(P,I3)
             => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I3),one_one_int)) ) )
       => ( ! [I3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I3),K2)
             => ( aa_int_o(P,I3)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I3),one_one_int)) ) )
         => aa_int_o(P,I2) ) ) ) ).

% int_induct
tff(fact_4343_frac__le__eq,axiom,
    ! [Y2: real,Z2: real,X: real,W: real] :
      ( ( Y2 != zero_zero_real )
     => ( ( Z2 != zero_zero_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(X),Y2)),aa_real_real(divide_divide_real(W),Z2))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(times_times_real(X),Z2)),aa_real_real(times_times_real(W),Y2))),aa_real_real(times_times_real(Y2),Z2))),zero_zero_real) ) ) ) ).

% frac_le_eq
tff(fact_4344_frac__le__eq,axiom,
    ! [Y2: rat,Z2: rat,X: rat,W: rat] :
      ( ( Y2 != zero_zero_rat )
     => ( ( Z2 != zero_zero_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(X),Y2)),aa_rat_rat(divide_divide_rat(W),Z2))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(X),Z2)),aa_rat_rat(times_times_rat(W),Y2))),aa_rat_rat(times_times_rat(Y2),Z2))),zero_zero_rat) ) ) ) ).

% frac_le_eq
tff(fact_4345_frac__less__eq,axiom,
    ! [Y2: real,Z2: real,X: real,W: real] :
      ( ( Y2 != zero_zero_real )
     => ( ( Z2 != zero_zero_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(X),Y2)),aa_real_real(divide_divide_real(W),Z2))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(times_times_real(X),Z2)),aa_real_real(times_times_real(W),Y2))),aa_real_real(times_times_real(Y2),Z2))),zero_zero_real) ) ) ) ).

% frac_less_eq
tff(fact_4346_frac__less__eq,axiom,
    ! [Y2: rat,Z2: rat,X: rat,W: rat] :
      ( ( Y2 != zero_zero_rat )
     => ( ( Z2 != zero_zero_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(X),Y2)),aa_rat_rat(divide_divide_rat(W),Z2))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(X),Z2)),aa_rat_rat(times_times_rat(W),Y2))),aa_rat_rat(times_times_rat(Y2),Z2))),zero_zero_rat) ) ) ) ).

% frac_less_eq
tff(fact_4347_power2__commute,axiom,
    ! [X: complex,Y2: complex] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(minus_minus_complex(X),Y2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(minus_minus_complex(Y2),X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_4348_power2__commute,axiom,
    ! [X: real,Y2: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(X),Y2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(Y2),X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_4349_power2__commute,axiom,
    ! [X: rat,Y2: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(minus_minus_rat(X),Y2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(aa_rat_rat(minus_minus_rat(Y2),X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_4350_power2__commute,axiom,
    ! [X: int,Y2: int] : aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Y2),X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_4351_even__diff__iff,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(minus_minus_int,K2),L))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),L)) ) ).

% even_diff_iff
tff(fact_4352_decr__mult__lemma,axiom,
    ! [D: int,P: fun_int_o,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ! [X4: int] :
            ( aa_int_o(P,X4)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D)) )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
         => ! [X2: int] :
              ( aa_int_o(P,X2)
             => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D))) ) ) ) ) ).

% decr_mult_lemma
tff(fact_4353_mod__pos__geq,axiom,
    ! [L: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),K2)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),L) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,K2),L)),L) ) ) ) ).

% mod_pos_geq
tff(fact_4354_of__bool__odd__eq__mod__2,axiom,
    ! [A: nat] : aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% of_bool_odd_eq_mod_2
tff(fact_4355_of__bool__odd__eq__mod__2,axiom,
    ! [A: int] : aa_o_int(zero_n2684676970156552555ol_int,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% of_bool_odd_eq_mod_2
tff(fact_4356_of__bool__odd__eq__mod__2,axiom,
    ! [A: code_integer] : zero_n356916108424825756nteger(~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)) = modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% of_bool_odd_eq_mod_2
tff(fact_4357_scaling__mono,axiom,
    ! [U: real,V: real,R: real,S: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,U),V)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),R)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,R),S)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(U),aa_real_real(divide_divide_real(aa_real_real(times_times_real(R),aa_real_real(minus_minus_real(V),U))),S))),V) ) ) ) ).

% scaling_mono
tff(fact_4358_scaling__mono,axiom,
    ! [U: rat,V: rat,R: rat,S: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,U),V)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),R)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,R),S)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(U),aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(R),aa_rat_rat(minus_minus_rat(V),U))),S))),V) ) ) ) ).

% scaling_mono
tff(fact_4359_bits__induct,axiom,
    ! [P: fun_nat_o,A: nat] :
      ( ! [A3: nat] :
          ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = A3 )
         => aa_nat_o(P,A3) )
     => ( ! [A3: nat,B2: $o] :
            ( aa_nat_o(P,A3)
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(B2))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = A3 )
             => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(B2))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A3))) ) )
       => aa_nat_o(P,A) ) ) ).

% bits_induct
tff(fact_4360_bits__induct,axiom,
    ! [P: fun_int_o,A: int] :
      ( ! [A3: int] :
          ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A3),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A3 )
         => aa_int_o(P,A3) )
     => ( ! [A3: int,B2: $o] :
            ( aa_int_o(P,A3)
           => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_o_int(zero_n2684676970156552555ol_int,(B2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A3))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A3 )
             => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_o_int(zero_n2684676970156552555ol_int,(B2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A3))) ) )
       => aa_int_o(P,A) ) ) ).

% bits_induct
tff(fact_4361_bits__induct,axiom,
    ! [P: fun_Code_integer_o,A: code_integer] :
      ( ! [A3: code_integer] :
          ( ( divide6298287555418463151nteger(A3,numera6620942414471956472nteger(aa_num_num(bit0,one))) = A3 )
         => aa_Code_integer_o(P,A3) )
     => ( ! [A3: code_integer,B2: $o] :
            ( aa_Code_integer_o(P,A3)
           => ( ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(zero_n356916108424825756nteger((B2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),A3)),numera6620942414471956472nteger(aa_num_num(bit0,one))) = A3 )
             => aa_Code_integer_o(P,plus_p5714425477246183910nteger(zero_n356916108424825756nteger((B2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),A3))) ) )
       => aa_Code_integer_o(P,A) ) ) ).

% bits_induct
tff(fact_4362_signed__take__bit__int__less__eq,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),K2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_ri631733984087533419it_int(Nb),K2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,K2),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,Nb)))) ) ).

% signed_take_bit_int_less_eq
tff(fact_4363_div__pos__geq,axiom,
    ! [L: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),K2)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,K2),L)),L)),one_one_int) ) ) ) ).

% div_pos_geq
tff(fact_4364_add__0__iff,axiom,
    ! [B: complex,A: complex] :
      ( ( B = aa_complex_complex(plus_plus_complex(B),A) )
    <=> ( A = zero_zero_complex ) ) ).

% add_0_iff
tff(fact_4365_add__0__iff,axiom,
    ! [B: real,A: real] :
      ( ( B = aa_real_real(plus_plus_real(B),A) )
    <=> ( A = zero_zero_real ) ) ).

% add_0_iff
tff(fact_4366_add__0__iff,axiom,
    ! [B: rat,A: rat] :
      ( ( B = aa_rat_rat(plus_plus_rat(B),A) )
    <=> ( A = zero_zero_rat ) ) ).

% add_0_iff
tff(fact_4367_add__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) )
    <=> ( A = zero_zero_nat ) ) ).

% add_0_iff
tff(fact_4368_add__0__iff,axiom,
    ! [B: int,A: int] :
      ( ( B = aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) )
    <=> ( A = zero_zero_int ) ) ).

% add_0_iff
tff(fact_4369_exp__mod__exp,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma)) ).

% exp_mod_exp
tff(fact_4370_exp__mod__exp,axiom,
    ! [Ma: nat,Nb: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Ma)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Ma)) ).

% exp_mod_exp
tff(fact_4371_exp__mod__exp,axiom,
    ! [Ma: nat,Nb: nat] : modulo364778990260209775nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Ma),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_n356916108424825756nteger(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Ma)) ).

% exp_mod_exp
tff(fact_4372_crossproduct__noteq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( A != B )
        & ( C != D ) )
    <=> ( aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),D)) != aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),D)),aa_real_real(times_times_real(B),C)) ) ) ).

% crossproduct_noteq
tff(fact_4373_crossproduct__noteq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ( A != B )
        & ( C != D ) )
    <=> ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D)) != aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),D)),aa_rat_rat(times_times_rat(B),C)) ) ) ).

% crossproduct_noteq
tff(fact_4374_crossproduct__noteq,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ( A != B )
        & ( C != D ) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D)) != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),D)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ) ).

% crossproduct_noteq
tff(fact_4375_crossproduct__noteq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ( A != B )
        & ( C != D ) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D)) != aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),D)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% crossproduct_noteq
tff(fact_4376_crossproduct__eq,axiom,
    ! [W: real,Y2: real,X: real,Z2: real] :
      ( ( aa_real_real(plus_plus_real(aa_real_real(times_times_real(W),Y2)),aa_real_real(times_times_real(X),Z2)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(W),Z2)),aa_real_real(times_times_real(X),Y2)) )
    <=> ( ( W = X )
        | ( Y2 = Z2 ) ) ) ).

% crossproduct_eq
tff(fact_4377_crossproduct__eq,axiom,
    ! [W: rat,Y2: rat,X: rat,Z2: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(W),Y2)),aa_rat_rat(times_times_rat(X),Z2)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(W),Z2)),aa_rat_rat(times_times_rat(X),Y2)) )
    <=> ( ( W = X )
        | ( Y2 = Z2 ) ) ) ).

% crossproduct_eq
tff(fact_4378_crossproduct__eq,axiom,
    ! [W: nat,Y2: nat,X: nat,Z2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,W),Y2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Z2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,W),Z2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Y2)) )
    <=> ( ( W = X )
        | ( Y2 = Z2 ) ) ) ).

% crossproduct_eq
tff(fact_4379_crossproduct__eq,axiom,
    ! [W: int,Y2: int,X: int,Z2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Y2)),aa_int_int(aa_int_fun_int_int(times_times_int,X),Z2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z2)),aa_int_int(aa_int_fun_int_int(times_times_int,X),Y2)) )
    <=> ( ( W = X )
        | ( Y2 = Z2 ) ) ) ).

% crossproduct_eq
tff(fact_4380_power2__diff,axiom,
    ! [X: complex,Y2: complex] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(minus_minus_complex(X),Y2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),X)),Y2)) ).

% power2_diff
tff(fact_4381_power2__diff,axiom,
    ! [X: real,Y2: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(X),Y2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X)),Y2)) ).

% power2_diff
tff(fact_4382_power2__diff,axiom,
    ! [X: rat,Y2: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(minus_minus_rat(X),Y2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),X)),Y2)) ).

% power2_diff
tff(fact_4383_power2__diff,axiom,
    ! [X: int,Y2: int] : aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X)),Y2)) ).

% power2_diff
tff(fact_4384_divmod__digit__1_I2_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B)))
         => ( minus_8373710615458151222nteger(modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B)),B) = modulo364778990260209775nteger(A,B) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_4385_divmod__digit__1_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_4386_divmod__digit__1_I2_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)))
         => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_4387_even__mask__div__iff_H,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Ma),one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ).

% even_mask_div_iff'
tff(fact_4388_even__mask__div__iff_H,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ).

% even_mask_div_iff'
tff(fact_4389_even__mask__div__iff_H,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Ma)),one_one_int)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ).

% even_mask_div_iff'
tff(fact_4390_even__mask__div__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Ma),one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)))
    <=> ( ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb) = zero_z3403309356797280102nteger )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ) ).

% even_mask_div_iff
tff(fact_4391_even__mask__div__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))
    <=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ) ).

% even_mask_div_iff
tff(fact_4392_even__mask__div__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Ma)),one_one_int)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)))
    <=> ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb) = zero_zero_int )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ) ).

% even_mask_div_iff
tff(fact_4393_neg__zmod__mult__2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),one_one_int)),A))),one_one_int) ) ) ).

% neg_zmod_mult_2
tff(fact_4394_vebt__buildup_Osimps_I3_J,axiom,
    ! [Va3: nat] :
      aa_nat_VEBT_VEBT(vEBT_vebt_buildup,aa_nat_nat(suc,aa_nat_nat(suc,Va3))) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va3))),
        $let(
          half: nat,
          half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
          aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half),aa_nat_VEBT_VEBT(vEBT_vebt_buildup,half))),aa_nat_VEBT_VEBT(vEBT_vebt_buildup,half)) ),
        $let(
          half: nat,
          half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
          aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va3)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half)),aa_nat_VEBT_VEBT(vEBT_vebt_buildup,half))),aa_nat_VEBT_VEBT(vEBT_vebt_buildup,aa_nat_nat(suc,half))) ) ) ).

% vebt_buildup.simps(3)
tff(fact_4395_divmod__step__eq,axiom,
    ! [L: num,Q: nat,R: nat] :
      unique5026877609467782581ep_nat(L,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Q),R)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,L)),R),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Q)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,R),aa_num_nat(numeral_numeral_nat,L))),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Q)),R)) ).

% divmod_step_eq
tff(fact_4396_divmod__step__eq,axiom,
    ! [L: num,Q: int,R: int] :
      aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q),R)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,L)),R),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Q)),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,R),aa_num_int(numeral_numeral_int,L))),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Q)),R)) ).

% divmod_step_eq
tff(fact_4397_divmod__step__eq,axiom,
    ! [L: num,Q: code_integer,R: code_integer] :
      unique4921790084139445826nteger(L,produc1086072967326762835nteger(Q,R)) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(L)),R),produc1086072967326762835nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Q),one_one_Code_integer),minus_8373710615458151222nteger(R,numera6620942414471956472nteger(L))),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Q),R)) ).

% divmod_step_eq
tff(fact_4398_triangle__def,axiom,
    ! [Nb: nat] : aa_nat_nat(nat_triangle,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(suc,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% triangle_def
tff(fact_4399_vebt__buildup_Opelims,axiom,
    ! [X: nat,Y2: vEBT_VEBT] :
      ( ( aa_nat_VEBT_VEBT(vEBT_vebt_buildup,X) = Y2 )
     => ( aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),X)
       => ( ( ( X = zero_zero_nat )
           => ( ( Y2 = aa_o_VEBT_VEBT(vEBT_Leaf($false),$false) )
             => ~ aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),zero_zero_nat) ) )
         => ( ( ( X = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y2 = aa_o_VEBT_VEBT(vEBT_Leaf($false),$false) )
               => ~ aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),aa_nat_nat(suc,zero_zero_nat)) ) )
           => ~ ! [Va: nat] :
                  ( ( X = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
                 => ( ( Y2 = $ite(
                          aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va))),
                          $let(
                            half: nat,
                            half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                            aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half),aa_nat_VEBT_VEBT(vEBT_vebt_buildup,half))),aa_nat_VEBT_VEBT(vEBT_vebt_buildup,half)) ),
                          $let(
                            half: nat,
                            half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                            aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half)),aa_nat_VEBT_VEBT(vEBT_vebt_buildup,half))),aa_nat_VEBT_VEBT(vEBT_vebt_buildup,aa_nat_nat(suc,half))) ) ) )
                   => ~ aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),aa_nat_nat(suc,aa_nat_nat(suc,Va))) ) ) ) ) ) ) ).

% vebt_buildup.pelims
tff(fact_4400_set__decode__Suc,axiom,
    ! [Nb: nat,X: nat] :
      ( aa_set_nat_o(member_nat(aa_nat_nat(suc,Nb)),nat_set_decode(X))
    <=> aa_set_nat_o(member_nat(Nb),nat_set_decode(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% set_decode_Suc
tff(fact_4401_set__decode__0,axiom,
    ! [X: nat] :
      ( aa_set_nat_o(member_nat(zero_zero_nat),nat_set_decode(X))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X) ) ).

% set_decode_0
tff(fact_4402_set__decode__def,axiom,
    ! [X: nat] : nat_set_decode(X) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bo,X)) ).

% set_decode_def
tff(fact_4403_even__mult__exp__div__exp__iff,axiom,
    ! [A: code_integer,Ma: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Ma)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Ma)
        | ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb) = zero_z3403309356797280102nteger )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Ma)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_4404_even__mult__exp__div__exp__iff,axiom,
    ! [A: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Ma)
        | ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Ma)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_4405_even__mult__exp__div__exp__iff,axiom,
    ! [A: int,Ma: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Ma))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Ma)
        | ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb) = zero_zero_int )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
          & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Ma)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_4406_idiff__infinity,axiom,
    ! [Nb: extended_enat] : minus_3235023915231533773d_enat(extend5688581933313929465d_enat,Nb) = extend5688581933313929465d_enat ).

% idiff_infinity
tff(fact_4407_idiff__0,axiom,
    ! [Nb: extended_enat] : minus_3235023915231533773d_enat(zero_z5237406670263579293d_enat,Nb) = zero_z5237406670263579293d_enat ).

% idiff_0
tff(fact_4408_idiff__0__right,axiom,
    ! [Nb: extended_enat] : minus_3235023915231533773d_enat(Nb,zero_z5237406670263579293d_enat) = Nb ).

% idiff_0_right
tff(fact_4409_Suc__diff__diff,axiom,
    ! [Ma: nat,Nb: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Ma)),Nb)),aa_nat_nat(suc,K2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb)),K2) ).

% Suc_diff_diff
tff(fact_4410_diff__Suc__Suc,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Ma)),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb) ).

% diff_Suc_Suc
tff(fact_4411_diff__0__eq__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,zero_zero_nat),Nb) = zero_zero_nat ).

% diff_0_eq_0
tff(fact_4412_diff__self__eq__0,axiom,
    ! [Ma: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Ma) = zero_zero_nat ).

% diff_self_eq_0
tff(fact_4413_diff__diff__cancel,axiom,
    ! [I2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),I2)) = I2 ) ) ).

% diff_diff_cancel
tff(fact_4414_diff__diff__left,axiom,
    ! [I2: nat,J: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I2),J)),K2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K2)) ).

% diff_diff_left
tff(fact_4415_idiff__enat__0,axiom,
    ! [Nb: extended_enat] : minus_3235023915231533773d_enat(aa_nat_Extended_enat(extended_enat2,zero_zero_nat),Nb) = aa_nat_Extended_enat(extended_enat2,zero_zero_nat) ).

% idiff_enat_0
tff(fact_4416_idiff__enat__0__right,axiom,
    ! [Nb: extended_enat] : minus_3235023915231533773d_enat(Nb,aa_nat_Extended_enat(extended_enat2,zero_zero_nat)) = Nb ).

% idiff_enat_0_right
tff(fact_4417_idiff__enat__enat,axiom,
    ! [A: nat,B: nat] : minus_3235023915231533773d_enat(aa_nat_Extended_enat(extended_enat2,A),aa_nat_Extended_enat(extended_enat2,B)) = aa_nat_Extended_enat(extended_enat2,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) ).

% idiff_enat_enat
tff(fact_4418_idiff__self,axiom,
    ! [Nb: extended_enat] :
      ( ( Nb != extend5688581933313929465d_enat )
     => ( minus_3235023915231533773d_enat(Nb,Nb) = zero_z5237406670263579293d_enat ) ) ).

% idiff_self
tff(fact_4419_add__diff__cancel__enat,axiom,
    ! [X: extended_enat,Y2: extended_enat] :
      ( ( X != extend5688581933313929465d_enat )
     => ( minus_3235023915231533773d_enat(plus_p3455044024723400733d_enat(X,Y2),X) = Y2 ) ) ).

% add_diff_cancel_enat
tff(fact_4420_zero__less__diff,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Ma))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb) ) ).

% zero_less_diff
tff(fact_4421_diff__is__0__eq_H,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb) = zero_zero_nat ) ) ).

% diff_is_0_eq'
tff(fact_4422_diff__is__0__eq,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb) = zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ).

% diff_is_0_eq
tff(fact_4423_Nat_Odiff__diff__right,axiom,
    ! [K2: nat,J: nat,I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),J) ) ) ).

% Nat.diff_diff_right
tff(fact_4424_Nat_Oadd__diff__assoc2,axiom,
    ! [K2: nat,J: nat,I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K2)),I2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I2)),K2) ) ) ).

% Nat.add_diff_assoc2
tff(fact_4425_Nat_Oadd__diff__assoc,axiom,
    ! [K2: nat,J: nat,I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),J)),K2) ) ) ).

% Nat.add_diff_assoc
tff(fact_4426_diff__Suc__1,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Nb)),one_one_nat) = Nb ).

% diff_Suc_1
tff(fact_4427_idiff__infinity__right,axiom,
    ! [A: nat] : minus_3235023915231533773d_enat(aa_nat_Extended_enat(extended_enat2,A),extend5688581933313929465d_enat) = zero_z5237406670263579293d_enat ).

% idiff_infinity_right
tff(fact_4428_triangle__Suc,axiom,
    ! [Nb: nat] : aa_nat_nat(nat_triangle,aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,Nb)),aa_nat_nat(suc,Nb)) ).

% triangle_Suc
tff(fact_4429_Suc__pred,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))) = Nb ) ) ).

% Suc_pred
tff(fact_4430_diff__Suc__diff__eq1,axiom,
    ! [K2: nat,J: nat,I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I2),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K2))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),aa_nat_nat(suc,J)) ) ) ).

% diff_Suc_diff_eq1
tff(fact_4431_diff__Suc__diff__eq2,axiom,
    ! [K2: nat,J: nat,I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K2))),I2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,J)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),I2)) ) ) ).

% diff_Suc_diff_eq2
tff(fact_4432_Suc__diff__1,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)) = Nb ) ) ).

% Suc_diff_1
tff(fact_4433_odd__Suc__minus__one,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))) = Nb ) ) ).

% odd_Suc_minus_one
tff(fact_4434_even__diff__nat,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb)) ) ) ).

% even_diff_nat
tff(fact_4435_odd__two__times__div__two__nat,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat) ) ) ).

% odd_two_times_div_two_nat
tff(fact_4436_diff__commute,axiom,
    ! [I2: nat,J: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I2),J)),K2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I2),K2)),J) ).

% diff_commute
tff(fact_4437_zero__induct__lemma,axiom,
    ! [P: fun_nat_o,K2: nat,I2: nat] :
      ( aa_nat_o(P,K2)
     => ( ! [N: nat] :
            ( aa_nat_o(P,aa_nat_nat(suc,N))
           => aa_nat_o(P,N) )
       => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),I2)) ) ) ).

% zero_induct_lemma
tff(fact_4438_Diff__mono,axiom,
    ! [A2: set_int,C4: set_int,D3: set_int,B3: set_int] :
      ( ord_less_eq_set_int(A2,C4)
     => ( ord_less_eq_set_int(D3,B3)
       => ord_less_eq_set_int(minus_minus_set_int(A2,B3),minus_minus_set_int(C4,D3)) ) ) ).

% Diff_mono
tff(fact_4439_Diff__subset,axiom,
    ! [A2: set_int,B3: set_int] : ord_less_eq_set_int(minus_minus_set_int(A2,B3),A2) ).

% Diff_subset
tff(fact_4440_double__diff,axiom,
    ! [A2: set_int,B3: set_int,C4: set_int] :
      ( ord_less_eq_set_int(A2,B3)
     => ( ord_less_eq_set_int(B3,C4)
       => ( minus_minus_set_int(B3,minus_minus_set_int(C4,A2)) = A2 ) ) ) ).

% double_diff
tff(fact_4441_minus__nat_Odiff__0,axiom,
    ! [Ma: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),zero_zero_nat) = Ma ).

% minus_nat.diff_0
tff(fact_4442_diffs0__imp__equal,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb) = zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Ma) = zero_zero_nat )
       => ( Ma = Nb ) ) ) ).

% diffs0_imp_equal
tff(fact_4443_diff__less__mono2,axiom,
    ! [Ma: nat,Nb: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),Ma)) ) ) ).

% diff_less_mono2
tff(fact_4444_less__imp__diff__less,axiom,
    ! [J: nat,K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),K2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),Nb)),K2) ) ).

% less_imp_diff_less
tff(fact_4445_diff__le__mono2,axiom,
    ! [Ma: nat,Nb: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),Ma)) ) ).

% diff_le_mono2
tff(fact_4446_le__diff__iff_H,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C),B))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ) ) ).

% le_diff_iff'
tff(fact_4447_diff__le__self,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb)),Ma) ).

% diff_le_self
tff(fact_4448_diff__le__mono,axiom,
    ! [Ma: nat,Nb: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),L)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),L)) ) ).

% diff_le_mono
tff(fact_4449_Nat_Odiff__diff__eq,axiom,
    ! [K2: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Ma)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),K2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb) ) ) ) ).

% Nat.diff_diff_eq
tff(fact_4450_le__diff__iff,axiom,
    ! [K2: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Ma)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),K2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ) ) ).

% le_diff_iff
tff(fact_4451_eq__diff__iff,axiom,
    ! [K2: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Ma)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),K2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2) )
        <=> ( Ma = Nb ) ) ) ) ).

% eq_diff_iff
tff(fact_4452_Nat_Odiff__cancel,axiom,
    ! [K2: nat,Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb) ).

% Nat.diff_cancel
tff(fact_4453_diff__cancel2,axiom,
    ! [Ma: nat,K2: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb) ).

% diff_cancel2
tff(fact_4454_diff__add__inverse,axiom,
    ! [Nb: nat,Ma: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Ma)),Nb) = Ma ).

% diff_add_inverse
tff(fact_4455_diff__add__inverse2,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb)),Nb) = Ma ).

% diff_add_inverse2
tff(fact_4456_diff__mult__distrib2,axiom,
    ! [K2: nat,Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Nb)) ).

% diff_mult_distrib2
tff(fact_4457_diff__mult__distrib,axiom,
    ! [Ma: nat,Nb: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb)),K2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),K2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K2)) ).

% diff_mult_distrib
tff(fact_4458_dvd__diff__nat,axiom,
    ! [K2: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K2),Ma)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K2),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb)) ) ) ).

% dvd_diff_nat
tff(fact_4459_subset__decode__imp__le,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(nat_set_decode(Ma)),nat_set_decode(Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ).

% subset_decode_imp_le
tff(fact_4460_diff__less__Suc,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb)),aa_nat_nat(suc,Ma)) ).

% diff_less_Suc
tff(fact_4461_Suc__diff__Suc,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Ma)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb) ) ) ).

% Suc_diff_Suc
tff(fact_4462_diff__less,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ma)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb)),Ma) ) ) ).

% diff_less
tff(fact_4463_Suc__diff__le,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Ma)),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb)) ) ) ).

% Suc_diff_le
tff(fact_4464_diff__less__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),C)) ) ) ).

% diff_less_mono
tff(fact_4465_less__diff__iff,axiom,
    ! [K2: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Ma)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),K2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb) ) ) ) ).

% less_diff_iff
tff(fact_4466_diff__add__0,axiom,
    ! [Nb: nat,Ma: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Ma)) = zero_zero_nat ).

% diff_add_0
tff(fact_4467_less__diff__conv,axiom,
    ! [I2: nat,J: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),J) ) ).

% less_diff_conv
tff(fact_4468_add__diff__inverse__nat,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb)) = Ma ) ) ).

% add_diff_inverse_nat
tff(fact_4469_Nat_Ole__imp__diff__is__add,axiom,
    ! [I2: nat,J: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I2) = K2 )
      <=> ( J = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),I2) ) ) ) ).

% Nat.le_imp_diff_is_add
tff(fact_4470_Nat_Odiff__add__assoc2,axiom,
    ! [K2: nat,J: nat,I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I2)),K2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K2)),I2) ) ) ).

% Nat.diff_add_assoc2
tff(fact_4471_Nat_Odiff__add__assoc,axiom,
    ! [K2: nat,J: nat,I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),J)),K2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K2)) ) ) ).

% Nat.diff_add_assoc
tff(fact_4472_Nat_Ole__diff__conv2,axiom,
    ! [K2: nat,J: nat,I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K2))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),J) ) ) ).

% Nat.le_diff_conv2
tff(fact_4473_le__diff__conv,axiom,
    ! [J: nat,K2: nat,I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K2)),I2)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)) ) ).

% le_diff_conv
tff(fact_4474_diff__Suc__eq__diff__pred,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),one_one_nat)),Nb) ).

% diff_Suc_eq_diff_pred
tff(fact_4475_dvd__minus__self,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Ma))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Ma)
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ma),Nb) ) ) ).

% dvd_minus_self
tff(fact_4476_dvd__diffD,axiom,
    ! [K2: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K2),Nb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma)
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K2),Ma) ) ) ) ).

% dvd_diffD
tff(fact_4477_dvd__diffD1,axiom,
    ! [K2: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K2),Ma)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma)
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K2),Nb) ) ) ) ).

% dvd_diffD1
tff(fact_4478_less__eq__dvd__minus,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ma),Nb)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Ma)) ) ) ).

% less_eq_dvd_minus
tff(fact_4479_bezout1__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D2: nat,X4: nat,Y4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),A)
      & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),B)
      & ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y4)) = D2 )
        | ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y4)) = D2 ) ) ) ).

% bezout1_nat
tff(fact_4480_mod__if,axiom,
    ! [Ma: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb),Ma,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb)),Nb)) ).

% mod_if
tff(fact_4481_mod__geq,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb)),Nb) ) ) ).

% mod_geq
tff(fact_4482_le__mod__geq,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb)),Nb) ) ) ).

% le_mod_geq
tff(fact_4483_diff__enat__def,axiom,
    ! [A: extended_enat,B: extended_enat] : minus_3235023915231533773d_enat(A,B) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(aa_fun2186929853097002313d_enat(extend3600170679010898289d_enat,aTP_Lamm_bq(B)),extend5688581933313929465d_enat),A) ).

% diff_enat_def
tff(fact_4484_add__diff__assoc__enat,axiom,
    ! [Z2: extended_enat,Y2: extended_enat,X: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(Z2),Y2)
     => ( plus_p3455044024723400733d_enat(X,minus_3235023915231533773d_enat(Y2,Z2)) = minus_3235023915231533773d_enat(plus_p3455044024723400733d_enat(X,Y2),Z2) ) ) ).

% add_diff_assoc_enat
tff(fact_4485_diff__Suc__less,axiom,
    ! [Nb: nat,I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,I2))),Nb) ) ).

% diff_Suc_less
tff(fact_4486_nat__diff__split,axiom,
    ! [P: fun_nat_o,A: nat,B: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B))
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
         => aa_nat_o(P,zero_zero_nat) )
        & ! [D4: nat] :
            ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D4) )
           => aa_nat_o(P,D4) ) ) ) ).

% nat_diff_split
tff(fact_4487_nat__diff__split__asm,axiom,
    ! [P: fun_nat_o,A: nat,B: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B))
    <=> ~ ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
            & ~ aa_nat_o(P,zero_zero_nat) )
          | ? [D4: nat] :
              ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D4) )
              & ~ aa_nat_o(P,D4) ) ) ) ).

% nat_diff_split_asm
tff(fact_4488_less__diff__conv2,axiom,
    ! [K2: nat,J: nat,I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K2)),I2)
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)) ) ) ).

% less_diff_conv2
tff(fact_4489_nat__eq__add__iff1,axiom,
    ! [J: nat,I2: nat,U: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I2)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),U)),Ma) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb) )
      <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I2),J)),U)),Ma) = Nb ) ) ) ).

% nat_eq_add_iff1
tff(fact_4490_nat__eq__add__iff2,axiom,
    ! [I2: nat,J: nat,U: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),U)),Ma) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb) )
      <=> ( Ma = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I2)),U)),Nb) ) ) ) ).

% nat_eq_add_iff2
tff(fact_4491_nat__le__add__iff1,axiom,
    ! [J: nat,I2: nat,U: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),U)),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I2),J)),U)),Ma)),Nb) ) ) ).

% nat_le_add_iff1
tff(fact_4492_nat__le__add__iff2,axiom,
    ! [I2: nat,J: nat,U: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),U)),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I2)),U)),Nb)) ) ) ).

% nat_le_add_iff2
tff(fact_4493_nat__diff__add__eq1,axiom,
    ! [J: nat,I2: nat,U: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),U)),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I2),J)),U)),Ma)),Nb) ) ) ).

% nat_diff_add_eq1
tff(fact_4494_nat__diff__add__eq2,axiom,
    ! [I2: nat,J: nat,U: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),U)),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I2)),U)),Nb)) ) ) ).

% nat_diff_add_eq2
tff(fact_4495_mod__eq__dvd__iff__nat,axiom,
    ! [Nb: nat,Ma: nat,Q: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),Q) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),Q) )
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Q),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb)) ) ) ).

% mod_eq_dvd_iff_nat
tff(fact_4496_modulo__nat__def,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),Nb)),Nb)) ).

% modulo_nat_def
tff(fact_4497_power__diff,axiom,
    ! [A: complex,Nb: nat,Ma: nat] :
      ( ( A != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma)
       => ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb)) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),Ma)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),Nb)) ) ) ) ).

% power_diff
tff(fact_4498_power__diff,axiom,
    ! [A: real,Nb: nat,Ma: nat] :
      ( ( A != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb)) = aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Ma)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ) ) ).

% power_diff
tff(fact_4499_power__diff,axiom,
    ! [A: rat,Nb: nat,Ma: nat] :
      ( ( A != zero_zero_rat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma)
       => ( aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb)) = aa_rat_rat(divide_divide_rat(aa_nat_rat(power_power_rat(A),Ma)),aa_nat_rat(power_power_rat(A),Nb)) ) ) ) ).

% power_diff
tff(fact_4500_power__diff,axiom,
    ! [A: nat,Nb: nat,Ma: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ) ) ).

% power_diff
tff(fact_4501_power__diff,axiom,
    ! [A: int,Nb: nat,Ma: nat] :
      ( ( A != zero_zero_int )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma)
       => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Ma)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)) ) ) ) ).

% power_diff
tff(fact_4502_Suc__diff__eq__diff__pred,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Ma)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)) ) ) ).

% Suc_diff_eq_diff_pred
tff(fact_4503_Suc__pred_H,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( Nb = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)) ) ) ).

% Suc_pred'
tff(fact_4504_div__if,axiom,
    ! [Ma: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),Nb) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
        | ( Nb = zero_zero_nat ) ),
        zero_zero_nat,
        aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb)),Nb)) ) ).

% div_if
tff(fact_4505_div__geq,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb)),Nb)) ) ) ) ).

% div_geq
tff(fact_4506_add__eq__if,axiom,
    ! [Ma: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb) = $ite(Ma = zero_zero_nat,Nb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),one_one_nat)),Nb))) ).

% add_eq_if
tff(fact_4507_nat__less__add__iff1,axiom,
    ! [J: nat,I2: nat,U: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),U)),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I2),J)),U)),Ma)),Nb) ) ) ).

% nat_less_add_iff1
tff(fact_4508_nat__less__add__iff2,axiom,
    ! [I2: nat,J: nat,U: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I2),U)),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I2)),U)),Nb)) ) ) ).

% nat_less_add_iff2
tff(fact_4509_mult__eq__if,axiom,
    ! [Ma: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Nb) = $ite(Ma = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),one_one_nat)),Nb))) ).

% mult_eq_if
tff(fact_4510_dvd__minus__add,axiom,
    ! [Q: nat,Nb: nat,R: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Q),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Q),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R),Ma))
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Q))
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R),Ma)),Q))) ) ) ) ).

% dvd_minus_add
tff(fact_4511_mod__nat__eqI,axiom,
    ! [R: nat,Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,R),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,R),Ma)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),R))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),Nb) = R ) ) ) ) ).

% mod_nat_eqI
tff(fact_4512_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [Nb: nat,Ma: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Ma)) != zero_zero_nat ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_4513_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [Nb: nat,Ma: nat] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb) != zero_zero_int )
     => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Ma)) != zero_zero_int ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_4514_power__diff__power__eq,axiom,
    ! [A: nat,Ma: nat,Nb: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Ma)))) ) ) ).

% power_diff_power_eq
tff(fact_4515_power__diff__power__eq,axiom,
    ! [A: int,Ma: nat,Nb: nat] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Ma)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb)),aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Ma)))) ) ) ).

% power_diff_power_eq
tff(fact_4516_power__eq__if,axiom,
    ! [P2: complex,Ma: nat] :
      aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,P2),Ma) = $ite(Ma = zero_zero_nat,one_one_complex,aa_complex_complex(times_times_complex(P2),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),one_one_nat)))) ).

% power_eq_if
tff(fact_4517_power__eq__if,axiom,
    ! [P2: real,Ma: nat] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,P2),Ma) = $ite(Ma = zero_zero_nat,one_one_real,aa_real_real(times_times_real(P2),aa_nat_real(aa_real_fun_nat_real(power_power_real,P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),one_one_nat)))) ).

% power_eq_if
tff(fact_4518_power__eq__if,axiom,
    ! [P2: rat,Ma: nat] :
      aa_nat_rat(power_power_rat(P2),Ma) = $ite(Ma = zero_zero_nat,one_one_rat,aa_rat_rat(times_times_rat(P2),aa_nat_rat(power_power_rat(P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),one_one_nat)))) ).

% power_eq_if
tff(fact_4519_power__eq__if,axiom,
    ! [P2: nat,Ma: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P2),Ma) = $ite(Ma = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,P2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),one_one_nat)))) ).

% power_eq_if
tff(fact_4520_power__eq__if,axiom,
    ! [P2: int,Ma: nat] :
      aa_nat_int(aa_int_fun_nat_int(power_power_int,P2),Ma) = $ite(Ma = zero_zero_nat,one_one_int,aa_int_int(aa_int_fun_int_int(times_times_int,P2),aa_nat_int(aa_int_fun_nat_int(power_power_int,P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),one_one_nat)))) ).

% power_eq_if
tff(fact_4521_power__minus__mult,axiom,
    ! [Nb: nat,A: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),A) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),Nb) ) ) ).

% power_minus_mult
tff(fact_4522_power__minus__mult,axiom,
    ! [Nb: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),A) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb) ) ) ).

% power_minus_mult
tff(fact_4523_power__minus__mult,axiom,
    ! [Nb: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),A) = aa_nat_rat(power_power_rat(A),Nb) ) ) ).

% power_minus_mult
tff(fact_4524_power__minus__mult,axiom,
    ! [Nb: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),A) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb) ) ) ).

% power_minus_mult
tff(fact_4525_power__minus__mult,axiom,
    ! [Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),A) = aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb) ) ) ).

% power_minus_mult
tff(fact_4526_diff__le__diff__pow,axiom,
    ! [K2: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K2),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K2),Nb))) ) ).

% diff_le_diff_pow
tff(fact_4527_le__div__geq,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb)),Nb)) ) ) ) ).

% le_div_geq
tff(fact_4528_mult__exp__mod__exp__eq,axiom,
    ! [Ma: nat,Nb: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Ma)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_4529_mult__exp__mod__exp__eq,axiom,
    ! [Ma: nat,Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Ma))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Ma)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Ma)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_4530_mult__exp__mod__exp__eq,axiom,
    ! [Ma: nat,Nb: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => ( modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Ma)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,modulo364778990260209775nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Ma)))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Ma)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_4531_int__power__div__base,axiom,
    ! [Ma: nat,K2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ma)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K2)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,K2),Ma)),K2) = aa_nat_int(aa_int_fun_nat_int(power_power_int,K2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),aa_nat_nat(suc,zero_zero_nat))) ) ) ) ).

% int_power_div_base
tff(fact_4532_even__mod__4__div__2,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_nat_nat(suc,zero_zero_nat) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% even_mod_4_div_2
tff(fact_4533_exp__div__exp__eq,axiom,
    ! [Ma: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,
          aa_o_nat(zero_n2687167440665602831ol_nat,
            ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma) != zero_zero_nat )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma) ))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb))) ).

% exp_div_exp_eq
tff(fact_4534_exp__div__exp__eq,axiom,
    ! [Ma: nat,Nb: nat] :
      aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Ma)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,
          aa_o_int(zero_n2684676970156552555ol_int,
            ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Ma) != zero_zero_int )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma) ))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb))) ).

% exp_div_exp_eq
tff(fact_4535_exp__div__exp__eq,axiom,
    ! [Ma: nat,Nb: nat] :
      divide6298287555418463151nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Ma),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,
          zero_n356916108424825756nteger(( ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Ma) != zero_z3403309356797280102nteger )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma) ))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb))) ).

% exp_div_exp_eq
tff(fact_4536_real__average__minus__second,axiom,
    ! [B: real,A: real] : aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(B),A)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),A) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(B),A)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% real_average_minus_second
tff(fact_4537_real__average__minus__first,axiom,
    ! [A: real,B: real] : aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(A),B)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),A) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(B),A)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% real_average_minus_first
tff(fact_4538_signed__take__bit__rec,axiom,
    ! [Nb: nat,A: code_integer] :
      bit_ri6519982836138164636nteger(Nb,A) = $ite(Nb = zero_zero_nat,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))),plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_ri6519982836138164636nteger(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat),divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))))))) ).

% signed_take_bit_rec
tff(fact_4539_signed__take__bit__rec,axiom,
    ! [Nb: nat,A: int] :
      aa_int_int(bit_ri631733984087533419it_int(Nb),A) = $ite(Nb = zero_zero_nat,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))))) ).

% signed_take_bit_rec
tff(fact_4540_Bolzano,axiom,
    ! [A: real,B: real,P: fun_real_fun_real_o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( ! [A3: real,B2: real,C3: real] :
            ( aa_real_o(aa_real_fun_real_o(P,A3),B2)
           => ( aa_real_o(aa_real_fun_real_o(P,B2),C3)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A3),B2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),C3)
                 => aa_real_o(aa_real_fun_real_o(P,A3),C3) ) ) ) )
       => ( ! [X4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X4)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),B)
               => ? [D5: real] :
                    ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D5)
                    & ! [A3: real,B2: real] :
                        ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A3),X4)
                          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),B2)
                          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(B2),A3)),D5) )
                       => aa_real_o(aa_real_fun_real_o(P,A3),B2) ) ) ) )
         => aa_real_o(aa_real_fun_real_o(P,A),B) ) ) ) ).

% Bolzano
tff(fact_4541_divmod__step__def,axiom,
    ! [L: num,Qr: product_prod_nat_nat] : unique5026877609467782581ep_nat(L,Qr) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_br(L)),Qr) ).

% divmod_step_def
tff(fact_4542_divmod__step__def,axiom,
    ! [L: num,Qr: product_prod_int_int] : aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(L),Qr) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_bs(L)),Qr) ).

% divmod_step_def
tff(fact_4543_divmod__step__def,axiom,
    ! [L: num,Qr: produc8923325533196201883nteger] : unique4921790084139445826nteger(L,Qr) = produc6916734918728496179nteger(aTP_Lamm_bt(L),Qr) ).

% divmod_step_def
tff(fact_4544_signed__take__bit__Suc__bit1,axiom,
    ! [Nb: nat,K2: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Nb)),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(Nb),aa_num_int(numeral_numeral_int,K2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% signed_take_bit_Suc_bit1
tff(fact_4545_take__bit__rec,axiom,
    ! [Nb: nat,A: code_integer] :
      bit_se1745604003318907178nteger(Nb,A) = $ite(Nb = zero_zero_nat,zero_z3403309356797280102nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,bit_se1745604003318907178nteger(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat),divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))))),numera6620942414471956472nteger(aa_num_num(bit0,one))),modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ).

% take_bit_rec
tff(fact_4546_take__bit__rec,axiom,
    ! [Nb: nat,A: int] :
      aa_int_int(bit_se2923211474154528505it_int(Nb),A) = $ite(Nb = zero_zero_nat,zero_zero_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ).

% take_bit_rec
tff(fact_4547_take__bit__rec,axiom,
    ! [Nb: nat,A: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A) = $ite(Nb = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% take_bit_rec
tff(fact_4548_verit__minus__simplify_I4_J,axiom,
    ! [B: real] : aa_real_real(uminus_uminus_real,aa_real_real(uminus_uminus_real,B)) = B ).

% verit_minus_simplify(4)
tff(fact_4549_verit__minus__simplify_I4_J,axiom,
    ! [B: int] : aa_int_int(uminus_uminus_int,aa_int_int(uminus_uminus_int,B)) = B ).

% verit_minus_simplify(4)
tff(fact_4550_verit__minus__simplify_I4_J,axiom,
    ! [B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(uminus1482373934393186551omplex,B)) = B ).

% verit_minus_simplify(4)
tff(fact_4551_verit__minus__simplify_I4_J,axiom,
    ! [B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(uminus_uminus_rat,B)) = B ).

% verit_minus_simplify(4)
tff(fact_4552_verit__minus__simplify_I4_J,axiom,
    ! [B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = B ).

% verit_minus_simplify(4)
tff(fact_4553_add_Oinverse__inverse,axiom,
    ! [A: real] : aa_real_real(uminus_uminus_real,aa_real_real(uminus_uminus_real,A)) = A ).

% add.inverse_inverse
tff(fact_4554_add_Oinverse__inverse,axiom,
    ! [A: int] : aa_int_int(uminus_uminus_int,aa_int_int(uminus_uminus_int,A)) = A ).

% add.inverse_inverse
tff(fact_4555_add_Oinverse__inverse,axiom,
    ! [A: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(uminus1482373934393186551omplex,A)) = A ).

% add.inverse_inverse
tff(fact_4556_add_Oinverse__inverse,axiom,
    ! [A: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(uminus_uminus_rat,A)) = A ).

% add.inverse_inverse
tff(fact_4557_add_Oinverse__inverse,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = A ).

% add.inverse_inverse
tff(fact_4558_neg__equal__iff__equal,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = aa_real_real(uminus_uminus_real,B) )
    <=> ( A = B ) ) ).

% neg_equal_iff_equal
tff(fact_4559_neg__equal__iff__equal,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = aa_int_int(uminus_uminus_int,B) )
    <=> ( A = B ) ) ).

% neg_equal_iff_equal
tff(fact_4560_neg__equal__iff__equal,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,A) = aa_complex_complex(uminus1482373934393186551omplex,B) )
    <=> ( A = B ) ) ).

% neg_equal_iff_equal
tff(fact_4561_neg__equal__iff__equal,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = aa_rat_rat(uminus_uminus_rat,B) )
    <=> ( A = B ) ) ).

% neg_equal_iff_equal
tff(fact_4562_neg__equal__iff__equal,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) )
    <=> ( A = B ) ) ).

% neg_equal_iff_equal
tff(fact_4563_Compl__subset__Compl__iff,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ord_less_eq_set_int(uminus1532241313380277803et_int(A2),uminus1532241313380277803et_int(B3))
    <=> ord_less_eq_set_int(B3,A2) ) ).

% Compl_subset_Compl_iff
tff(fact_4564_Compl__anti__mono,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ord_less_eq_set_int(A2,B3)
     => ord_less_eq_set_int(uminus1532241313380277803et_int(B3),uminus1532241313380277803et_int(A2)) ) ).

% Compl_anti_mono
tff(fact_4565_verit__eq__simplify_I9_J,axiom,
    ! [X32: num,Y32: num] :
      ( ( aa_num_num(bit1,X32) = aa_num_num(bit1,Y32) )
    <=> ( X32 = Y32 ) ) ).

% verit_eq_simplify(9)
tff(fact_4566_semiring__norm_I90_J,axiom,
    ! [Ma: num,Nb: num] :
      ( ( aa_num_num(bit1,Ma) = aa_num_num(bit1,Nb) )
    <=> ( Ma = Nb ) ) ).

% semiring_norm(90)
tff(fact_4567_neg__le__iff__le,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% neg_le_iff_le
tff(fact_4568_neg__le__iff__le,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ).

% neg_le_iff_le
tff(fact_4569_neg__le__iff__le,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% neg_le_iff_le
tff(fact_4570_neg__le__iff__le,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% neg_le_iff_le
tff(fact_4571_add_Oinverse__neutral,axiom,
    aa_real_real(uminus_uminus_real,zero_zero_real) = zero_zero_real ).

% add.inverse_neutral
tff(fact_4572_add_Oinverse__neutral,axiom,
    aa_int_int(uminus_uminus_int,zero_zero_int) = zero_zero_int ).

% add.inverse_neutral
tff(fact_4573_add_Oinverse__neutral,axiom,
    aa_complex_complex(uminus1482373934393186551omplex,zero_zero_complex) = zero_zero_complex ).

% add.inverse_neutral
tff(fact_4574_add_Oinverse__neutral,axiom,
    aa_rat_rat(uminus_uminus_rat,zero_zero_rat) = zero_zero_rat ).

% add.inverse_neutral
tff(fact_4575_add_Oinverse__neutral,axiom,
    aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% add.inverse_neutral
tff(fact_4576_neg__0__equal__iff__equal,axiom,
    ! [A: real] :
      ( ( zero_zero_real = aa_real_real(uminus_uminus_real,A) )
    <=> ( zero_zero_real = A ) ) ).

% neg_0_equal_iff_equal
tff(fact_4577_neg__0__equal__iff__equal,axiom,
    ! [A: int] :
      ( ( zero_zero_int = aa_int_int(uminus_uminus_int,A) )
    <=> ( zero_zero_int = A ) ) ).

% neg_0_equal_iff_equal
tff(fact_4578_neg__0__equal__iff__equal,axiom,
    ! [A: complex] :
      ( ( zero_zero_complex = aa_complex_complex(uminus1482373934393186551omplex,A) )
    <=> ( zero_zero_complex = A ) ) ).

% neg_0_equal_iff_equal
tff(fact_4579_neg__0__equal__iff__equal,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = aa_rat_rat(uminus_uminus_rat,A) )
    <=> ( zero_zero_rat = A ) ) ).

% neg_0_equal_iff_equal
tff(fact_4580_neg__0__equal__iff__equal,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) )
    <=> ( zero_z3403309356797280102nteger = A ) ) ).

% neg_0_equal_iff_equal
tff(fact_4581_neg__equal__0__iff__equal,axiom,
    ! [A: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% neg_equal_0_iff_equal
tff(fact_4582_neg__equal__0__iff__equal,axiom,
    ! [A: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% neg_equal_0_iff_equal
tff(fact_4583_neg__equal__0__iff__equal,axiom,
    ! [A: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,A) = zero_zero_complex )
    <=> ( A = zero_zero_complex ) ) ).

% neg_equal_0_iff_equal
tff(fact_4584_neg__equal__0__iff__equal,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% neg_equal_0_iff_equal
tff(fact_4585_neg__equal__0__iff__equal,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_0_iff_equal
tff(fact_4586_equal__neg__zero,axiom,
    ! [A: real] :
      ( ( A = aa_real_real(uminus_uminus_real,A) )
    <=> ( A = zero_zero_real ) ) ).

% equal_neg_zero
tff(fact_4587_equal__neg__zero,axiom,
    ! [A: int] :
      ( ( A = aa_int_int(uminus_uminus_int,A) )
    <=> ( A = zero_zero_int ) ) ).

% equal_neg_zero
tff(fact_4588_equal__neg__zero,axiom,
    ! [A: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,A) )
    <=> ( A = zero_zero_rat ) ) ).

% equal_neg_zero
tff(fact_4589_equal__neg__zero,axiom,
    ! [A: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% equal_neg_zero
tff(fact_4590_neg__equal__zero,axiom,
    ! [A: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = A )
    <=> ( A = zero_zero_real ) ) ).

% neg_equal_zero
tff(fact_4591_neg__equal__zero,axiom,
    ! [A: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = A )
    <=> ( A = zero_zero_int ) ) ).

% neg_equal_zero
tff(fact_4592_neg__equal__zero,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = A )
    <=> ( A = zero_zero_rat ) ) ).

% neg_equal_zero
tff(fact_4593_neg__equal__zero,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = A )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_zero
tff(fact_4594_neg__less__iff__less,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ).

% neg_less_iff_less
tff(fact_4595_neg__less__iff__less,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% neg_less_iff_less
tff(fact_4596_neg__less__iff__less,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% neg_less_iff_less
tff(fact_4597_neg__less__iff__less,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B) ) ).

% neg_less_iff_less
tff(fact_4598_neg__numeral__eq__iff,axiom,
    ! [Ma: num,Nb: num] :
      ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Ma)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)) )
    <=> ( Ma = Nb ) ) ).

% neg_numeral_eq_iff
tff(fact_4599_neg__numeral__eq__iff,axiom,
    ! [Ma: num,Nb: num] :
      ( ( aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Ma)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) )
    <=> ( Ma = Nb ) ) ).

% neg_numeral_eq_iff
tff(fact_4600_neg__numeral__eq__iff,axiom,
    ! [Ma: num,Nb: num] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Ma)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb)) )
    <=> ( Ma = Nb ) ) ).

% neg_numeral_eq_iff
tff(fact_4601_neg__numeral__eq__iff,axiom,
    ! [Ma: num,Nb: num] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Ma)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)) )
    <=> ( Ma = Nb ) ) ).

% neg_numeral_eq_iff
tff(fact_4602_neg__numeral__eq__iff,axiom,
    ! [Ma: num,Nb: num] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Ma)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb)) )
    <=> ( Ma = Nb ) ) ).

% neg_numeral_eq_iff
tff(fact_4603_mult__minus__right,axiom,
    ! [A: real,B: real] : aa_real_real(times_times_real(A),aa_real_real(uminus_uminus_real,B)) = aa_real_real(uminus_uminus_real,aa_real_real(times_times_real(A),B)) ).

% mult_minus_right
tff(fact_4604_mult__minus__right,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ).

% mult_minus_right
tff(fact_4605_mult__minus__right,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(times_times_complex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(times_times_complex(A),B)) ).

% mult_minus_right
tff(fact_4606_mult__minus__right,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(times_times_rat(A),B)) ).

% mult_minus_right
tff(fact_4607_mult__minus__right,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) ).

% mult_minus_right
tff(fact_4608_minus__mult__minus,axiom,
    ! [A: real,B: real] : aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) = aa_real_real(times_times_real(A),B) ).

% minus_mult_minus
tff(fact_4609_minus__mult__minus,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),B) ).

% minus_mult_minus
tff(fact_4610_minus__mult__minus,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(times_times_complex(A),B) ).

% minus_mult_minus
tff(fact_4611_minus__mult__minus,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(times_times_rat(A),B) ).

% minus_mult_minus
tff(fact_4612_minus__mult__minus,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) ).

% minus_mult_minus
tff(fact_4613_mult__minus__left,axiom,
    ! [A: real,B: real] : aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,A)),B) = aa_real_real(uminus_uminus_real,aa_real_real(times_times_real(A),B)) ).

% mult_minus_left
tff(fact_4614_mult__minus__left,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ).

% mult_minus_left
tff(fact_4615_mult__minus__left,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),B) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(times_times_complex(A),B)) ).

% mult_minus_left
tff(fact_4616_mult__minus__left,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,A)),B) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(times_times_rat(A),B)) ).

% mult_minus_left
tff(fact_4617_mult__minus__left,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) ).

% mult_minus_left
tff(fact_4618_minus__add__distrib,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(plus_plus_real(A),B)) = aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) ).

% minus_add_distrib
tff(fact_4619_minus__add__distrib,axiom,
    ! [A: int,B: int] : aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B)) ).

% minus_add_distrib
tff(fact_4620_minus__add__distrib,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(plus_plus_complex(A),B)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(uminus1482373934393186551omplex,B)) ).

% minus_add_distrib
tff(fact_4621_minus__add__distrib,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(plus_plus_rat(A),B)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B)) ).

% minus_add_distrib
tff(fact_4622_minus__add__distrib,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,plus_p5714425477246183910nteger(A,B)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ).

% minus_add_distrib
tff(fact_4623_minus__add__cancel,axiom,
    ! [A: real,B: real] : aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,A)),aa_real_real(plus_plus_real(A),B)) = B ).

% minus_add_cancel
tff(fact_4624_minus__add__cancel,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = B ).

% minus_add_cancel
tff(fact_4625_minus__add__cancel,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(plus_plus_complex(A),B)) = B ).

% minus_add_cancel
tff(fact_4626_minus__add__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(plus_plus_rat(A),B)) = B ).

% minus_add_cancel
tff(fact_4627_minus__add__cancel,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),plus_p5714425477246183910nteger(A,B)) = B ).

% minus_add_cancel
tff(fact_4628_add__minus__cancel,axiom,
    ! [A: real,B: real] : aa_real_real(plus_plus_real(A),aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,A)),B)) = B ).

% add_minus_cancel
tff(fact_4629_add__minus__cancel,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),B)) = B ).

% add_minus_cancel
tff(fact_4630_add__minus__cancel,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(plus_plus_complex(A),aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),B)) = B ).

% add_minus_cancel
tff(fact_4631_add__minus__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),B)) = B ).

% add_minus_cancel
tff(fact_4632_add__minus__cancel,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(A,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B)) = B ).

% add_minus_cancel
tff(fact_4633_minus__diff__eq,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(minus_minus_real(A),B)) = aa_real_real(minus_minus_real(B),A) ).

% minus_diff_eq
tff(fact_4634_minus__diff__eq,axiom,
    ! [A: int,B: int] : aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A) ).

% minus_diff_eq
tff(fact_4635_minus__diff__eq,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(minus_minus_complex(A),B)) = aa_complex_complex(minus_minus_complex(B),A) ).

% minus_diff_eq
tff(fact_4636_minus__diff__eq,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(minus_minus_rat(A),B)) = aa_rat_rat(minus_minus_rat(B),A) ).

% minus_diff_eq
tff(fact_4637_minus__diff__eq,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,minus_8373710615458151222nteger(A,B)) = minus_8373710615458151222nteger(B,A) ).

% minus_diff_eq
tff(fact_4638_div__minus__minus,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ).

% div_minus_minus
tff(fact_4639_div__minus__minus,axiom,
    ! [A: code_integer,B: code_integer] : divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = divide6298287555418463151nteger(A,B) ).

% div_minus_minus
tff(fact_4640_semiring__norm_I88_J,axiom,
    ! [Ma: num,Nb: num] : aa_num_num(bit0,Ma) != aa_num_num(bit1,Nb) ).

% semiring_norm(88)
tff(fact_4641_semiring__norm_I89_J,axiom,
    ! [Ma: num,Nb: num] : aa_num_num(bit1,Ma) != aa_num_num(bit0,Nb) ).

% semiring_norm(89)
tff(fact_4642_semiring__norm_I84_J,axiom,
    ! [Nb: num] : one != aa_num_num(bit1,Nb) ).

% semiring_norm(84)
tff(fact_4643_semiring__norm_I86_J,axiom,
    ! [Ma: num] : aa_num_num(bit1,Ma) != one ).

% semiring_norm(86)
tff(fact_4644_mod__minus__minus,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ).

% mod_minus_minus
tff(fact_4645_mod__minus__minus,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(A,B)) ).

% mod_minus_minus
tff(fact_4646_take__bit__of__0,axiom,
    ! [Nb: nat] : aa_int_int(bit_se2923211474154528505it_int(Nb),zero_zero_int) = zero_zero_int ).

% take_bit_of_0
tff(fact_4647_take__bit__of__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),zero_zero_nat) = zero_zero_nat ).

% take_bit_of_0
tff(fact_4648_real__add__minus__iff,axiom,
    ! [X: real,A: real] :
      ( ( aa_real_real(plus_plus_real(X),aa_real_real(uminus_uminus_real,A)) = zero_zero_real )
    <=> ( X = A ) ) ).

% real_add_minus_iff
tff(fact_4649_concat__bit__of__zero__2,axiom,
    ! [Nb: nat,K2: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(Nb),K2),zero_zero_int) = aa_int_int(bit_se2923211474154528505it_int(Nb),K2) ).

% concat_bit_of_zero_2
tff(fact_4650_semiring__norm_I73_J,axiom,
    ! [Ma: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit1,Ma)),aa_num_num(bit1,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ma),Nb) ) ).

% semiring_norm(73)
tff(fact_4651_semiring__norm_I80_J,axiom,
    ! [Ma: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit1,Ma)),aa_num_num(bit1,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Ma),Nb) ) ).

% semiring_norm(80)
tff(fact_4652_neg__0__le__iff__le,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% neg_0_le_iff_le
tff(fact_4653_neg__0__le__iff__le,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger) ) ).

% neg_0_le_iff_le
tff(fact_4654_neg__0__le__iff__le,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% neg_0_le_iff_le
tff(fact_4655_neg__0__le__iff__le,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% neg_0_le_iff_le
tff(fact_4656_neg__le__0__iff__le,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% neg_le_0_iff_le
tff(fact_4657_neg__le__0__iff__le,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),zero_z3403309356797280102nteger)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A) ) ).

% neg_le_0_iff_le
tff(fact_4658_neg__le__0__iff__le,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% neg_le_0_iff_le
tff(fact_4659_neg__le__0__iff__le,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).

% neg_le_0_iff_le
tff(fact_4660_less__eq__neg__nonpos,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% less_eq_neg_nonpos
tff(fact_4661_less__eq__neg__nonpos,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger) ) ).

% less_eq_neg_nonpos
tff(fact_4662_less__eq__neg__nonpos,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% less_eq_neg_nonpos
tff(fact_4663_less__eq__neg__nonpos,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% less_eq_neg_nonpos
tff(fact_4664_neg__less__eq__nonneg,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% neg_less_eq_nonneg
tff(fact_4665_neg__less__eq__nonneg,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),A)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A) ) ).

% neg_less_eq_nonneg
tff(fact_4666_neg__less__eq__nonneg,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% neg_less_eq_nonneg
tff(fact_4667_neg__less__eq__nonneg,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),A)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).

% neg_less_eq_nonneg
tff(fact_4668_neg__less__0__iff__less,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% neg_less_0_iff_less
tff(fact_4669_neg__less__0__iff__less,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,A)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ).

% neg_less_0_iff_less
tff(fact_4670_neg__less__0__iff__less,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% neg_less_0_iff_less
tff(fact_4671_neg__less__0__iff__less,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),zero_z3403309356797280102nteger)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A) ) ).

% neg_less_0_iff_less
tff(fact_4672_neg__0__less__iff__less,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% neg_0_less_iff_less
tff(fact_4673_neg__0__less__iff__less,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% neg_0_less_iff_less
tff(fact_4674_neg__0__less__iff__less,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% neg_0_less_iff_less
tff(fact_4675_neg__0__less__iff__less,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ).

% neg_0_less_iff_less
tff(fact_4676_neg__less__pos,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,A)),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% neg_less_pos
tff(fact_4677_neg__less__pos,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,A)),A)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ).

% neg_less_pos
tff(fact_4678_neg__less__pos,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A)),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% neg_less_pos
tff(fact_4679_neg__less__pos,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),A)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A) ) ).

% neg_less_pos
tff(fact_4680_less__neg__neg,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(uminus_uminus_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% less_neg_neg
tff(fact_4681_less__neg__neg,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% less_neg_neg
tff(fact_4682_less__neg__neg,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% less_neg_neg
tff(fact_4683_less__neg__neg,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ).

% less_neg_neg
tff(fact_4684_ab__left__minus,axiom,
    ! [A: real] : aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,A)),A) = zero_zero_real ).

% ab_left_minus
tff(fact_4685_ab__left__minus,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),A) = zero_zero_int ).

% ab_left_minus
tff(fact_4686_ab__left__minus,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),A) = zero_zero_complex ).

% ab_left_minus
tff(fact_4687_ab__left__minus,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),A) = zero_zero_rat ).

% ab_left_minus
tff(fact_4688_ab__left__minus,axiom,
    ! [A: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),A) = zero_z3403309356797280102nteger ).

% ab_left_minus
tff(fact_4689_add_Oright__inverse,axiom,
    ! [A: real] : aa_real_real(plus_plus_real(A),aa_real_real(uminus_uminus_real,A)) = zero_zero_real ).

% add.right_inverse
tff(fact_4690_add_Oright__inverse,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(uminus_uminus_int,A)) = zero_zero_int ).

% add.right_inverse
tff(fact_4691_add_Oright__inverse,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(A),aa_complex_complex(uminus1482373934393186551omplex,A)) = zero_zero_complex ).

% add.right_inverse
tff(fact_4692_add_Oright__inverse,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(A),aa_rat_rat(uminus_uminus_rat,A)) = zero_zero_rat ).

% add.right_inverse
tff(fact_4693_add_Oright__inverse,axiom,
    ! [A: code_integer] : plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = zero_z3403309356797280102nteger ).

% add.right_inverse
tff(fact_4694_diff__0,axiom,
    ! [A: real] : aa_real_real(minus_minus_real(zero_zero_real),A) = aa_real_real(uminus_uminus_real,A) ).

% diff_0
tff(fact_4695_diff__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,zero_zero_int),A) = aa_int_int(uminus_uminus_int,A) ).

% diff_0
tff(fact_4696_diff__0,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(zero_zero_complex),A) = aa_complex_complex(uminus1482373934393186551omplex,A) ).

% diff_0
tff(fact_4697_diff__0,axiom,
    ! [A: rat] : aa_rat_rat(minus_minus_rat(zero_zero_rat),A) = aa_rat_rat(uminus_uminus_rat,A) ).

% diff_0
tff(fact_4698_diff__0,axiom,
    ! [A: code_integer] : minus_8373710615458151222nteger(zero_z3403309356797280102nteger,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ).

% diff_0
tff(fact_4699_verit__minus__simplify_I3_J,axiom,
    ! [B: real] : aa_real_real(minus_minus_real(zero_zero_real),B) = aa_real_real(uminus_uminus_real,B) ).

% verit_minus_simplify(3)
tff(fact_4700_verit__minus__simplify_I3_J,axiom,
    ! [B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,zero_zero_int),B) = aa_int_int(uminus_uminus_int,B) ).

% verit_minus_simplify(3)
tff(fact_4701_verit__minus__simplify_I3_J,axiom,
    ! [B: complex] : aa_complex_complex(minus_minus_complex(zero_zero_complex),B) = aa_complex_complex(uminus1482373934393186551omplex,B) ).

% verit_minus_simplify(3)
tff(fact_4702_verit__minus__simplify_I3_J,axiom,
    ! [B: rat] : aa_rat_rat(minus_minus_rat(zero_zero_rat),B) = aa_rat_rat(uminus_uminus_rat,B) ).

% verit_minus_simplify(3)
tff(fact_4703_verit__minus__simplify_I3_J,axiom,
    ! [B: code_integer] : minus_8373710615458151222nteger(zero_z3403309356797280102nteger,B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ).

% verit_minus_simplify(3)
tff(fact_4704_add__neg__numeral__simps_I3_J,axiom,
    ! [Ma: num,Nb: num] : aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Ma))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_real_real(uminus_uminus_real,aa_real_real(plus_plus_real(aa_num_real(numeral_numeral_real,Ma)),aa_num_real(numeral_numeral_real,Nb))) ).

% add_neg_numeral_simps(3)
tff(fact_4705_add__neg__numeral__simps_I3_J,axiom,
    ! [Ma: num,Nb: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Ma))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Ma)),aa_num_int(numeral_numeral_int,Nb))) ).

% add_neg_numeral_simps(3)
tff(fact_4706_add__neg__numeral__simps_I3_J,axiom,
    ! [Ma: num,Nb: num] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Ma))),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb))) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Ma)),numera6690914467698888265omplex(Nb))) ).

% add_neg_numeral_simps(3)
tff(fact_4707_add__neg__numeral__simps_I3_J,axiom,
    ! [Ma: num,Nb: num] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Ma))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,Ma)),aa_num_rat(numeral_numeral_rat,Nb))) ).

% add_neg_numeral_simps(3)
tff(fact_4708_add__neg__numeral__simps_I3_J,axiom,
    ! [Ma: num,Nb: num] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Ma)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,plus_p5714425477246183910nteger(numera6620942414471956472nteger(Ma),numera6620942414471956472nteger(Nb))) ).

% add_neg_numeral_simps(3)
tff(fact_4709_mult__minus1__right,axiom,
    ! [Z2: real] : aa_real_real(times_times_real(Z2),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,Z2) ).

% mult_minus1_right
tff(fact_4710_mult__minus1__right,axiom,
    ! [Z2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,Z2),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,Z2) ).

% mult_minus1_right
tff(fact_4711_mult__minus1__right,axiom,
    ! [Z2: complex] : aa_complex_complex(times_times_complex(Z2),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,Z2) ).

% mult_minus1_right
tff(fact_4712_mult__minus1__right,axiom,
    ! [Z2: rat] : aa_rat_rat(times_times_rat(Z2),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,Z2) ).

% mult_minus1_right
tff(fact_4713_mult__minus1__right,axiom,
    ! [Z2: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Z2),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Z2) ).

% mult_minus1_right
tff(fact_4714_mult__minus1,axiom,
    ! [Z2: real] : aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,one_one_real)),Z2) = aa_real_real(uminus_uminus_real,Z2) ).

% mult_minus1
tff(fact_4715_mult__minus1,axiom,
    ! [Z2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,one_one_int)),Z2) = aa_int_int(uminus_uminus_int,Z2) ).

% mult_minus1
tff(fact_4716_mult__minus1,axiom,
    ! [Z2: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Z2) = aa_complex_complex(uminus1482373934393186551omplex,Z2) ).

% mult_minus1
tff(fact_4717_mult__minus1,axiom,
    ! [Z2: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Z2) = aa_rat_rat(uminus_uminus_rat,Z2) ).

% mult_minus1
tff(fact_4718_mult__minus1,axiom,
    ! [Z2: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Z2) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Z2) ).

% mult_minus1
tff(fact_4719_diff__minus__eq__add,axiom,
    ! [A: real,B: real] : aa_real_real(minus_minus_real(A),aa_real_real(uminus_uminus_real,B)) = aa_real_real(plus_plus_real(A),B) ).

% diff_minus_eq_add
tff(fact_4720_diff__minus__eq__add,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) ).

% diff_minus_eq_add
tff(fact_4721_diff__minus__eq__add,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(minus_minus_complex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(plus_plus_complex(A),B) ).

% diff_minus_eq_add
tff(fact_4722_diff__minus__eq__add,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(A),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(plus_plus_rat(A),B) ).

% diff_minus_eq_add
tff(fact_4723_diff__minus__eq__add,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = plus_p5714425477246183910nteger(A,B) ).

% diff_minus_eq_add
tff(fact_4724_uminus__add__conv__diff,axiom,
    ! [A: real,B: real] : aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,A)),B) = aa_real_real(minus_minus_real(B),A) ).

% uminus_add_conv_diff
tff(fact_4725_uminus__add__conv__diff,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A) ).

% uminus_add_conv_diff
tff(fact_4726_uminus__add__conv__diff,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),B) = aa_complex_complex(minus_minus_complex(B),A) ).

% uminus_add_conv_diff
tff(fact_4727_uminus__add__conv__diff,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),B) = aa_rat_rat(minus_minus_rat(B),A) ).

% uminus_add_conv_diff
tff(fact_4728_uminus__add__conv__diff,axiom,
    ! [A: code_integer,B: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) = minus_8373710615458151222nteger(B,A) ).

% uminus_add_conv_diff
tff(fact_4729_div__minus1__right,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,A) ).

% div_minus1_right
tff(fact_4730_div__minus1__right,axiom,
    ! [A: code_integer] : divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ).

% div_minus1_right
tff(fact_4731_divide__minus1,axiom,
    ! [X: real] : aa_real_real(divide_divide_real(X),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,X) ).

% divide_minus1
tff(fact_4732_divide__minus1,axiom,
    ! [X: complex] : aa_complex_complex(divide1717551699836669952omplex(X),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,X) ).

% divide_minus1
tff(fact_4733_divide__minus1,axiom,
    ! [X: rat] : aa_rat_rat(divide_divide_rat(X),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,X) ).

% divide_minus1
tff(fact_4734_minus__mod__self1,axiom,
    ! [B: int,A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A)),B) ).

% minus_mod_self1
tff(fact_4735_minus__mod__self1,axiom,
    ! [B: code_integer,A: code_integer] : modulo364778990260209775nteger(minus_8373710615458151222nteger(B,A),B) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) ).

% minus_mod_self1
tff(fact_4736_take__bit__0,axiom,
    ! [A: int] : aa_int_int(bit_se2923211474154528505it_int(zero_zero_nat),A) = zero_zero_int ).

% take_bit_0
tff(fact_4737_take__bit__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,zero_zero_nat),A) = zero_zero_nat ).

% take_bit_0
tff(fact_4738_take__bit__Suc__1,axiom,
    ! [Nb: nat] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),one_one_int) = one_one_int ).

% take_bit_Suc_1
tff(fact_4739_take__bit__Suc__1,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,Nb)),one_one_nat) = one_one_nat ).

% take_bit_Suc_1
tff(fact_4740_take__bit__numeral__1,axiom,
    ! [L: num] : aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,L)),one_one_int) = one_one_int ).

% take_bit_numeral_1
tff(fact_4741_take__bit__numeral__1,axiom,
    ! [L: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_num_nat(numeral_numeral_nat,L)),one_one_nat) = one_one_nat ).

% take_bit_numeral_1
tff(fact_4742_signed__take__bit__of__minus__1,axiom,
    ! [Nb: nat] : bit_ri6519982836138164636nteger(Nb,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% signed_take_bit_of_minus_1
tff(fact_4743_signed__take__bit__of__minus__1,axiom,
    ! [Nb: nat] : aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% signed_take_bit_of_minus_1
tff(fact_4744_semiring__norm_I9_J,axiom,
    ! [Ma: num,Nb: num] : aa_num_num(plus_plus_num(aa_num_num(bit1,Ma)),aa_num_num(bit0,Nb)) = aa_num_num(bit1,aa_num_num(plus_plus_num(Ma),Nb)) ).

% semiring_norm(9)
tff(fact_4745_semiring__norm_I7_J,axiom,
    ! [Ma: num,Nb: num] : aa_num_num(plus_plus_num(aa_num_num(bit0,Ma)),aa_num_num(bit1,Nb)) = aa_num_num(bit1,aa_num_num(plus_plus_num(Ma),Nb)) ).

% semiring_norm(7)
tff(fact_4746_semiring__norm_I15_J,axiom,
    ! [Ma: num,Nb: num] : aa_num_num(times_times_num(aa_num_num(bit1,Ma)),aa_num_num(bit0,Nb)) = aa_num_num(bit0,aa_num_num(times_times_num(aa_num_num(bit1,Ma)),Nb)) ).

% semiring_norm(15)
tff(fact_4747_semiring__norm_I14_J,axiom,
    ! [Ma: num,Nb: num] : aa_num_num(times_times_num(aa_num_num(bit0,Ma)),aa_num_num(bit1,Nb)) = aa_num_num(bit0,aa_num_num(times_times_num(Ma),aa_num_num(bit1,Nb))) ).

% semiring_norm(14)
tff(fact_4748_semiring__norm_I72_J,axiom,
    ! [Ma: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit0,Ma)),aa_num_num(bit1,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ma),Nb) ) ).

% semiring_norm(72)
tff(fact_4749_semiring__norm_I81_J,axiom,
    ! [Ma: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit1,Ma)),aa_num_num(bit0,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Ma),Nb) ) ).

% semiring_norm(81)
tff(fact_4750_semiring__norm_I70_J,axiom,
    ! [Ma: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit1,Ma)),one) ).

% semiring_norm(70)
tff(fact_4751_semiring__norm_I77_J,axiom,
    ! [Nb: num] : aa_num_o(aa_num_fun_num_o(ord_less_num,one),aa_num_num(bit1,Nb)) ).

% semiring_norm(77)
tff(fact_4752_dbl__simps_I1_J,axiom,
    ! [K2: num] : aa_real_real(neg_numeral_dbl_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,K2))) = aa_real_real(uminus_uminus_real,aa_real_real(neg_numeral_dbl_real,aa_num_real(numeral_numeral_real,K2))) ).

% dbl_simps(1)
tff(fact_4753_dbl__simps_I1_J,axiom,
    ! [K2: num] : aa_int_int(neg_numeral_dbl_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))) = aa_int_int(uminus_uminus_int,aa_int_int(neg_numeral_dbl_int,aa_num_int(numeral_numeral_int,K2))) ).

% dbl_simps(1)
tff(fact_4754_dbl__simps_I1_J,axiom,
    ! [K2: num] : aa_complex_complex(neg_nu7009210354673126013omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(K2))) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(neg_nu7009210354673126013omplex,numera6690914467698888265omplex(K2))) ).

% dbl_simps(1)
tff(fact_4755_dbl__simps_I1_J,axiom,
    ! [K2: num] : aa_rat_rat(neg_numeral_dbl_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,K2))) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(neg_numeral_dbl_rat,aa_num_rat(numeral_numeral_rat,K2))) ).

% dbl_simps(1)
tff(fact_4756_dbl__simps_I1_J,axiom,
    ! [K2: num] : neg_nu8804712462038260780nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(K2))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,neg_nu8804712462038260780nteger(numera6620942414471956472nteger(K2))) ).

% dbl_simps(1)
tff(fact_4757_add__neg__numeral__special_I8_J,axiom,
    aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,one_one_real)),one_one_real) = zero_zero_real ).

% add_neg_numeral_special(8)
tff(fact_4758_add__neg__numeral__special_I8_J,axiom,
    aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) = zero_zero_int ).

% add_neg_numeral_special(8)
tff(fact_4759_add__neg__numeral__special_I8_J,axiom,
    aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),one_one_complex) = zero_zero_complex ).

% add_neg_numeral_special(8)
tff(fact_4760_add__neg__numeral__special_I8_J,axiom,
    aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat) = zero_zero_rat ).

% add_neg_numeral_special(8)
tff(fact_4761_add__neg__numeral__special_I8_J,axiom,
    plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),one_one_Code_integer) = zero_z3403309356797280102nteger ).

% add_neg_numeral_special(8)
tff(fact_4762_add__neg__numeral__special_I7_J,axiom,
    aa_real_real(plus_plus_real(one_one_real),aa_real_real(uminus_uminus_real,one_one_real)) = zero_zero_real ).

% add_neg_numeral_special(7)
tff(fact_4763_add__neg__numeral__special_I7_J,axiom,
    aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int)) = zero_zero_int ).

% add_neg_numeral_special(7)
tff(fact_4764_add__neg__numeral__special_I7_J,axiom,
    aa_complex_complex(plus_plus_complex(one_one_complex),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = zero_zero_complex ).

% add_neg_numeral_special(7)
tff(fact_4765_add__neg__numeral__special_I7_J,axiom,
    aa_rat_rat(plus_plus_rat(one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = zero_zero_rat ).

% add_neg_numeral_special(7)
tff(fact_4766_add__neg__numeral__special_I7_J,axiom,
    plus_p5714425477246183910nteger(one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% add_neg_numeral_special(7)
tff(fact_4767_diff__numeral__special_I12_J,axiom,
    aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,one_one_real)) = zero_zero_real ).

% diff_numeral_special(12)
tff(fact_4768_diff__numeral__special_I12_J,axiom,
    aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,one_one_int)) = zero_zero_int ).

% diff_numeral_special(12)
tff(fact_4769_diff__numeral__special_I12_J,axiom,
    aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = zero_zero_complex ).

% diff_numeral_special(12)
tff(fact_4770_diff__numeral__special_I12_J,axiom,
    aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = zero_zero_rat ).

% diff_numeral_special(12)
tff(fact_4771_diff__numeral__special_I12_J,axiom,
    minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% diff_numeral_special(12)
tff(fact_4772_numeral__eq__neg__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)) = aa_real_real(uminus_uminus_real,one_one_real) )
    <=> ( Nb = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_4773_numeral__eq__neg__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,one_one_int) )
    <=> ( Nb = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_4774_numeral__eq__neg__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) )
    <=> ( Nb = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_4775_numeral__eq__neg__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)) = aa_rat_rat(uminus_uminus_rat,one_one_rat) )
    <=> ( Nb = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_4776_numeral__eq__neg__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
    <=> ( Nb = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_4777_neg__one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( aa_real_real(uminus_uminus_real,one_one_real) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)) )
    <=> ( Nb = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_4778_neg__one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( aa_int_int(uminus_uminus_int,one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) )
    <=> ( Nb = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_4779_neg__one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb)) )
    <=> ( Nb = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_4780_neg__one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( aa_rat_rat(uminus_uminus_rat,one_one_rat) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)) )
    <=> ( Nb = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_4781_neg__one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb)) )
    <=> ( Nb = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_4782_left__minus__one__mult__self,axiom,
    ! [Nb: nat,A: real] : aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb)),aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb)),A)) = A ).

% left_minus_one_mult_self
tff(fact_4783_left__minus__one__mult__self,axiom,
    ! [Nb: nat,A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),Nb)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),Nb)),A)) = A ).

% left_minus_one_mult_self
tff(fact_4784_left__minus__one__mult__self,axiom,
    ! [Nb: nat,A: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb)),aa_complex_complex(times_times_complex(aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb)),A)) = A ).

% left_minus_one_mult_self
tff(fact_4785_left__minus__one__mult__self,axiom,
    ! [Nb: nat,A: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb)),aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb)),A)) = A ).

% left_minus_one_mult_self
tff(fact_4786_left__minus__one__mult__self,axiom,
    ! [Nb: nat,A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb)),A)) = A ).

% left_minus_one_mult_self
tff(fact_4787_minus__one__mult__self,axiom,
    ! [Nb: nat] : aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb)) = one_one_real ).

% minus_one_mult_self
tff(fact_4788_minus__one__mult__self,axiom,
    ! [Nb: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),Nb)) = one_one_int ).

% minus_one_mult_self
tff(fact_4789_minus__one__mult__self,axiom,
    ! [Nb: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb)) = one_one_complex ).

% minus_one_mult_self
tff(fact_4790_minus__one__mult__self,axiom,
    ! [Nb: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb)) = one_one_rat ).

% minus_one_mult_self
tff(fact_4791_minus__one__mult__self,axiom,
    ! [Nb: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb)) = one_one_Code_integer ).

% minus_one_mult_self
tff(fact_4792_mod__minus1__right,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(uminus_uminus_int,one_one_int)) = zero_zero_int ).

% mod_minus1_right
tff(fact_4793_mod__minus1__right,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% mod_minus1_right
tff(fact_4794_take__bit__of__1__eq__0__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Nb),one_one_int) = zero_zero_int )
    <=> ( Nb = zero_zero_nat ) ) ).

% take_bit_of_1_eq_0_iff
tff(fact_4795_take__bit__of__1__eq__0__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),one_one_nat) = zero_zero_nat )
    <=> ( Nb = zero_zero_nat ) ) ).

% take_bit_of_1_eq_0_iff
tff(fact_4796_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y2: real] : aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),Y2)) = aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(V),W)))),Y2) ).

% semiring_norm(168)
tff(fact_4797_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),Y2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(V),W)))),Y2) ).

% semiring_norm(168)
tff(fact_4798_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y2: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(V))),aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),Y2)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(plus_plus_num(V),W)))),Y2) ).

% semiring_norm(168)
tff(fact_4799_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y2: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),Y2)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(V),W)))),Y2) ).

% semiring_norm(168)
tff(fact_4800_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y2: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V)),plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(W)),Y2)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(plus_plus_num(V),W))),Y2) ).

% semiring_norm(168)
tff(fact_4801_diff__numeral__simps_I2_J,axiom,
    ! [Ma: num,Nb: num] : aa_real_real(minus_minus_real(aa_num_real(numeral_numeral_real,Ma)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(Ma),Nb)) ).

% diff_numeral_simps(2)
tff(fact_4802_diff__numeral__simps_I2_J,axiom,
    ! [Ma: num,Nb: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Ma)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(Ma),Nb)) ).

% diff_numeral_simps(2)
tff(fact_4803_diff__numeral__simps_I2_J,axiom,
    ! [Ma: num,Nb: num] : aa_complex_complex(minus_minus_complex(numera6690914467698888265omplex(Ma)),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb))) = numera6690914467698888265omplex(aa_num_num(plus_plus_num(Ma),Nb)) ).

% diff_numeral_simps(2)
tff(fact_4804_diff__numeral__simps_I2_J,axiom,
    ! [Ma: num,Nb: num] : aa_rat_rat(minus_minus_rat(aa_num_rat(numeral_numeral_rat,Ma)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(Ma),Nb)) ).

% diff_numeral_simps(2)
tff(fact_4805_diff__numeral__simps_I2_J,axiom,
    ! [Ma: num,Nb: num] : minus_8373710615458151222nteger(numera6620942414471956472nteger(Ma),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) = numera6620942414471956472nteger(aa_num_num(plus_plus_num(Ma),Nb)) ).

% diff_numeral_simps(2)
tff(fact_4806_diff__numeral__simps_I3_J,axiom,
    ! [Ma: num,Nb: num] : aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Ma))),aa_num_real(numeral_numeral_real,Nb)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(Ma),Nb))) ).

% diff_numeral_simps(3)
tff(fact_4807_diff__numeral__simps_I3_J,axiom,
    ! [Ma: num,Nb: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Ma))),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(Ma),Nb))) ).

% diff_numeral_simps(3)
tff(fact_4808_diff__numeral__simps_I3_J,axiom,
    ! [Ma: num,Nb: num] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Ma))),numera6690914467698888265omplex(Nb)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(plus_plus_num(Ma),Nb))) ).

% diff_numeral_simps(3)
tff(fact_4809_diff__numeral__simps_I3_J,axiom,
    ! [Ma: num,Nb: num] : aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Ma))),aa_num_rat(numeral_numeral_rat,Nb)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(Ma),Nb))) ).

% diff_numeral_simps(3)
tff(fact_4810_diff__numeral__simps_I3_J,axiom,
    ! [Ma: num,Nb: num] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Ma)),numera6620942414471956472nteger(Nb)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(plus_plus_num(Ma),Nb))) ).

% diff_numeral_simps(3)
tff(fact_4811_zdiv__numeral__Bit1,axiom,
    ! [V: num,W: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,V))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,W)) ).

% zdiv_numeral_Bit1
tff(fact_4812_semiring__norm_I3_J,axiom,
    ! [Nb: num] : aa_num_num(plus_plus_num(one),aa_num_num(bit0,Nb)) = aa_num_num(bit1,Nb) ).

% semiring_norm(3)
tff(fact_4813_semiring__norm_I4_J,axiom,
    ! [Nb: num] : aa_num_num(plus_plus_num(one),aa_num_num(bit1,Nb)) = aa_num_num(bit0,aa_num_num(plus_plus_num(Nb),one)) ).

% semiring_norm(4)
tff(fact_4814_semiring__norm_I5_J,axiom,
    ! [Ma: num] : aa_num_num(plus_plus_num(aa_num_num(bit0,Ma)),one) = aa_num_num(bit1,Ma) ).

% semiring_norm(5)
tff(fact_4815_semiring__norm_I8_J,axiom,
    ! [Ma: num] : aa_num_num(plus_plus_num(aa_num_num(bit1,Ma)),one) = aa_num_num(bit0,aa_num_num(plus_plus_num(Ma),one)) ).

% semiring_norm(8)
tff(fact_4816_semiring__norm_I10_J,axiom,
    ! [Ma: num,Nb: num] : aa_num_num(plus_plus_num(aa_num_num(bit1,Ma)),aa_num_num(bit1,Nb)) = aa_num_num(bit0,aa_num_num(plus_plus_num(aa_num_num(plus_plus_num(Ma),Nb)),one)) ).

% semiring_norm(10)
tff(fact_4817_mult__neg__numeral__simps_I1_J,axiom,
    ! [Ma: num,Nb: num] : aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Ma))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_num_real(numeral_numeral_real,aa_num_num(times_times_num(Ma),Nb)) ).

% mult_neg_numeral_simps(1)
tff(fact_4818_mult__neg__numeral__simps_I1_J,axiom,
    ! [Ma: num,Nb: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Ma))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(Ma),Nb)) ).

% mult_neg_numeral_simps(1)
tff(fact_4819_mult__neg__numeral__simps_I1_J,axiom,
    ! [Ma: num,Nb: num] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Ma))),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb))) = numera6690914467698888265omplex(aa_num_num(times_times_num(Ma),Nb)) ).

% mult_neg_numeral_simps(1)
tff(fact_4820_mult__neg__numeral__simps_I1_J,axiom,
    ! [Ma: num,Nb: num] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Ma))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(Ma),Nb)) ).

% mult_neg_numeral_simps(1)
tff(fact_4821_mult__neg__numeral__simps_I1_J,axiom,
    ! [Ma: num,Nb: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Ma))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) = numera6620942414471956472nteger(aa_num_num(times_times_num(Ma),Nb)) ).

% mult_neg_numeral_simps(1)
tff(fact_4822_mult__neg__numeral__simps_I2_J,axiom,
    ! [Ma: num,Nb: num] : aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Ma))),aa_num_real(numeral_numeral_real,Nb)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(times_times_num(Ma),Nb))) ).

% mult_neg_numeral_simps(2)
tff(fact_4823_mult__neg__numeral__simps_I2_J,axiom,
    ! [Ma: num,Nb: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Ma))),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(Ma),Nb))) ).

% mult_neg_numeral_simps(2)
tff(fact_4824_mult__neg__numeral__simps_I2_J,axiom,
    ! [Ma: num,Nb: num] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Ma))),numera6690914467698888265omplex(Nb)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(times_times_num(Ma),Nb))) ).

% mult_neg_numeral_simps(2)
tff(fact_4825_mult__neg__numeral__simps_I2_J,axiom,
    ! [Ma: num,Nb: num] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Ma))),aa_num_rat(numeral_numeral_rat,Nb)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(Ma),Nb))) ).

% mult_neg_numeral_simps(2)
tff(fact_4826_mult__neg__numeral__simps_I2_J,axiom,
    ! [Ma: num,Nb: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Ma))),numera6620942414471956472nteger(Nb)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(times_times_num(Ma),Nb))) ).

% mult_neg_numeral_simps(2)
tff(fact_4827_mult__neg__numeral__simps_I3_J,axiom,
    ! [Ma: num,Nb: num] : aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,Ma)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(times_times_num(Ma),Nb))) ).

% mult_neg_numeral_simps(3)
tff(fact_4828_mult__neg__numeral__simps_I3_J,axiom,
    ! [Ma: num,Nb: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,Ma)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(Ma),Nb))) ).

% mult_neg_numeral_simps(3)
tff(fact_4829_mult__neg__numeral__simps_I3_J,axiom,
    ! [Ma: num,Nb: num] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(Ma)),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb))) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(times_times_num(Ma),Nb))) ).

% mult_neg_numeral_simps(3)
tff(fact_4830_mult__neg__numeral__simps_I3_J,axiom,
    ! [Ma: num,Nb: num] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,Ma)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(Ma),Nb))) ).

% mult_neg_numeral_simps(3)
tff(fact_4831_mult__neg__numeral__simps_I3_J,axiom,
    ! [Ma: num,Nb: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(Ma)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(times_times_num(Ma),Nb))) ).

% mult_neg_numeral_simps(3)
tff(fact_4832_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y2: real] : aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,W)),Y2)) = aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(times_times_num(V),W)))),Y2) ).

% semiring_norm(170)
tff(fact_4833_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,W)),Y2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(V),W)))),Y2) ).

% semiring_norm(170)
tff(fact_4834_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y2: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(V))),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),Y2)) = aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(times_times_num(V),W)))),Y2) ).

% semiring_norm(170)
tff(fact_4835_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y2: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),Y2)) = aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(V),W)))),Y2) ).

% semiring_norm(170)
tff(fact_4836_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y2: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(W)),Y2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(times_times_num(V),W)))),Y2) ).

% semiring_norm(170)
tff(fact_4837_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y2: real] : aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,V)),aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),Y2)) = aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(times_times_num(V),W)))),Y2) ).

% semiring_norm(171)
tff(fact_4838_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),Y2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(V),W)))),Y2) ).

% semiring_norm(171)
tff(fact_4839_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y2: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),Y2)) = aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(times_times_num(V),W)))),Y2) ).

% semiring_norm(171)
tff(fact_4840_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y2: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),Y2)) = aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(V),W)))),Y2) ).

% semiring_norm(171)
tff(fact_4841_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y2: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(V)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(W))),Y2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(times_times_num(V),W)))),Y2) ).

% semiring_norm(171)
tff(fact_4842_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y2: real] : aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),Y2)) = aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(times_times_num(V),W))),Y2) ).

% semiring_norm(172)
tff(fact_4843_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),Y2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(V),W))),Y2) ).

% semiring_norm(172)
tff(fact_4844_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y2: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(V))),aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),Y2)) = aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(times_times_num(V),W))),Y2) ).

% semiring_norm(172)
tff(fact_4845_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y2: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),Y2)) = aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(V),W))),Y2) ).

% semiring_norm(172)
tff(fact_4846_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y2: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(W))),Y2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(times_times_num(V),W))),Y2) ).

% semiring_norm(172)
tff(fact_4847_neg__numeral__le__iff,axiom,
    ! [Ma: num,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Ma))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),Ma) ) ).

% neg_numeral_le_iff
tff(fact_4848_neg__numeral__le__iff,axiom,
    ! [Ma: num,Nb: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Ma))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),Ma) ) ).

% neg_numeral_le_iff
tff(fact_4849_neg__numeral__le__iff,axiom,
    ! [Ma: num,Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Ma))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),Ma) ) ).

% neg_numeral_le_iff
tff(fact_4850_neg__numeral__le__iff,axiom,
    ! [Ma: num,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Ma))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),Ma) ) ).

% neg_numeral_le_iff
tff(fact_4851_neg__numeral__less__iff,axiom,
    ! [Ma: num,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Ma))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Nb),Ma) ) ).

% neg_numeral_less_iff
tff(fact_4852_neg__numeral__less__iff,axiom,
    ! [Ma: num,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Ma))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Nb),Ma) ) ).

% neg_numeral_less_iff
tff(fact_4853_neg__numeral__less__iff,axiom,
    ! [Ma: num,Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Ma))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Nb),Ma) ) ).

% neg_numeral_less_iff
tff(fact_4854_neg__numeral__less__iff,axiom,
    ! [Ma: num,Nb: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Ma))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Nb),Ma) ) ).

% neg_numeral_less_iff
tff(fact_4855_take__bit__of__Suc__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).

% take_bit_of_Suc_0
tff(fact_4856_semiring__norm_I16_J,axiom,
    ! [Ma: num,Nb: num] : aa_num_num(times_times_num(aa_num_num(bit1,Ma)),aa_num_num(bit1,Nb)) = aa_num_num(bit1,aa_num_num(plus_plus_num(aa_num_num(plus_plus_num(Ma),Nb)),aa_num_num(bit0,aa_num_num(times_times_num(Ma),Nb)))) ).

% semiring_norm(16)
tff(fact_4857_semiring__norm_I79_J,axiom,
    ! [Ma: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit0,Ma)),aa_num_num(bit1,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ma),Nb) ) ).

% semiring_norm(79)
tff(fact_4858_semiring__norm_I74_J,axiom,
    ! [Ma: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit1,Ma)),aa_num_num(bit0,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Ma),Nb) ) ).

% semiring_norm(74)
tff(fact_4859_not__neg__one__le__neg__numeral__iff,axiom,
    ! [Ma: num] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Ma)))
    <=> ( Ma != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_4860_not__neg__one__le__neg__numeral__iff,axiom,
    ! [Ma: num] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Ma)))
    <=> ( Ma != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_4861_not__neg__one__le__neg__numeral__iff,axiom,
    ! [Ma: num] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Ma)))
    <=> ( Ma != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_4862_not__neg__one__le__neg__numeral__iff,axiom,
    ! [Ma: num] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Ma)))
    <=> ( Ma != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_4863_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(divide_divide_real(B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(times_times_real(A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))) ) ).

% le_divide_eq_numeral1(2)
tff(fact_4864_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))) ) ).

% le_divide_eq_numeral1(2)
tff(fact_4865_divide__le__eq__numeral1_I2_J,axiom,
    ! [B: real,W: num,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))),B) ) ).

% divide_le_eq_numeral1(2)
tff(fact_4866_divide__le__eq__numeral1_I2_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))),B) ) ).

% divide_le_eq_numeral1(2)
tff(fact_4867_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W: num] :
      ( ( A = aa_real_real(divide_divide_real(B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) )
    <=> $ite(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) != zero_zero_real,aa_real_real(times_times_real(A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) = B,A = zero_zero_real) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_4868_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: complex,B: complex,W: num] :
      ( ( A = aa_complex_complex(divide1717551699836669952omplex(B),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) )
    <=> $ite(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) != zero_zero_complex,aa_complex_complex(times_times_complex(A),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) = B,A = zero_zero_complex) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_4869_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( A = aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))) )
    <=> $ite(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) != zero_zero_rat,aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))) = B,A = zero_zero_rat) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_4870_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: real,W: num,A: real] :
      ( ( aa_real_real(divide_divide_real(B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) = A )
    <=> $ite(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) != zero_zero_real,B = aa_real_real(times_times_real(A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),A = zero_zero_real) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_4871_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: complex,W: num,A: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) = A )
    <=> $ite(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) != zero_zero_complex,B = aa_complex_complex(times_times_complex(A),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),A = zero_zero_complex) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_4872_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))) = A )
    <=> $ite(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) != zero_zero_rat,B = aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),A = zero_zero_rat) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_4873_neg__numeral__less__neg__one__iff,axiom,
    ! [Ma: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Ma))),aa_real_real(uminus_uminus_real,one_one_real))
    <=> ( Ma != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_4874_neg__numeral__less__neg__one__iff,axiom,
    ! [Ma: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Ma))),aa_int_int(uminus_uminus_int,one_one_int))
    <=> ( Ma != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_4875_neg__numeral__less__neg__one__iff,axiom,
    ! [Ma: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Ma))),aa_rat_rat(uminus_uminus_rat,one_one_rat))
    <=> ( Ma != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_4876_neg__numeral__less__neg__one__iff,axiom,
    ! [Ma: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Ma))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))
    <=> ( Ma != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_4877_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(divide_divide_real(B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(times_times_real(A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))) ) ).

% less_divide_eq_numeral1(2)
tff(fact_4878_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))) ) ).

% less_divide_eq_numeral1(2)
tff(fact_4879_divide__less__eq__numeral1_I2_J,axiom,
    ! [B: real,W: num,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))),B) ) ).

% divide_less_eq_numeral1(2)
tff(fact_4880_divide__less__eq__numeral1_I2_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))),B) ) ).

% divide_less_eq_numeral1(2)
tff(fact_4881_power2__minus,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_minus
tff(fact_4882_power2__minus,axiom,
    ! [A: int] : aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_minus
tff(fact_4883_power2__minus,axiom,
    ! [A: complex] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_minus
tff(fact_4884_power2__minus,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_minus
tff(fact_4885_power2__minus,axiom,
    ! [A: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_minus
tff(fact_4886_take__bit__of__1,axiom,
    ! [Nb: nat] : bit_se1745604003318907178nteger(Nb,one_one_Code_integer) = zero_n356916108424825756nteger(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).

% take_bit_of_1
tff(fact_4887_take__bit__of__1,axiom,
    ! [Nb: nat] : aa_int_int(bit_se2923211474154528505it_int(Nb),one_one_int) = aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).

% take_bit_of_1
tff(fact_4888_take__bit__of__1,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),one_one_nat) = aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).

% take_bit_of_1
tff(fact_4889_add__neg__numeral__special_I9_J,axiom,
    aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_4890_add__neg__numeral__special_I9_J,axiom,
    aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_4891_add__neg__numeral__special_I9_J,axiom,
    aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_4892_add__neg__numeral__special_I9_J,axiom,
    aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_4893_add__neg__numeral__special_I9_J,axiom,
    plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_4894_diff__numeral__special_I10_J,axiom,
    aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,one_one_real)),one_one_real) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_4895_diff__numeral__special_I10_J,axiom,
    aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_4896_diff__numeral__special_I10_J,axiom,
    aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),one_one_complex) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_4897_diff__numeral__special_I10_J,axiom,
    aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_4898_diff__numeral__special_I10_J,axiom,
    minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_4899_diff__numeral__special_I11_J,axiom,
    aa_real_real(minus_minus_real(one_one_real),aa_real_real(uminus_uminus_real,one_one_real)) = aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_4900_diff__numeral__special_I11_J,axiom,
    aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_4901_diff__numeral__special_I11_J,axiom,
    aa_complex_complex(minus_minus_complex(one_one_complex),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = numera6690914467698888265omplex(aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_4902_diff__numeral__special_I11_J,axiom,
    aa_rat_rat(minus_minus_rat(one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_4903_diff__numeral__special_I11_J,axiom,
    minus_8373710615458151222nteger(one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = numera6620942414471956472nteger(aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_4904_minus__1__div__2__eq,axiom,
    aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(uminus_uminus_int,one_one_int) ).

% minus_1_div_2_eq
tff(fact_4905_minus__1__div__2__eq,axiom,
    divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),numera6620942414471956472nteger(aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% minus_1_div_2_eq
tff(fact_4906_minus__1__mod__2__eq,axiom,
    aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ).

% minus_1_mod_2_eq
tff(fact_4907_minus__1__mod__2__eq,axiom,
    modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),numera6620942414471956472nteger(aa_num_num(bit0,one))) = one_one_Code_integer ).

% minus_1_mod_2_eq
tff(fact_4908_bits__minus__1__mod__2__eq,axiom,
    aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ).

% bits_minus_1_mod_2_eq
tff(fact_4909_bits__minus__1__mod__2__eq,axiom,
    modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),numera6620942414471956472nteger(aa_num_num(bit0,one))) = one_one_Code_integer ).

% bits_minus_1_mod_2_eq
tff(fact_4910_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% Power.ring_1_class.power_minus_even
tff(fact_4911_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% Power.ring_1_class.power_minus_even
tff(fact_4912_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% Power.ring_1_class.power_minus_even
tff(fact_4913_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% Power.ring_1_class.power_minus_even
tff(fact_4914_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: code_integer,Nb: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% Power.ring_1_class.power_minus_even
tff(fact_4915_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [Nb: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),Nb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_4916_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,A)),Nb) = aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_4917_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [Nb: nat,A: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,A)),Nb) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),Nb) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_4918_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [Nb: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),Nb) = aa_nat_rat(power_power_rat(A),Nb) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_4919_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [Nb: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),Nb) = aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_4920_power__minus__odd,axiom,
    ! [Nb: nat,A: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),Nb) = aa_real_real(uminus_uminus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ) ).

% power_minus_odd
tff(fact_4921_power__minus__odd,axiom,
    ! [Nb: nat,A: int] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,A)),Nb) = aa_int_int(uminus_uminus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)) ) ) ).

% power_minus_odd
tff(fact_4922_power__minus__odd,axiom,
    ! [Nb: nat,A: complex] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,A)),Nb) = aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),Nb)) ) ) ).

% power_minus_odd
tff(fact_4923_power__minus__odd,axiom,
    ! [Nb: nat,A: rat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),Nb) = aa_rat_rat(uminus_uminus_rat,aa_nat_rat(power_power_rat(A),Nb)) ) ) ).

% power_minus_odd
tff(fact_4924_power__minus__odd,axiom,
    ! [Nb: nat,A: code_integer] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),Nb) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)) ) ) ).

% power_minus_odd
tff(fact_4925_even__take__bit__eq,axiom,
    ! [Nb: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se1745604003318907178nteger(Nb,A))
    <=> ( ( Nb = zero_zero_nat )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A) ) ) ).

% even_take_bit_eq
tff(fact_4926_even__take__bit__eq,axiom,
    ! [Nb: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se2923211474154528505it_int(Nb),A))
    <=> ( ( Nb = zero_zero_nat )
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) ) ) ).

% even_take_bit_eq
tff(fact_4927_even__take__bit__eq,axiom,
    ! [Nb: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A))
    <=> ( ( Nb = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A) ) ) ).

% even_take_bit_eq
tff(fact_4928_diff__numeral__special_I4_J,axiom,
    ! [Ma: num] : aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Ma))),one_one_real) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(Ma),one))) ).

% diff_numeral_special(4)
tff(fact_4929_diff__numeral__special_I4_J,axiom,
    ! [Ma: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Ma))),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(Ma),one))) ).

% diff_numeral_special(4)
tff(fact_4930_diff__numeral__special_I4_J,axiom,
    ! [Ma: num] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Ma))),one_one_complex) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(plus_plus_num(Ma),one))) ).

% diff_numeral_special(4)
tff(fact_4931_diff__numeral__special_I4_J,axiom,
    ! [Ma: num] : aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Ma))),one_one_rat) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(Ma),one))) ).

% diff_numeral_special(4)
tff(fact_4932_diff__numeral__special_I4_J,axiom,
    ! [Ma: num] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Ma)),one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(plus_plus_num(Ma),one))) ).

% diff_numeral_special(4)
tff(fact_4933_diff__numeral__special_I3_J,axiom,
    ! [Nb: num] : aa_real_real(minus_minus_real(one_one_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(one),Nb)) ).

% diff_numeral_special(3)
tff(fact_4934_diff__numeral__special_I3_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(one),Nb)) ).

% diff_numeral_special(3)
tff(fact_4935_diff__numeral__special_I3_J,axiom,
    ! [Nb: num] : aa_complex_complex(minus_minus_complex(one_one_complex),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb))) = numera6690914467698888265omplex(aa_num_num(plus_plus_num(one),Nb)) ).

% diff_numeral_special(3)
tff(fact_4936_diff__numeral__special_I3_J,axiom,
    ! [Nb: num] : aa_rat_rat(minus_minus_rat(one_one_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(one),Nb)) ).

% diff_numeral_special(3)
tff(fact_4937_diff__numeral__special_I3_J,axiom,
    ! [Nb: num] : minus_8373710615458151222nteger(one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) = numera6620942414471956472nteger(aa_num_num(plus_plus_num(one),Nb)) ).

% diff_numeral_special(3)
tff(fact_4938_Suc__div__eq__add3__div__numeral,axiom,
    ! [Ma: nat,V: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Ma)))),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Ma)),aa_num_nat(numeral_numeral_nat,V)) ).

% Suc_div_eq_add3_div_numeral
tff(fact_4939_div__Suc__eq__div__add3,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Nb)))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Nb)) ).

% div_Suc_eq_div_add3
tff(fact_4940_Suc__mod__eq__add3__mod__numeral,axiom,
    ! [Ma: nat,V: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Ma)))),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Ma)),aa_num_nat(numeral_numeral_nat,V)) ).

% Suc_mod_eq_add3_mod_numeral
tff(fact_4941_mod__Suc__eq__mod__add3,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Nb)))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Nb)) ).

% mod_Suc_eq_mod_add3
tff(fact_4942_signed__take__bit__Suc__minus__bit0,axiom,
    ! [Nb: nat,K2: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K2)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% signed_take_bit_Suc_minus_bit0
tff(fact_4943_dbl__simps_I4_J,axiom,
    aa_real_real(neg_numeral_dbl_real,aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% dbl_simps(4)
tff(fact_4944_dbl__simps_I4_J,axiom,
    aa_int_int(neg_numeral_dbl_int,aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% dbl_simps(4)
tff(fact_4945_dbl__simps_I4_J,axiom,
    aa_complex_complex(neg_nu7009210354673126013omplex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% dbl_simps(4)
tff(fact_4946_dbl__simps_I4_J,axiom,
    aa_rat_rat(neg_numeral_dbl_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))) ).

% dbl_simps(4)
tff(fact_4947_dbl__simps_I4_J,axiom,
    neg_nu8804712462038260780nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% dbl_simps(4)
tff(fact_4948_power__minus1__even,axiom,
    ! [Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = one_one_real ).

% power_minus1_even
tff(fact_4949_power__minus1__even,axiom,
    ! [Nb: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = one_one_int ).

% power_minus1_even
tff(fact_4950_power__minus1__even,axiom,
    ! [Nb: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = one_one_complex ).

% power_minus1_even
tff(fact_4951_power__minus1__even,axiom,
    ! [Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = one_one_rat ).

% power_minus1_even
tff(fact_4952_power__minus1__even,axiom,
    ! [Nb: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = one_one_Code_integer ).

% power_minus1_even
tff(fact_4953_neg__one__even__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb) = one_one_real ) ) ).

% neg_one_even_power
tff(fact_4954_neg__one__even__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),Nb) = one_one_int ) ) ).

% neg_one_even_power
tff(fact_4955_neg__one__even__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb) = one_one_complex ) ) ).

% neg_one_even_power
tff(fact_4956_neg__one__even__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb) = one_one_rat ) ) ).

% neg_one_even_power
tff(fact_4957_neg__one__even__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb) = one_one_Code_integer ) ) ).

% neg_one_even_power
tff(fact_4958_neg__one__odd__power,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb) = aa_real_real(uminus_uminus_real,one_one_real) ) ) ).

% neg_one_odd_power
tff(fact_4959_neg__one__odd__power,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),Nb) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ).

% neg_one_odd_power
tff(fact_4960_neg__one__odd__power,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ).

% neg_one_odd_power
tff(fact_4961_neg__one__odd__power,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ).

% neg_one_odd_power
tff(fact_4962_neg__one__odd__power,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ).

% neg_one_odd_power
tff(fact_4963_take__bit__Suc__0,axiom,
    ! [A: code_integer] : bit_se1745604003318907178nteger(aa_nat_nat(suc,zero_zero_nat),A) = modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% take_bit_Suc_0
tff(fact_4964_take__bit__Suc__0,axiom,
    ! [A: int] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,zero_zero_nat)),A) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_Suc_0
tff(fact_4965_take__bit__Suc__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,zero_zero_nat)),A) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% take_bit_Suc_0
tff(fact_4966_signed__take__bit__0,axiom,
    ! [A: code_integer] : bit_ri6519982836138164636nteger(zero_zero_nat,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))) ).

% signed_take_bit_0
tff(fact_4967_signed__take__bit__0,axiom,
    ! [A: int] : aa_int_int(bit_ri631733984087533419it_int(zero_zero_nat),A) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% signed_take_bit_0
tff(fact_4968_take__bit__of__exp,axiom,
    ! [Ma: nat,Nb: nat] : bit_se1745604003318907178nteger(Ma,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_n356916108424825756nteger(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Ma))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)) ).

% take_bit_of_exp
tff(fact_4969_take__bit__of__exp,axiom,
    ! [Ma: nat,Nb: nat] : aa_int_int(bit_se2923211474154528505it_int(Ma),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Ma))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% take_bit_of_exp
tff(fact_4970_take__bit__of__exp,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Ma))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% take_bit_of_exp
tff(fact_4971_take__bit__of__2,axiom,
    ! [Nb: nat] : bit_se1745604003318907178nteger(Nb,numera6620942414471956472nteger(aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_n356916108424825756nteger(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% take_bit_of_2
tff(fact_4972_take__bit__of__2,axiom,
    ! [Nb: nat] : aa_int_int(bit_se2923211474154528505it_int(Nb),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_of_2
tff(fact_4973_take__bit__of__2,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% take_bit_of_2
tff(fact_4974_zmod__numeral__Bit1,axiom,
    ! [V: num,W: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,V))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,W)))),one_one_int) ).

% zmod_numeral_Bit1
tff(fact_4975_signed__take__bit__Suc__minus__bit1,axiom,
    ! [Nb: nat,K2: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K2)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% signed_take_bit_Suc_minus_bit1
tff(fact_4976_minus__set__def,axiom,
    ! [A2: set_real,B3: set_real] : minus_minus_set_real(A2,B3) = collect_real(minus_404528632214381484real_o(aTP_Lamm_ab(A2),aTP_Lamm_ab(B3))) ).

% minus_set_def
tff(fact_4977_minus__set__def,axiom,
    ! [A2: set_complex,B3: set_complex] : minus_811609699411566653omplex(A2,B3) = collect_complex(minus_8169481405280535082plex_o(aTP_Lamm_ac(A2),aTP_Lamm_ac(B3))) ).

% minus_set_def
tff(fact_4978_minus__set__def,axiom,
    ! [A2: set_list_nat,B3: set_list_nat] : minus_7954133019191499631st_nat(A2,B3) = collect_list_nat(minus_5761417318614613624_nat_o(aTP_Lamm_ad(A2),aTP_Lamm_ad(B3))) ).

% minus_set_def
tff(fact_4979_minus__set__def,axiom,
    ! [A2: set_set_nat,B3: set_set_nat] : minus_2163939370556025621et_nat(A2,B3) = collect_set_nat(minus_6087510627544308050_nat_o(aTP_Lamm_ae(A2),aTP_Lamm_ae(B3))) ).

% minus_set_def
tff(fact_4980_minus__set__def,axiom,
    ! [A2: set_nat,B3: set_nat] : aa_set_nat_set_nat(minus_minus_set_nat(A2),B3) = collect_nat(minus_8972599623528345224_nat_o(aTP_Lamm_af(A2),aTP_Lamm_af(B3))) ).

% minus_set_def
tff(fact_4981_minus__set__def,axiom,
    ! [A2: set_int,B3: set_int] : minus_minus_set_int(A2,B3) = collect_int(minus_7151557278095657260_int_o(aTP_Lamm_ag(A2),aTP_Lamm_ag(B3))) ).

% minus_set_def
tff(fact_4982_set__diff__eq,axiom,
    ! [A2: set_real,B3: set_real] : minus_minus_set_real(A2,B3) = collect_real(aa_set2323979645938365349real_o(aTP_Lamm_bu(A2),B3)) ).

% set_diff_eq
tff(fact_4983_set__diff__eq,axiom,
    ! [A2: set_complex,B3: set_complex] : minus_811609699411566653omplex(A2,B3) = collect_complex(aa_set1742385355640705825plex_o(aTP_Lamm_bv(A2),B3)) ).

% set_diff_eq
tff(fact_4984_set__diff__eq,axiom,
    ! [A2: set_list_nat,B3: set_list_nat] : minus_7954133019191499631st_nat(A2,B3) = collect_list_nat(aa_set6675653781276780861_nat_o(aTP_Lamm_bw(A2),B3)) ).

% set_diff_eq
tff(fact_4985_set__diff__eq,axiom,
    ! [A2: set_set_nat,B3: set_set_nat] : minus_2163939370556025621et_nat(A2,B3) = collect_set_nat(aa_set4404905041378927473_nat_o(aTP_Lamm_bx(A2),B3)) ).

% set_diff_eq
tff(fact_4986_set__diff__eq,axiom,
    ! [A2: set_nat,B3: set_nat] : aa_set_nat_set_nat(minus_minus_set_nat(A2),B3) = collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_by(A2),B3)) ).

% set_diff_eq
tff(fact_4987_set__diff__eq,axiom,
    ! [A2: set_int,B3: set_int] : minus_minus_set_int(A2,B3) = collect_int(aa_set_int_fun_int_o(aTP_Lamm_bz(A2),B3)) ).

% set_diff_eq
tff(fact_4988_verit__negate__coefficient_I3_J,axiom,
    ! [A: real,B: real] :
      ( ( A = B )
     => ( aa_real_real(uminus_uminus_real,A) = aa_real_real(uminus_uminus_real,B) ) ) ).

% verit_negate_coefficient(3)
tff(fact_4989_verit__negate__coefficient_I3_J,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
     => ( aa_int_int(uminus_uminus_int,A) = aa_int_int(uminus_uminus_int,B) ) ) ).

% verit_negate_coefficient(3)
tff(fact_4990_verit__negate__coefficient_I3_J,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
     => ( aa_rat_rat(uminus_uminus_rat,A) = aa_rat_rat(uminus_uminus_rat,B) ) ) ).

% verit_negate_coefficient(3)
tff(fact_4991_verit__negate__coefficient_I3_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = B )
     => ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ) ) ).

% verit_negate_coefficient(3)
tff(fact_4992_equation__minus__iff,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(uminus_uminus_real,B) )
    <=> ( B = aa_real_real(uminus_uminus_real,A) ) ) ).

% equation_minus_iff
tff(fact_4993_equation__minus__iff,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(uminus_uminus_int,B) )
    <=> ( B = aa_int_int(uminus_uminus_int,A) ) ) ).

% equation_minus_iff
tff(fact_4994_equation__minus__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( A = aa_complex_complex(uminus1482373934393186551omplex,B) )
    <=> ( B = aa_complex_complex(uminus1482373934393186551omplex,A) ) ) ).

% equation_minus_iff
tff(fact_4995_equation__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,B) )
    <=> ( B = aa_rat_rat(uminus_uminus_rat,A) ) ) ).

% equation_minus_iff
tff(fact_4996_equation__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) )
    <=> ( B = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).

% equation_minus_iff
tff(fact_4997_minus__equation__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = B )
    <=> ( aa_real_real(uminus_uminus_real,B) = A ) ) ).

% minus_equation_iff
tff(fact_4998_minus__equation__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = B )
    <=> ( aa_int_int(uminus_uminus_int,B) = A ) ) ).

% minus_equation_iff
tff(fact_4999_minus__equation__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,A) = B )
    <=> ( aa_complex_complex(uminus1482373934393186551omplex,B) = A ) ) ).

% minus_equation_iff
tff(fact_5000_minus__equation__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = B )
    <=> ( aa_rat_rat(uminus_uminus_rat,B) = A ) ) ).

% minus_equation_iff
tff(fact_5001_minus__equation__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = B )
    <=> ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) = A ) ) ).

% minus_equation_iff
tff(fact_5002_power__minus__Bit1,axiom,
    ! [X: real,K2: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K2))) = aa_real_real(uminus_uminus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K2)))) ).

% power_minus_Bit1
tff(fact_5003_power__minus__Bit1,axiom,
    ! [X: int,K2: num] : aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K2))) = aa_int_int(uminus_uminus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K2)))) ).

% power_minus_Bit1
tff(fact_5004_power__minus__Bit1,axiom,
    ! [X: complex,K2: num] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K2))) = aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K2)))) ).

% power_minus_Bit1
tff(fact_5005_power__minus__Bit1,axiom,
    ! [X: rat,K2: num] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K2))) = aa_rat_rat(uminus_uminus_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K2)))) ).

% power_minus_Bit1
tff(fact_5006_power__minus__Bit1,axiom,
    ! [X: code_integer,K2: num] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K2))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K2)))) ).

% power_minus_Bit1
tff(fact_5007_take__bit__add,axiom,
    ! [Nb: nat,A: int,B: int] : aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(bit_se2923211474154528505it_int(Nb),A)),aa_int_int(bit_se2923211474154528505it_int(Nb),B))) = aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ).

% take_bit_add
tff(fact_5008_take__bit__add,axiom,
    ! [Nb: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),B))) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ).

% take_bit_add
tff(fact_5009_take__bit__tightened,axiom,
    ! [Nb: nat,A: int,B: int,Ma: nat] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Nb),A) = aa_int_int(bit_se2923211474154528505it_int(Nb),B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
       => ( aa_int_int(bit_se2923211474154528505it_int(Ma),A) = aa_int_int(bit_se2923211474154528505it_int(Ma),B) ) ) ) ).

% take_bit_tightened
tff(fact_5010_take__bit__tightened,axiom,
    ! [Nb: nat,A: nat,B: nat,Ma: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Ma),A) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Ma),B) ) ) ) ).

% take_bit_tightened
tff(fact_5011_take__bit__nat__less__eq__self,axiom,
    ! [Nb: nat,Ma: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Ma)),Ma) ).

% take_bit_nat_less_eq_self
tff(fact_5012_take__bit__tightened__less__eq__nat,axiom,
    ! [Ma: nat,Nb: nat,Q: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Ma),Q)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Q)) ) ).

% take_bit_tightened_less_eq_nat
tff(fact_5013_take__bit__mult,axiom,
    ! [Nb: nat,K2: int,L: int] : aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(Nb),K2)),aa_int_int(bit_se2923211474154528505it_int(Nb),L))) = aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(aa_int_fun_int_int(times_times_int,K2),L)) ).

% take_bit_mult
tff(fact_5014_le__imp__neg__le,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A)) ) ).

% le_imp_neg_le
tff(fact_5015_le__imp__neg__le,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ).

% le_imp_neg_le
tff(fact_5016_le__imp__neg__le,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A)) ) ).

% le_imp_neg_le
tff(fact_5017_le__imp__neg__le,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A)) ) ).

% le_imp_neg_le
tff(fact_5018_minus__le__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),A) ) ).

% minus_le_iff
tff(fact_5019_minus__le__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),A) ) ).

% minus_le_iff
tff(fact_5020_minus__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),A) ) ).

% minus_le_iff
tff(fact_5021_minus__le__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),A) ) ).

% minus_le_iff
tff(fact_5022_le__minus__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(uminus_uminus_real,A)) ) ).

% le_minus_iff
tff(fact_5023_le__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ).

% le_minus_iff
tff(fact_5024_le__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(uminus_uminus_rat,A)) ) ).

% le_minus_iff
tff(fact_5025_le__minus__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(uminus_uminus_int,B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(uminus_uminus_int,A)) ) ).

% le_minus_iff
tff(fact_5026_minus__less__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,A)),B)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),A) ) ).

% minus_less_iff
tff(fact_5027_minus__less__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,A)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,B)),A) ) ).

% minus_less_iff
tff(fact_5028_minus__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),A) ) ).

% minus_less_iff
tff(fact_5029_minus__less__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),A) ) ).

% minus_less_iff
tff(fact_5030_less__minus__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(uminus_uminus_real,B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(uminus_uminus_real,A)) ) ).

% less_minus_iff
tff(fact_5031_less__minus__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(uminus_uminus_int,B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(uminus_uminus_int,A)) ) ).

% less_minus_iff
tff(fact_5032_less__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(uminus_uminus_rat,A)) ) ).

% less_minus_iff
tff(fact_5033_less__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ).

% less_minus_iff
tff(fact_5034_verit__negate__coefficient_I2_J,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A)) ) ).

% verit_negate_coefficient(2)
tff(fact_5035_verit__negate__coefficient_I2_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A)) ) ).

% verit_negate_coefficient(2)
tff(fact_5036_verit__negate__coefficient_I2_J,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A)) ) ).

% verit_negate_coefficient(2)
tff(fact_5037_verit__negate__coefficient_I2_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ).

% verit_negate_coefficient(2)
tff(fact_5038_verit__eq__simplify_I14_J,axiom,
    ! [X23: num,X32: num] : aa_num_num(bit0,X23) != aa_num_num(bit1,X32) ).

% verit_eq_simplify(14)
tff(fact_5039_verit__eq__simplify_I12_J,axiom,
    ! [X32: num] : one != aa_num_num(bit1,X32) ).

% verit_eq_simplify(12)
tff(fact_5040_numeral__neq__neg__numeral,axiom,
    ! [Ma: num,Nb: num] : aa_num_real(numeral_numeral_real,Ma) != aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)) ).

% numeral_neq_neg_numeral
tff(fact_5041_numeral__neq__neg__numeral,axiom,
    ! [Ma: num,Nb: num] : aa_num_int(numeral_numeral_int,Ma) != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) ).

% numeral_neq_neg_numeral
tff(fact_5042_numeral__neq__neg__numeral,axiom,
    ! [Ma: num,Nb: num] : numera6690914467698888265omplex(Ma) != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb)) ).

% numeral_neq_neg_numeral
tff(fact_5043_numeral__neq__neg__numeral,axiom,
    ! [Ma: num,Nb: num] : aa_num_rat(numeral_numeral_rat,Ma) != aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)) ).

% numeral_neq_neg_numeral
tff(fact_5044_numeral__neq__neg__numeral,axiom,
    ! [Ma: num,Nb: num] : numera6620942414471956472nteger(Ma) != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb)) ).

% numeral_neq_neg_numeral
tff(fact_5045_neg__numeral__neq__numeral,axiom,
    ! [Ma: num,Nb: num] : aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Ma)) != aa_num_real(numeral_numeral_real,Nb) ).

% neg_numeral_neq_numeral
tff(fact_5046_neg__numeral__neq__numeral,axiom,
    ! [Ma: num,Nb: num] : aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Ma)) != aa_num_int(numeral_numeral_int,Nb) ).

% neg_numeral_neq_numeral
tff(fact_5047_neg__numeral__neq__numeral,axiom,
    ! [Ma: num,Nb: num] : aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Ma)) != numera6690914467698888265omplex(Nb) ).

% neg_numeral_neq_numeral
tff(fact_5048_neg__numeral__neq__numeral,axiom,
    ! [Ma: num,Nb: num] : aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Ma)) != aa_num_rat(numeral_numeral_rat,Nb) ).

% neg_numeral_neq_numeral
tff(fact_5049_neg__numeral__neq__numeral,axiom,
    ! [Ma: num,Nb: num] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Ma)) != numera6620942414471956472nteger(Nb) ).

% neg_numeral_neq_numeral
tff(fact_5050_minus__mult__commute,axiom,
    ! [A: real,B: real] : aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,A)),B) = aa_real_real(times_times_real(A),aa_real_real(uminus_uminus_real,B)) ).

% minus_mult_commute
tff(fact_5051_minus__mult__commute,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(uminus_uminus_int,B)) ).

% minus_mult_commute
tff(fact_5052_minus__mult__commute,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),B) = aa_complex_complex(times_times_complex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) ).

% minus_mult_commute
tff(fact_5053_minus__mult__commute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,A)),B) = aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,B)) ).

% minus_mult_commute
tff(fact_5054_minus__mult__commute,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ).

% minus_mult_commute
tff(fact_5055_square__eq__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(times_times_real(A),A) = aa_real_real(times_times_real(B),B) )
    <=> ( ( A = B )
        | ( A = aa_real_real(uminus_uminus_real,B) ) ) ) ).

% square_eq_iff
tff(fact_5056_square__eq__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),A) = aa_int_int(aa_int_fun_int_int(times_times_int,B),B) )
    <=> ( ( A = B )
        | ( A = aa_int_int(uminus_uminus_int,B) ) ) ) ).

% square_eq_iff
tff(fact_5057_square__eq__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(A),A) = aa_complex_complex(times_times_complex(B),B) )
    <=> ( ( A = B )
        | ( A = aa_complex_complex(uminus1482373934393186551omplex,B) ) ) ) ).

% square_eq_iff
tff(fact_5058_square__eq__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(A),A) = aa_rat_rat(times_times_rat(B),B) )
    <=> ( ( A = B )
        | ( A = aa_rat_rat(uminus_uminus_rat,B) ) ) ) ).

% square_eq_iff
tff(fact_5059_square__eq__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),B) )
    <=> ( ( A = B )
        | ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ) ) ) ).

% square_eq_iff
tff(fact_5060_one__neq__neg__one,axiom,
    one_one_real != aa_real_real(uminus_uminus_real,one_one_real) ).

% one_neq_neg_one
tff(fact_5061_one__neq__neg__one,axiom,
    one_one_int != aa_int_int(uminus_uminus_int,one_one_int) ).

% one_neq_neg_one
tff(fact_5062_one__neq__neg__one,axiom,
    one_one_complex != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% one_neq_neg_one
tff(fact_5063_one__neq__neg__one,axiom,
    one_one_rat != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% one_neq_neg_one
tff(fact_5064_one__neq__neg__one,axiom,
    one_one_Code_integer != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% one_neq_neg_one
tff(fact_5065_add_Oinverse__distrib__swap,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(plus_plus_real(A),B)) = aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A)) ).

% add.inverse_distrib_swap
tff(fact_5066_add_Oinverse__distrib__swap,axiom,
    ! [A: int,B: int] : aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A)) ).

% add.inverse_distrib_swap
tff(fact_5067_add_Oinverse__distrib__swap,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(plus_plus_complex(A),B)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,B)),aa_complex_complex(uminus1482373934393186551omplex,A)) ).

% add.inverse_distrib_swap
tff(fact_5068_add_Oinverse__distrib__swap,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(plus_plus_rat(A),B)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A)) ).

% add.inverse_distrib_swap
tff(fact_5069_add_Oinverse__distrib__swap,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,plus_p5714425477246183910nteger(A,B)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ).

% add.inverse_distrib_swap
tff(fact_5070_group__cancel_Oneg1,axiom,
    ! [A2: real,K2: real,A: real] :
      ( ( A2 = aa_real_real(plus_plus_real(K2),A) )
     => ( aa_real_real(uminus_uminus_real,A2) = aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,K2)),aa_real_real(uminus_uminus_real,A)) ) ) ).

% group_cancel.neg1
tff(fact_5071_group__cancel_Oneg1,axiom,
    ! [A2: int,K2: int,A: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),A) )
     => ( aa_int_int(uminus_uminus_int,A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,K2)),aa_int_int(uminus_uminus_int,A)) ) ) ).

% group_cancel.neg1
tff(fact_5072_group__cancel_Oneg1,axiom,
    ! [A2: complex,K2: complex,A: complex] :
      ( ( A2 = aa_complex_complex(plus_plus_complex(K2),A) )
     => ( aa_complex_complex(uminus1482373934393186551omplex,A2) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,K2)),aa_complex_complex(uminus1482373934393186551omplex,A)) ) ) ).

% group_cancel.neg1
tff(fact_5073_group__cancel_Oneg1,axiom,
    ! [A2: rat,K2: rat,A: rat] :
      ( ( A2 = aa_rat_rat(plus_plus_rat(K2),A) )
     => ( aa_rat_rat(uminus_uminus_rat,A2) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,K2)),aa_rat_rat(uminus_uminus_rat,A)) ) ) ).

% group_cancel.neg1
tff(fact_5074_group__cancel_Oneg1,axiom,
    ! [A2: code_integer,K2: code_integer,A: code_integer] :
      ( ( A2 = plus_p5714425477246183910nteger(K2,A) )
     => ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K2),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ) ).

% group_cancel.neg1
tff(fact_5075_is__num__normalize_I8_J,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(plus_plus_real(A),B)) = aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A)) ).

% is_num_normalize(8)
tff(fact_5076_is__num__normalize_I8_J,axiom,
    ! [A: int,B: int] : aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A)) ).

% is_num_normalize(8)
tff(fact_5077_is__num__normalize_I8_J,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(plus_plus_complex(A),B)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,B)),aa_complex_complex(uminus1482373934393186551omplex,A)) ).

% is_num_normalize(8)
tff(fact_5078_is__num__normalize_I8_J,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(plus_plus_rat(A),B)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A)) ).

% is_num_normalize(8)
tff(fact_5079_is__num__normalize_I8_J,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,plus_p5714425477246183910nteger(A,B)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ).

% is_num_normalize(8)
tff(fact_5080_take__bit__diff,axiom,
    ! [Nb: nat,K2: int,L: int] : aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(bit_se2923211474154528505it_int(Nb),K2)),aa_int_int(bit_se2923211474154528505it_int(Nb),L))) = aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(aa_int_fun_int_int(minus_minus_int,K2),L)) ).

% take_bit_diff
tff(fact_5081_minus__diff__minus,axiom,
    ! [A: real,B: real] : aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) = aa_real_real(uminus_uminus_real,aa_real_real(minus_minus_real(A),B)) ).

% minus_diff_minus
tff(fact_5082_minus__diff__minus,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) ).

% minus_diff_minus
tff(fact_5083_minus__diff__minus,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(minus_minus_complex(A),B)) ).

% minus_diff_minus
tff(fact_5084_minus__diff__minus,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(minus_minus_rat(A),B)) ).

% minus_diff_minus
tff(fact_5085_minus__diff__minus,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,minus_8373710615458151222nteger(A,B)) ).

% minus_diff_minus
tff(fact_5086_minus__diff__commute,axiom,
    ! [B: real,A: real] : aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,B)),A) = aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,A)),B) ).

% minus_diff_commute
tff(fact_5087_minus__diff__commute,axiom,
    ! [B: int,A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,B)),A) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,A)),B) ).

% minus_diff_commute
tff(fact_5088_minus__diff__commute,axiom,
    ! [B: complex,A: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,B)),A) = aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),B) ).

% minus_diff_commute
tff(fact_5089_minus__diff__commute,axiom,
    ! [B: rat,A: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,B)),A) = aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,A)),B) ).

% minus_diff_commute
tff(fact_5090_minus__diff__commute,axiom,
    ! [B: code_integer,A: code_integer] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B),A) = minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) ).

% minus_diff_commute
tff(fact_5091_div__minus__right,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A)),B) ).

% div_minus_right
tff(fact_5092_div__minus__right,axiom,
    ! [A: code_integer,B: code_integer] : divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) ).

% div_minus_right
tff(fact_5093_minus__divide__left,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(A),B)) = aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,A)),B) ).

% minus_divide_left
tff(fact_5094_minus__divide__left,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(uminus1482373934393186551omplex,A)),B) ).

% minus_divide_left
tff(fact_5095_minus__divide__left,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(uminus_uminus_rat,A)),B) ).

% minus_divide_left
tff(fact_5096_minus__divide__divide,axiom,
    ! [A: real,B: real] : aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) = aa_real_real(divide_divide_real(A),B) ).

% minus_divide_divide
tff(fact_5097_minus__divide__divide,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ).

% minus_divide_divide
tff(fact_5098_minus__divide__divide,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(divide_divide_rat(A),B) ).

% minus_divide_divide
tff(fact_5099_minus__divide__right,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(A),B)) = aa_real_real(divide_divide_real(A),aa_real_real(uminus_uminus_real,B)) ).

% minus_divide_right
tff(fact_5100_minus__divide__right,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) ).

% minus_divide_right
tff(fact_5101_minus__divide__right,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B)) = aa_rat_rat(divide_divide_rat(A),aa_rat_rat(uminus_uminus_rat,B)) ).

% minus_divide_right
tff(fact_5102_mod__minus__eq,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A)),B) ).

% mod_minus_eq
tff(fact_5103_mod__minus__eq,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(A,B)),B) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) ).

% mod_minus_eq
tff(fact_5104_mod__minus__cong,axiom,
    ! [A: int,B: int,A4: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A4),B) )
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A4)),B) ) ) ).

% mod_minus_cong
tff(fact_5105_mod__minus__cong,axiom,
    ! [A: code_integer,B: code_integer,A4: code_integer] :
      ( ( modulo364778990260209775nteger(A,B) = modulo364778990260209775nteger(A4,B) )
     => ( modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A4),B) ) ) ).

% mod_minus_cong
tff(fact_5106_mod__minus__right,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A)),B)) ).

% mod_minus_right
tff(fact_5107_mod__minus__right,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B)) ).

% mod_minus_right
tff(fact_5108_minus__real__def,axiom,
    ! [X: real,Y2: real] : aa_real_real(minus_minus_real(X),Y2) = aa_real_real(plus_plus_real(X),aa_real_real(uminus_uminus_real,Y2)) ).

% minus_real_def
tff(fact_5109_concat__bit__eq__iff,axiom,
    ! [Nb: nat,K2: int,L: int,R: int,S: int] :
      ( ( aa_int_int(aa_int_fun_int_int(bit_concat_bit(Nb),K2),L) = aa_int_int(aa_int_fun_int_int(bit_concat_bit(Nb),R),S) )
    <=> ( ( aa_int_int(bit_se2923211474154528505it_int(Nb),K2) = aa_int_int(bit_se2923211474154528505it_int(Nb),R) )
        & ( L = S ) ) ) ).

% concat_bit_eq_iff
tff(fact_5110_concat__bit__take__bit__eq,axiom,
    ! [Nb: nat,B: int] : aa_int_fun_int_int(bit_concat_bit(Nb),aa_int_int(bit_se2923211474154528505it_int(Nb),B)) = aa_int_fun_int_int(bit_concat_bit(Nb),B) ).

% concat_bit_take_bit_eq
tff(fact_5111_signed__take__bit__minus,axiom,
    ! [Nb: nat,K2: int] : aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(uminus_uminus_int,aa_int_int(bit_ri631733984087533419it_int(Nb),K2))) = aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(uminus_uminus_int,K2)) ).

% signed_take_bit_minus
tff(fact_5112_take__bit__tightened__less__eq__int,axiom,
    ! [Ma: nat,Nb: nat,K2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se2923211474154528505it_int(Ma),K2)),aa_int_int(bit_se2923211474154528505it_int(Nb),K2)) ) ).

% take_bit_tightened_less_eq_int
tff(fact_5113_take__bit__int__less__eq__self__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se2923211474154528505it_int(Nb),K2)),K2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2) ) ).

% take_bit_int_less_eq_self_iff
tff(fact_5114_take__bit__nonnegative,axiom,
    ! [Nb: nat,K2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se2923211474154528505it_int(Nb),K2)) ).

% take_bit_nonnegative
tff(fact_5115_take__bit__int__greater__self__iff,axiom,
    ! [K2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),aa_int_int(bit_se2923211474154528505it_int(Nb),K2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int) ) ).

% take_bit_int_greater_self_iff
tff(fact_5116_not__take__bit__negative,axiom,
    ! [Nb: nat,K2: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se2923211474154528505it_int(Nb),K2)),zero_zero_int) ).

% not_take_bit_negative
tff(fact_5117_signed__take__bit__eq__iff__take__bit__eq,axiom,
    ! [Nb: nat,A: int,B: int] :
      ( ( aa_int_int(bit_ri631733984087533419it_int(Nb),A) = aa_int_int(bit_ri631733984087533419it_int(Nb),B) )
    <=> ( aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),A) = aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),B) ) ) ).

% signed_take_bit_eq_iff_take_bit_eq
tff(fact_5118_signed__take__bit__take__bit,axiom,
    ! [Ma: nat,Nb: nat,A: int] :
      aa_int_int(bit_ri631733984087533419it_int(Ma),aa_int_int(bit_se2923211474154528505it_int(Nb),A)) = aa_int_int($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma),bit_se2923211474154528505it_int(Nb),bit_ri631733984087533419it_int(Ma)),A) ).

% signed_take_bit_take_bit
tff(fact_5119_not__numeral__le__neg__numeral,axiom,
    ! [Ma: num,Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Ma)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) ).

% not_numeral_le_neg_numeral
tff(fact_5120_not__numeral__le__neg__numeral,axiom,
    ! [Ma: num,Nb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(Ma)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) ).

% not_numeral_le_neg_numeral
tff(fact_5121_not__numeral__le__neg__numeral,axiom,
    ! [Ma: num,Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Ma)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) ).

% not_numeral_le_neg_numeral
tff(fact_5122_not__numeral__le__neg__numeral,axiom,
    ! [Ma: num,Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Ma)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) ).

% not_numeral_le_neg_numeral
tff(fact_5123_neg__numeral__le__numeral,axiom,
    ! [Ma: num,Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Ma))),aa_num_real(numeral_numeral_real,Nb)) ).

% neg_numeral_le_numeral
tff(fact_5124_neg__numeral__le__numeral,axiom,
    ! [Ma: num,Nb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Ma))),numera6620942414471956472nteger(Nb)) ).

% neg_numeral_le_numeral
tff(fact_5125_neg__numeral__le__numeral,axiom,
    ! [Ma: num,Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Ma))),aa_num_rat(numeral_numeral_rat,Nb)) ).

% neg_numeral_le_numeral
tff(fact_5126_neg__numeral__le__numeral,axiom,
    ! [Ma: num,Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Ma))),aa_num_int(numeral_numeral_int,Nb)) ).

% neg_numeral_le_numeral
tff(fact_5127_zero__neq__neg__numeral,axiom,
    ! [Nb: num] : zero_zero_real != aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)) ).

% zero_neq_neg_numeral
tff(fact_5128_zero__neq__neg__numeral,axiom,
    ! [Nb: num] : zero_zero_int != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) ).

% zero_neq_neg_numeral
tff(fact_5129_zero__neq__neg__numeral,axiom,
    ! [Nb: num] : zero_zero_complex != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb)) ).

% zero_neq_neg_numeral
tff(fact_5130_zero__neq__neg__numeral,axiom,
    ! [Nb: num] : zero_zero_rat != aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)) ).

% zero_neq_neg_numeral
tff(fact_5131_zero__neq__neg__numeral,axiom,
    ! [Nb: num] : zero_z3403309356797280102nteger != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb)) ).

% zero_neq_neg_numeral
tff(fact_5132_not__numeral__less__neg__numeral,axiom,
    ! [Ma: num,Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,Ma)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) ).

% not_numeral_less_neg_numeral
tff(fact_5133_not__numeral__less__neg__numeral,axiom,
    ! [Ma: num,Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Ma)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) ).

% not_numeral_less_neg_numeral
tff(fact_5134_not__numeral__less__neg__numeral,axiom,
    ! [Ma: num,Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,Ma)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) ).

% not_numeral_less_neg_numeral
tff(fact_5135_not__numeral__less__neg__numeral,axiom,
    ! [Ma: num,Nb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,numera6620942414471956472nteger(Ma)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) ).

% not_numeral_less_neg_numeral
tff(fact_5136_neg__numeral__less__numeral,axiom,
    ! [Ma: num,Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Ma))),aa_num_real(numeral_numeral_real,Nb)) ).

% neg_numeral_less_numeral
tff(fact_5137_neg__numeral__less__numeral,axiom,
    ! [Ma: num,Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Ma))),aa_num_int(numeral_numeral_int,Nb)) ).

% neg_numeral_less_numeral
tff(fact_5138_neg__numeral__less__numeral,axiom,
    ! [Ma: num,Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Ma))),aa_num_rat(numeral_numeral_rat,Nb)) ).

% neg_numeral_less_numeral
tff(fact_5139_neg__numeral__less__numeral,axiom,
    ! [Ma: num,Nb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Ma))),numera6620942414471956472nteger(Nb)) ).

% neg_numeral_less_numeral
tff(fact_5140_le__minus__one__simps_I4_J,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(uminus_uminus_real,one_one_real)) ).

% le_minus_one_simps(4)
tff(fact_5141_le__minus__one__simps_I4_J,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% le_minus_one_simps(4)
tff(fact_5142_le__minus__one__simps_I4_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% le_minus_one_simps(4)
tff(fact_5143_le__minus__one__simps_I4_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int)) ).

% le_minus_one_simps(4)
tff(fact_5144_le__minus__one__simps_I2_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),one_one_real) ).

% le_minus_one_simps(2)
tff(fact_5145_le__minus__one__simps_I2_J,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),one_one_Code_integer) ).

% le_minus_one_simps(2)
tff(fact_5146_le__minus__one__simps_I2_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat) ).

% le_minus_one_simps(2)
tff(fact_5147_le__minus__one__simps_I2_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) ).

% le_minus_one_simps(2)
tff(fact_5148_zero__neq__neg__one,axiom,
    zero_zero_real != aa_real_real(uminus_uminus_real,one_one_real) ).

% zero_neq_neg_one
tff(fact_5149_zero__neq__neg__one,axiom,
    zero_zero_int != aa_int_int(uminus_uminus_int,one_one_int) ).

% zero_neq_neg_one
tff(fact_5150_zero__neq__neg__one,axiom,
    zero_zero_complex != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% zero_neq_neg_one
tff(fact_5151_zero__neq__neg__one,axiom,
    zero_zero_rat != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% zero_neq_neg_one
tff(fact_5152_zero__neq__neg__one,axiom,
    zero_z3403309356797280102nteger != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% zero_neq_neg_one
tff(fact_5153_neg__eq__iff__add__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = B )
    <=> ( aa_real_real(plus_plus_real(A),B) = zero_zero_real ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_5154_neg__eq__iff__add__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = B )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = zero_zero_int ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_5155_neg__eq__iff__add__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,A) = B )
    <=> ( aa_complex_complex(plus_plus_complex(A),B) = zero_zero_complex ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_5156_neg__eq__iff__add__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = B )
    <=> ( aa_rat_rat(plus_plus_rat(A),B) = zero_zero_rat ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_5157_neg__eq__iff__add__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = B )
    <=> ( plus_p5714425477246183910nteger(A,B) = zero_z3403309356797280102nteger ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_5158_eq__neg__iff__add__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(uminus_uminus_real,B) )
    <=> ( aa_real_real(plus_plus_real(A),B) = zero_zero_real ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_5159_eq__neg__iff__add__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(uminus_uminus_int,B) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = zero_zero_int ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_5160_eq__neg__iff__add__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( A = aa_complex_complex(uminus1482373934393186551omplex,B) )
    <=> ( aa_complex_complex(plus_plus_complex(A),B) = zero_zero_complex ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_5161_eq__neg__iff__add__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,B) )
    <=> ( aa_rat_rat(plus_plus_rat(A),B) = zero_zero_rat ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_5162_eq__neg__iff__add__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) )
    <=> ( plus_p5714425477246183910nteger(A,B) = zero_z3403309356797280102nteger ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_5163_add_Oinverse__unique,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(plus_plus_real(A),B) = zero_zero_real )
     => ( aa_real_real(uminus_uminus_real,A) = B ) ) ).

% add.inverse_unique
tff(fact_5164_add_Oinverse__unique,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = zero_zero_int )
     => ( aa_int_int(uminus_uminus_int,A) = B ) ) ).

% add.inverse_unique
tff(fact_5165_add_Oinverse__unique,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(plus_plus_complex(A),B) = zero_zero_complex )
     => ( aa_complex_complex(uminus1482373934393186551omplex,A) = B ) ) ).

% add.inverse_unique
tff(fact_5166_add_Oinverse__unique,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),B) = zero_zero_rat )
     => ( aa_rat_rat(uminus_uminus_rat,A) = B ) ) ).

% add.inverse_unique
tff(fact_5167_add_Oinverse__unique,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger(A,B) = zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = B ) ) ).

% add.inverse_unique
tff(fact_5168_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: real] : aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,A)),A) = zero_zero_real ).

% ab_group_add_class.ab_left_minus
tff(fact_5169_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),A) = zero_zero_int ).

% ab_group_add_class.ab_left_minus
tff(fact_5170_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),A) = zero_zero_complex ).

% ab_group_add_class.ab_left_minus
tff(fact_5171_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),A) = zero_zero_rat ).

% ab_group_add_class.ab_left_minus
tff(fact_5172_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),A) = zero_z3403309356797280102nteger ).

% ab_group_add_class.ab_left_minus
tff(fact_5173_add__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(plus_plus_real(A),B) = zero_zero_real )
    <=> ( B = aa_real_real(uminus_uminus_real,A) ) ) ).

% add_eq_0_iff
tff(fact_5174_add__eq__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = zero_zero_int )
    <=> ( B = aa_int_int(uminus_uminus_int,A) ) ) ).

% add_eq_0_iff
tff(fact_5175_add__eq__0__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(plus_plus_complex(A),B) = zero_zero_complex )
    <=> ( B = aa_complex_complex(uminus1482373934393186551omplex,A) ) ) ).

% add_eq_0_iff
tff(fact_5176_add__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),B) = zero_zero_rat )
    <=> ( B = aa_rat_rat(uminus_uminus_rat,A) ) ) ).

% add_eq_0_iff
tff(fact_5177_add__eq__0__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger(A,B) = zero_z3403309356797280102nteger )
    <=> ( B = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).

% add_eq_0_iff
tff(fact_5178_xor__num_Ocases,axiom,
    ! [X: product_prod_num_num] :
      ( ( X != aa_num3156681643174609981um_num(aa_num8681986212680877308um_num(product_Pair_num_num,one),one) )
     => ( ! [N: num] : X != aa_num3156681643174609981um_num(aa_num8681986212680877308um_num(product_Pair_num_num,one),aa_num_num(bit0,N))
       => ( ! [N: num] : X != aa_num3156681643174609981um_num(aa_num8681986212680877308um_num(product_Pair_num_num,one),aa_num_num(bit1,N))
         => ( ! [M2: num] : X != aa_num3156681643174609981um_num(aa_num8681986212680877308um_num(product_Pair_num_num,aa_num_num(bit0,M2)),one)
           => ( ! [M2: num,N: num] : X != aa_num3156681643174609981um_num(aa_num8681986212680877308um_num(product_Pair_num_num,aa_num_num(bit0,M2)),aa_num_num(bit0,N))
             => ( ! [M2: num,N: num] : X != aa_num3156681643174609981um_num(aa_num8681986212680877308um_num(product_Pair_num_num,aa_num_num(bit0,M2)),aa_num_num(bit1,N))
               => ( ! [M2: num] : X != aa_num3156681643174609981um_num(aa_num8681986212680877308um_num(product_Pair_num_num,aa_num_num(bit1,M2)),one)
                 => ( ! [M2: num,N: num] : X != aa_num3156681643174609981um_num(aa_num8681986212680877308um_num(product_Pair_num_num,aa_num_num(bit1,M2)),aa_num_num(bit0,N))
                   => ~ ! [M2: num,N: num] : X != aa_num3156681643174609981um_num(aa_num8681986212680877308um_num(product_Pair_num_num,aa_num_num(bit1,M2)),aa_num_num(bit1,N)) ) ) ) ) ) ) ) ) ).

% xor_num.cases
tff(fact_5179_num_Oexhaust,axiom,
    ! [Y2: num] :
      ( ( Y2 != one )
     => ( ! [X24: num] : Y2 != aa_num_num(bit0,X24)
       => ~ ! [X33: num] : Y2 != aa_num_num(bit1,X33) ) ) ).

% num.exhaust
tff(fact_5180_less__minus__one__simps_I2_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),one_one_real) ).

% less_minus_one_simps(2)
tff(fact_5181_less__minus__one__simps_I2_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) ).

% less_minus_one_simps(2)
tff(fact_5182_less__minus__one__simps_I2_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat) ).

% less_minus_one_simps(2)
tff(fact_5183_less__minus__one__simps_I2_J,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),one_one_Code_integer) ).

% less_minus_one_simps(2)
tff(fact_5184_less__minus__one__simps_I4_J,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(uminus_uminus_real,one_one_real)) ).

% less_minus_one_simps(4)
tff(fact_5185_less__minus__one__simps_I4_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int)) ).

% less_minus_one_simps(4)
tff(fact_5186_less__minus__one__simps_I4_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% less_minus_one_simps(4)
tff(fact_5187_less__minus__one__simps_I4_J,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% less_minus_one_simps(4)
tff(fact_5188_numeral__times__minus__swap,axiom,
    ! [W: num,X: real] : aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,W)),aa_real_real(uminus_uminus_real,X)) = aa_real_real(times_times_real(X),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) ).

% numeral_times_minus_swap
tff(fact_5189_numeral__times__minus__swap,axiom,
    ! [W: num,X: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,W)),aa_int_int(uminus_uminus_int,X)) = aa_int_int(aa_int_fun_int_int(times_times_int,X),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))) ).

% numeral_times_minus_swap
tff(fact_5190_numeral__times__minus__swap,axiom,
    ! [W: num,X: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),aa_complex_complex(uminus1482373934393186551omplex,X)) = aa_complex_complex(times_times_complex(X),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) ).

% numeral_times_minus_swap
tff(fact_5191_numeral__times__minus__swap,axiom,
    ! [W: num,X: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),aa_rat_rat(uminus_uminus_rat,X)) = aa_rat_rat(times_times_rat(X),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))) ).

% numeral_times_minus_swap
tff(fact_5192_numeral__times__minus__swap,axiom,
    ! [W: num,X: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(W)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(W))) ).

% numeral_times_minus_swap
tff(fact_5193_nonzero__minus__divide__divide,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) = aa_real_real(divide_divide_real(A),B) ) ) ).

% nonzero_minus_divide_divide
tff(fact_5194_nonzero__minus__divide__divide,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(divide1717551699836669952omplex(A),B) ) ) ).

% nonzero_minus_divide_divide
tff(fact_5195_nonzero__minus__divide__divide,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ).

% nonzero_minus_divide_divide
tff(fact_5196_nonzero__minus__divide__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(A),B)) = aa_real_real(divide_divide_real(A),aa_real_real(uminus_uminus_real,B)) ) ) ).

% nonzero_minus_divide_right
tff(fact_5197_nonzero__minus__divide__right,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) ) ) ).

% nonzero_minus_divide_right
tff(fact_5198_nonzero__minus__divide__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B)) = aa_rat_rat(divide_divide_rat(A),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% nonzero_minus_divide_right
tff(fact_5199_one__neq__neg__numeral,axiom,
    ! [Nb: num] : one_one_real != aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)) ).

% one_neq_neg_numeral
tff(fact_5200_one__neq__neg__numeral,axiom,
    ! [Nb: num] : one_one_int != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) ).

% one_neq_neg_numeral
tff(fact_5201_one__neq__neg__numeral,axiom,
    ! [Nb: num] : one_one_complex != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb)) ).

% one_neq_neg_numeral
tff(fact_5202_one__neq__neg__numeral,axiom,
    ! [Nb: num] : one_one_rat != aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)) ).

% one_neq_neg_numeral
tff(fact_5203_one__neq__neg__numeral,axiom,
    ! [Nb: num] : one_one_Code_integer != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb)) ).

% one_neq_neg_numeral
tff(fact_5204_numeral__neq__neg__one,axiom,
    ! [Nb: num] : aa_num_real(numeral_numeral_real,Nb) != aa_real_real(uminus_uminus_real,one_one_real) ).

% numeral_neq_neg_one
tff(fact_5205_numeral__neq__neg__one,axiom,
    ! [Nb: num] : aa_num_int(numeral_numeral_int,Nb) != aa_int_int(uminus_uminus_int,one_one_int) ).

% numeral_neq_neg_one
tff(fact_5206_numeral__neq__neg__one,axiom,
    ! [Nb: num] : numera6690914467698888265omplex(Nb) != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% numeral_neq_neg_one
tff(fact_5207_numeral__neq__neg__one,axiom,
    ! [Nb: num] : aa_num_rat(numeral_numeral_rat,Nb) != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% numeral_neq_neg_one
tff(fact_5208_numeral__neq__neg__one,axiom,
    ! [Nb: num] : numera6620942414471956472nteger(Nb) != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% numeral_neq_neg_one
tff(fact_5209_square__eq__1__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(times_times_real(X),X) = one_one_real )
    <=> ( ( X = one_one_real )
        | ( X = aa_real_real(uminus_uminus_real,one_one_real) ) ) ) ).

% square_eq_1_iff
tff(fact_5210_square__eq__1__iff,axiom,
    ! [X: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,X),X) = one_one_int )
    <=> ( ( X = one_one_int )
        | ( X = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% square_eq_1_iff
tff(fact_5211_square__eq__1__iff,axiom,
    ! [X: complex] :
      ( ( aa_complex_complex(times_times_complex(X),X) = one_one_complex )
    <=> ( ( X = one_one_complex )
        | ( X = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ) ).

% square_eq_1_iff
tff(fact_5212_square__eq__1__iff,axiom,
    ! [X: rat] :
      ( ( aa_rat_rat(times_times_rat(X),X) = one_one_rat )
    <=> ( ( X = one_one_rat )
        | ( X = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ) ).

% square_eq_1_iff
tff(fact_5213_square__eq__1__iff,axiom,
    ! [X: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),X) = one_one_Code_integer )
    <=> ( ( X = one_one_Code_integer )
        | ( X = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ) ).

% square_eq_1_iff
tff(fact_5214_group__cancel_Osub2,axiom,
    ! [B3: real,K2: real,B: real,A: real] :
      ( ( B3 = aa_real_real(plus_plus_real(K2),B) )
     => ( aa_real_real(minus_minus_real(A),B3) = aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,K2)),aa_real_real(minus_minus_real(A),B)) ) ) ).

% group_cancel.sub2
tff(fact_5215_group__cancel_Osub2,axiom,
    ! [B3: int,K2: int,B: int,A: int] :
      ( ( B3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),B) )
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B3) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,K2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) ) ) ).

% group_cancel.sub2
tff(fact_5216_group__cancel_Osub2,axiom,
    ! [B3: complex,K2: complex,B: complex,A: complex] :
      ( ( B3 = aa_complex_complex(plus_plus_complex(K2),B) )
     => ( aa_complex_complex(minus_minus_complex(A),B3) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,K2)),aa_complex_complex(minus_minus_complex(A),B)) ) ) ).

% group_cancel.sub2
tff(fact_5217_group__cancel_Osub2,axiom,
    ! [B3: rat,K2: rat,B: rat,A: rat] :
      ( ( B3 = aa_rat_rat(plus_plus_rat(K2),B) )
     => ( aa_rat_rat(minus_minus_rat(A),B3) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,K2)),aa_rat_rat(minus_minus_rat(A),B)) ) ) ).

% group_cancel.sub2
tff(fact_5218_group__cancel_Osub2,axiom,
    ! [B3: code_integer,K2: code_integer,B: code_integer,A: code_integer] :
      ( ( B3 = plus_p5714425477246183910nteger(K2,B) )
     => ( minus_8373710615458151222nteger(A,B3) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K2),minus_8373710615458151222nteger(A,B)) ) ) ).

% group_cancel.sub2
tff(fact_5219_diff__conv__add__uminus,axiom,
    ! [A: real,B: real] : aa_real_real(minus_minus_real(A),B) = aa_real_real(plus_plus_real(A),aa_real_real(uminus_uminus_real,B)) ).

% diff_conv_add_uminus
tff(fact_5220_diff__conv__add__uminus,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(uminus_uminus_int,B)) ).

% diff_conv_add_uminus
tff(fact_5221_diff__conv__add__uminus,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(minus_minus_complex(A),B) = aa_complex_complex(plus_plus_complex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) ).

% diff_conv_add_uminus
tff(fact_5222_diff__conv__add__uminus,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(uminus_uminus_rat,B)) ).

% diff_conv_add_uminus
tff(fact_5223_diff__conv__add__uminus,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,B) = plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ).

% diff_conv_add_uminus
tff(fact_5224_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: real,B: real] : aa_real_real(minus_minus_real(A),B) = aa_real_real(plus_plus_real(A),aa_real_real(uminus_uminus_real,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_5225_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(uminus_uminus_int,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_5226_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(minus_minus_complex(A),B) = aa_complex_complex(plus_plus_complex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_5227_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(uminus_uminus_rat,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_5228_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,B) = plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_5229_take__bit__unset__bit__eq,axiom,
    ! [Nb: nat,Ma: nat,A: int] :
      aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Ma),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma),aa_int_int(bit_se2923211474154528505it_int(Nb),A),aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Ma),aa_int_int(bit_se2923211474154528505it_int(Nb),A))) ).

% take_bit_unset_bit_eq
tff(fact_5230_take__bit__unset__bit__eq,axiom,
    ! [Nb: nat,Ma: nat,A: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,Ma),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A))) ).

% take_bit_unset_bit_eq
tff(fact_5231_take__bit__set__bit__eq,axiom,
    ! [Nb: nat,Ma: nat,A: int] :
      aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,Ma),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma),aa_int_int(bit_se2923211474154528505it_int(Nb),A),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,Ma),aa_int_int(bit_se2923211474154528505it_int(Nb),A))) ).

% take_bit_set_bit_eq
tff(fact_5232_take__bit__set__bit__eq,axiom,
    ! [Nb: nat,Ma: nat,A: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,Ma),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A))) ).

% take_bit_set_bit_eq
tff(fact_5233_take__bit__flip__bit__eq,axiom,
    ! [Nb: nat,Ma: nat,A: int] :
      aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(bit_se2159334234014336723it_int(Ma),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma),aa_int_int(bit_se2923211474154528505it_int(Nb),A),aa_int_int(bit_se2159334234014336723it_int(Ma),aa_int_int(bit_se2923211474154528505it_int(Nb),A))) ).

% take_bit_flip_bit_eq
tff(fact_5234_take__bit__flip__bit__eq,axiom,
    ! [Nb: nat,Ma: nat,A: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,Ma),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A))) ).

% take_bit_flip_bit_eq
tff(fact_5235_dvd__div__neg,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A)
     => ( aa_real_real(divide_divide_real(A),aa_real_real(uminus_uminus_real,B)) = aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(A),B)) ) ) ).

% dvd_div_neg
tff(fact_5236_dvd__div__neg,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% dvd_div_neg
tff(fact_5237_dvd__div__neg,axiom,
    ! [B: complex,A: complex] :
      ( aa_complex_o(dvd_dvd_complex(B),A)
     => ( aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(uminus1482373934393186551omplex,B)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) ) ) ).

% dvd_div_neg
tff(fact_5238_dvd__div__neg,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A)
     => ( aa_rat_rat(divide_divide_rat(A),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B)) ) ) ).

% dvd_div_neg
tff(fact_5239_dvd__div__neg,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
     => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,divide6298287555418463151nteger(A,B)) ) ) ).

% dvd_div_neg
tff(fact_5240_dvd__neg__div,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A)
     => ( aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,A)),B) = aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(A),B)) ) ) ).

% dvd_neg_div
tff(fact_5241_dvd__neg__div,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% dvd_neg_div
tff(fact_5242_dvd__neg__div,axiom,
    ! [B: complex,A: complex] :
      ( aa_complex_o(dvd_dvd_complex(B),A)
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(uminus1482373934393186551omplex,A)),B) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) ) ) ).

% dvd_neg_div
tff(fact_5243_dvd__neg__div,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A)
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(uminus_uminus_rat,A)),B) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B)) ) ) ).

% dvd_neg_div
tff(fact_5244_dvd__neg__div,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
     => ( divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,divide6298287555418463151nteger(A,B)) ) ) ).

% dvd_neg_div
tff(fact_5245_real__minus__mult__self__le,axiom,
    ! [U: real,X: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(times_times_real(U),U))),aa_real_real(times_times_real(X),X)) ).

% real_minus_mult_self_le
tff(fact_5246_zmult__eq__1__iff,axiom,
    ! [Ma: int,Nb: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Ma),Nb) = one_one_int )
    <=> ( ( ( Ma = one_one_int )
          & ( Nb = one_one_int ) )
        | ( ( Ma = aa_int_int(uminus_uminus_int,one_one_int) )
          & ( Nb = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ) ).

% zmult_eq_1_iff
tff(fact_5247_pos__zmult__eq__1__iff__lemma,axiom,
    ! [Ma: int,Nb: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Ma),Nb) = one_one_int )
     => ( ( Ma = one_one_int )
        | ( Ma = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% pos_zmult_eq_1_iff_lemma
tff(fact_5248_zmod__zminus2__not__zero,axiom,
    ! [K2: int,L: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),aa_int_int(uminus_uminus_int,L)) != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),L) != zero_zero_int ) ) ).

% zmod_zminus2_not_zero
tff(fact_5249_zmod__zminus1__not__zero,axiom,
    ! [K2: int,L: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,K2)),L) != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),L) != zero_zero_int ) ) ).

% zmod_zminus1_not_zero
tff(fact_5250_take__bit__Suc__minus__bit0,axiom,
    ! [Nb: nat,K2: num] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K2)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_Suc_minus_bit0
tff(fact_5251_take__bit__signed__take__bit,axiom,
    ! [Ma: nat,Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),aa_nat_nat(suc,Nb))
     => ( aa_int_int(bit_se2923211474154528505it_int(Ma),aa_int_int(bit_ri631733984087533419it_int(Nb),A)) = aa_int_int(bit_se2923211474154528505it_int(Ma),A) ) ) ).

% take_bit_signed_take_bit
tff(fact_5252_not__zero__le__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) ).

% not_zero_le_neg_numeral
tff(fact_5253_not__zero__le__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) ).

% not_zero_le_neg_numeral
tff(fact_5254_not__zero__le__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) ).

% not_zero_le_neg_numeral
tff(fact_5255_not__zero__le__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) ).

% not_zero_le_neg_numeral
tff(fact_5256_neg__numeral__le__zero,axiom,
    ! [Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))),zero_zero_real) ).

% neg_numeral_le_zero
tff(fact_5257_neg__numeral__le__zero,axiom,
    ! [Nb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))),zero_z3403309356797280102nteger) ).

% neg_numeral_le_zero
tff(fact_5258_neg__numeral__le__zero,axiom,
    ! [Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))),zero_zero_rat) ).

% neg_numeral_le_zero
tff(fact_5259_neg__numeral__le__zero,axiom,
    ! [Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))),zero_zero_int) ).

% neg_numeral_le_zero
tff(fact_5260_not__zero__less__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) ).

% not_zero_less_neg_numeral
tff(fact_5261_not__zero__less__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) ).

% not_zero_less_neg_numeral
tff(fact_5262_not__zero__less__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) ).

% not_zero_less_neg_numeral
tff(fact_5263_not__zero__less__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) ).

% not_zero_less_neg_numeral
tff(fact_5264_neg__numeral__less__zero,axiom,
    ! [Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))),zero_zero_real) ).

% neg_numeral_less_zero
tff(fact_5265_neg__numeral__less__zero,axiom,
    ! [Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))),zero_zero_int) ).

% neg_numeral_less_zero
tff(fact_5266_neg__numeral__less__zero,axiom,
    ! [Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))),zero_zero_rat) ).

% neg_numeral_less_zero
tff(fact_5267_neg__numeral__less__zero,axiom,
    ! [Nb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))),zero_z3403309356797280102nteger) ).

% neg_numeral_less_zero
tff(fact_5268_le__minus__one__simps_I1_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),zero_zero_real) ).

% le_minus_one_simps(1)
tff(fact_5269_le__minus__one__simps_I1_J,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),zero_z3403309356797280102nteger) ).

% le_minus_one_simps(1)
tff(fact_5270_le__minus__one__simps_I1_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),zero_zero_rat) ).

% le_minus_one_simps(1)
tff(fact_5271_le__minus__one__simps_I1_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),zero_zero_int) ).

% le_minus_one_simps(1)
tff(fact_5272_le__minus__one__simps_I3_J,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,one_one_real)) ).

% le_minus_one_simps(3)
tff(fact_5273_le__minus__one__simps_I3_J,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% le_minus_one_simps(3)
tff(fact_5274_le__minus__one__simps_I3_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% le_minus_one_simps(3)
tff(fact_5275_le__minus__one__simps_I3_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,one_one_int)) ).

% le_minus_one_simps(3)
tff(fact_5276_numeral__Bit1,axiom,
    ! [Nb: num] : numera6690914467698888265omplex(aa_num_num(bit1,Nb)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Nb)),numera6690914467698888265omplex(Nb))),one_one_complex) ).

% numeral_Bit1
tff(fact_5277_numeral__Bit1,axiom,
    ! [Nb: num] : aa_num_real(numeral_numeral_real,aa_num_num(bit1,Nb)) = aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(aa_num_real(numeral_numeral_real,Nb)),aa_num_real(numeral_numeral_real,Nb))),one_one_real) ).

% numeral_Bit1
tff(fact_5278_numeral__Bit1,axiom,
    ! [Nb: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,Nb)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,Nb)),aa_num_rat(numeral_numeral_rat,Nb))),one_one_rat) ).

% numeral_Bit1
tff(fact_5279_numeral__Bit1,axiom,
    ! [Nb: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,Nb))),one_one_nat) ).

% numeral_Bit1
tff(fact_5280_numeral__Bit1,axiom,
    ! [Nb: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,Nb))),one_one_int) ).

% numeral_Bit1
tff(fact_5281_take__bit__decr__eq,axiom,
    ! [Nb: nat,K2: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Nb),K2) != zero_zero_int )
     => ( aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(aa_int_fun_int_int(minus_minus_int,K2),one_one_int)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(bit_se2923211474154528505it_int(Nb),K2)),one_one_int) ) ) ).

% take_bit_decr_eq
tff(fact_5282_less__minus__one__simps_I3_J,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,one_one_real)) ).

% less_minus_one_simps(3)
tff(fact_5283_less__minus__one__simps_I3_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,one_one_int)) ).

% less_minus_one_simps(3)
tff(fact_5284_less__minus__one__simps_I3_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% less_minus_one_simps(3)
tff(fact_5285_less__minus__one__simps_I3_J,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% less_minus_one_simps(3)
tff(fact_5286_less__minus__one__simps_I1_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),zero_zero_real) ).

% less_minus_one_simps(1)
tff(fact_5287_less__minus__one__simps_I1_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),zero_zero_int) ).

% less_minus_one_simps(1)
tff(fact_5288_less__minus__one__simps_I1_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),zero_zero_rat) ).

% less_minus_one_simps(1)
tff(fact_5289_less__minus__one__simps_I1_J,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),zero_z3403309356797280102nteger) ).

% less_minus_one_simps(1)
tff(fact_5290_not__one__le__neg__numeral,axiom,
    ! [Ma: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Ma))) ).

% not_one_le_neg_numeral
tff(fact_5291_not__one__le__neg__numeral,axiom,
    ! [Ma: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Ma))) ).

% not_one_le_neg_numeral
tff(fact_5292_not__one__le__neg__numeral,axiom,
    ! [Ma: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Ma))) ).

% not_one_le_neg_numeral
tff(fact_5293_not__one__le__neg__numeral,axiom,
    ! [Ma: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Ma))) ).

% not_one_le_neg_numeral
tff(fact_5294_not__numeral__le__neg__one,axiom,
    ! [Ma: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Ma)),aa_real_real(uminus_uminus_real,one_one_real)) ).

% not_numeral_le_neg_one
tff(fact_5295_not__numeral__le__neg__one,axiom,
    ! [Ma: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(Ma)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% not_numeral_le_neg_one
tff(fact_5296_not__numeral__le__neg__one,axiom,
    ! [Ma: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Ma)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% not_numeral_le_neg_one
tff(fact_5297_not__numeral__le__neg__one,axiom,
    ! [Ma: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Ma)),aa_int_int(uminus_uminus_int,one_one_int)) ).

% not_numeral_le_neg_one
tff(fact_5298_neg__numeral__le__neg__one,axiom,
    ! [Ma: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Ma))),aa_real_real(uminus_uminus_real,one_one_real)) ).

% neg_numeral_le_neg_one
tff(fact_5299_neg__numeral__le__neg__one,axiom,
    ! [Ma: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Ma))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% neg_numeral_le_neg_one
tff(fact_5300_neg__numeral__le__neg__one,axiom,
    ! [Ma: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Ma))),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% neg_numeral_le_neg_one
tff(fact_5301_neg__numeral__le__neg__one,axiom,
    ! [Ma: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Ma))),aa_int_int(uminus_uminus_int,one_one_int)) ).

% neg_numeral_le_neg_one
tff(fact_5302_neg__one__le__numeral,axiom,
    ! [Ma: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_num_real(numeral_numeral_real,Ma)) ).

% neg_one_le_numeral
tff(fact_5303_neg__one__le__numeral,axiom,
    ! [Ma: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),numera6620942414471956472nteger(Ma)) ).

% neg_one_le_numeral
tff(fact_5304_neg__one__le__numeral,axiom,
    ! [Ma: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_num_rat(numeral_numeral_rat,Ma)) ).

% neg_one_le_numeral
tff(fact_5305_neg__one__le__numeral,axiom,
    ! [Ma: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,Ma)) ).

% neg_one_le_numeral
tff(fact_5306_neg__numeral__le__one,axiom,
    ! [Ma: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Ma))),one_one_real) ).

% neg_numeral_le_one
tff(fact_5307_neg__numeral__le__one,axiom,
    ! [Ma: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Ma))),one_one_Code_integer) ).

% neg_numeral_le_one
tff(fact_5308_neg__numeral__le__one,axiom,
    ! [Ma: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Ma))),one_one_rat) ).

% neg_numeral_le_one
tff(fact_5309_neg__numeral__le__one,axiom,
    ! [Ma: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Ma))),one_one_int) ).

% neg_numeral_le_one
tff(fact_5310_not__neg__one__less__neg__numeral,axiom,
    ! [Ma: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Ma))) ).

% not_neg_one_less_neg_numeral
tff(fact_5311_not__neg__one__less__neg__numeral,axiom,
    ! [Ma: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Ma))) ).

% not_neg_one_less_neg_numeral
tff(fact_5312_not__neg__one__less__neg__numeral,axiom,
    ! [Ma: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Ma))) ).

% not_neg_one_less_neg_numeral
tff(fact_5313_not__neg__one__less__neg__numeral,axiom,
    ! [Ma: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Ma))) ).

% not_neg_one_less_neg_numeral
tff(fact_5314_not__one__less__neg__numeral,axiom,
    ! [Ma: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Ma))) ).

% not_one_less_neg_numeral
tff(fact_5315_not__one__less__neg__numeral,axiom,
    ! [Ma: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Ma))) ).

% not_one_less_neg_numeral
tff(fact_5316_not__one__less__neg__numeral,axiom,
    ! [Ma: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Ma))) ).

% not_one_less_neg_numeral
tff(fact_5317_not__one__less__neg__numeral,axiom,
    ! [Ma: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Ma))) ).

% not_one_less_neg_numeral
tff(fact_5318_not__numeral__less__neg__one,axiom,
    ! [Ma: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,Ma)),aa_real_real(uminus_uminus_real,one_one_real)) ).

% not_numeral_less_neg_one
tff(fact_5319_not__numeral__less__neg__one,axiom,
    ! [Ma: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Ma)),aa_int_int(uminus_uminus_int,one_one_int)) ).

% not_numeral_less_neg_one
tff(fact_5320_not__numeral__less__neg__one,axiom,
    ! [Ma: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,Ma)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% not_numeral_less_neg_one
tff(fact_5321_not__numeral__less__neg__one,axiom,
    ! [Ma: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,numera6620942414471956472nteger(Ma)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% not_numeral_less_neg_one
tff(fact_5322_neg__one__less__numeral,axiom,
    ! [Ma: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_num_real(numeral_numeral_real,Ma)) ).

% neg_one_less_numeral
tff(fact_5323_neg__one__less__numeral,axiom,
    ! [Ma: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,Ma)) ).

% neg_one_less_numeral
tff(fact_5324_neg__one__less__numeral,axiom,
    ! [Ma: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_num_rat(numeral_numeral_rat,Ma)) ).

% neg_one_less_numeral
tff(fact_5325_neg__one__less__numeral,axiom,
    ! [Ma: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),numera6620942414471956472nteger(Ma)) ).

% neg_one_less_numeral
tff(fact_5326_neg__numeral__less__one,axiom,
    ! [Ma: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Ma))),one_one_real) ).

% neg_numeral_less_one
tff(fact_5327_neg__numeral__less__one,axiom,
    ! [Ma: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Ma))),one_one_int) ).

% neg_numeral_less_one
tff(fact_5328_neg__numeral__less__one,axiom,
    ! [Ma: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Ma))),one_one_rat) ).

% neg_numeral_less_one
tff(fact_5329_neg__numeral__less__one,axiom,
    ! [Ma: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Ma))),one_one_Code_integer) ).

% neg_numeral_less_one
tff(fact_5330_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: real,C: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( C = aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(A),B)) )
      <=> ( aa_real_real(times_times_real(C),B) = aa_real_real(uminus_uminus_real,A) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_5331_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: complex,C: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( C = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) )
      <=> ( aa_complex_complex(times_times_complex(C),B) = aa_complex_complex(uminus1482373934393186551omplex,A) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_5332_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( C = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B)) )
      <=> ( aa_rat_rat(times_times_rat(C),B) = aa_rat_rat(uminus_uminus_rat,A) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_5333_nonzero__neg__divide__eq__eq,axiom,
    ! [B: real,A: real,C: real] :
      ( ( B != zero_zero_real )
     => ( ( aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(A),B)) = C )
      <=> ( aa_real_real(uminus_uminus_real,A) = aa_real_real(times_times_real(C),B) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_5334_nonzero__neg__divide__eq__eq,axiom,
    ! [B: complex,A: complex,C: complex] :
      ( ( B != zero_zero_complex )
     => ( ( aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = C )
      <=> ( aa_complex_complex(uminus1482373934393186551omplex,A) = aa_complex_complex(times_times_complex(C),B) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_5335_nonzero__neg__divide__eq__eq,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( B != zero_zero_rat )
     => ( ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B)) = C )
      <=> ( aa_rat_rat(uminus_uminus_rat,A) = aa_rat_rat(times_times_rat(C),B) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_5336_minus__divide__eq__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C)) = A )
    <=> $ite(C != zero_zero_real,aa_real_real(uminus_uminus_real,B) = aa_real_real(times_times_real(A),C),A = zero_zero_real) ) ).

% minus_divide_eq_eq
tff(fact_5337_minus__divide__eq__eq,axiom,
    ! [B: complex,C: complex,A: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(B),C)) = A )
    <=> $ite(C != zero_zero_complex,aa_complex_complex(uminus1482373934393186551omplex,B) = aa_complex_complex(times_times_complex(A),C),A = zero_zero_complex) ) ).

% minus_divide_eq_eq
tff(fact_5338_minus__divide__eq__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C)) = A )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(uminus_uminus_rat,B) = aa_rat_rat(times_times_rat(A),C),A = zero_zero_rat) ) ).

% minus_divide_eq_eq
tff(fact_5339_eq__minus__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A = aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C)) )
    <=> $ite(C != zero_zero_real,aa_real_real(times_times_real(A),C) = aa_real_real(uminus_uminus_real,B),A = zero_zero_real) ) ).

% eq_minus_divide_eq
tff(fact_5340_eq__minus__divide__eq,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( A = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(B),C)) )
    <=> $ite(C != zero_zero_complex,aa_complex_complex(times_times_complex(A),C) = aa_complex_complex(uminus1482373934393186551omplex,B),A = zero_zero_complex) ) ).

% eq_minus_divide_eq
tff(fact_5341_eq__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C)) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(times_times_rat(A),C) = aa_rat_rat(uminus_uminus_rat,B),A = zero_zero_rat) ) ).

% eq_minus_divide_eq
tff(fact_5342_divide__eq__minus__1__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(divide_divide_real(A),B) = aa_real_real(uminus_uminus_real,one_one_real) )
    <=> ( ( B != zero_zero_real )
        & ( A = aa_real_real(uminus_uminus_real,B) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_5343_divide__eq__minus__1__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A),B) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) )
    <=> ( ( B != zero_zero_complex )
        & ( A = aa_complex_complex(uminus1482373934393186551omplex,B) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_5344_divide__eq__minus__1__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(divide_divide_rat(A),B) = aa_rat_rat(uminus_uminus_rat,one_one_rat) )
    <=> ( ( B != zero_zero_rat )
        & ( A = aa_rat_rat(uminus_uminus_rat,B) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_5345_mult__1s__ring__1_I1_J,axiom,
    ! [B: real] : aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,one))),B) = aa_real_real(uminus_uminus_real,B) ).

% mult_1s_ring_1(1)
tff(fact_5346_mult__1s__ring__1_I1_J,axiom,
    ! [B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,one))),B) = aa_int_int(uminus_uminus_int,B) ).

% mult_1s_ring_1(1)
tff(fact_5347_mult__1s__ring__1_I1_J,axiom,
    ! [B: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(one))),B) = aa_complex_complex(uminus1482373934393186551omplex,B) ).

% mult_1s_ring_1(1)
tff(fact_5348_mult__1s__ring__1_I1_J,axiom,
    ! [B: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,one))),B) = aa_rat_rat(uminus_uminus_rat,B) ).

% mult_1s_ring_1(1)
tff(fact_5349_mult__1s__ring__1_I1_J,axiom,
    ! [B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(one))),B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ).

% mult_1s_ring_1(1)
tff(fact_5350_mult__1s__ring__1_I2_J,axiom,
    ! [B: real] : aa_real_real(times_times_real(B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,one))) = aa_real_real(uminus_uminus_real,B) ).

% mult_1s_ring_1(2)
tff(fact_5351_mult__1s__ring__1_I2_J,axiom,
    ! [B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,one))) = aa_int_int(uminus_uminus_int,B) ).

% mult_1s_ring_1(2)
tff(fact_5352_mult__1s__ring__1_I2_J,axiom,
    ! [B: complex] : aa_complex_complex(times_times_complex(B),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(one))) = aa_complex_complex(uminus1482373934393186551omplex,B) ).

% mult_1s_ring_1(2)
tff(fact_5353_mult__1s__ring__1_I2_J,axiom,
    ! [B: rat] : aa_rat_rat(times_times_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,one))) = aa_rat_rat(uminus_uminus_rat,B) ).

% mult_1s_ring_1(2)
tff(fact_5354_mult__1s__ring__1_I2_J,axiom,
    ! [B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(one))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ).

% mult_1s_ring_1(2)
tff(fact_5355_uminus__numeral__One,axiom,
    aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,one)) = aa_real_real(uminus_uminus_real,one_one_real) ).

% uminus_numeral_One
tff(fact_5356_uminus__numeral__One,axiom,
    aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,one)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% uminus_numeral_One
tff(fact_5357_uminus__numeral__One,axiom,
    aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(one)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% uminus_numeral_One
tff(fact_5358_uminus__numeral__One,axiom,
    aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,one)) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% uminus_numeral_One
tff(fact_5359_uminus__numeral__One,axiom,
    aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(one)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% uminus_numeral_One
tff(fact_5360_eval__nat__numeral_I3_J,axiom,
    ! [Nb: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb)) = aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb))) ).

% eval_nat_numeral(3)
tff(fact_5361_power__minus,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),Nb) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ).

% power_minus
tff(fact_5362_power__minus,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,A)),Nb) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)) ).

% power_minus
tff(fact_5363_power__minus,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,A)),Nb) = aa_complex_complex(times_times_complex(aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),Nb)) ).

% power_minus
tff(fact_5364_power__minus,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),Nb) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb)),aa_nat_rat(power_power_rat(A),Nb)) ).

% power_minus
tff(fact_5365_power__minus,axiom,
    ! [A: code_integer,Nb: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),Nb) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)) ).

% power_minus
tff(fact_5366_cong__exp__iff__simps_I10_J,axiom,
    ! [Ma: num,Q: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Ma))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q))) ).

% cong_exp_iff_simps(10)
tff(fact_5367_cong__exp__iff__simps_I10_J,axiom,
    ! [Ma: num,Q: num,Nb: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Ma))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q))) != aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q))) ).

% cong_exp_iff_simps(10)
tff(fact_5368_cong__exp__iff__simps_I10_J,axiom,
    ! [Ma: num,Q: num,Nb: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,Ma)),numera6620942414471956472nteger(aa_num_num(bit0,Q))) != modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,Nb)),numera6620942414471956472nteger(aa_num_num(bit0,Q))) ).

% cong_exp_iff_simps(10)
tff(fact_5369_cong__exp__iff__simps_I12_J,axiom,
    ! [Ma: num,Q: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Ma))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q))) ).

% cong_exp_iff_simps(12)
tff(fact_5370_cong__exp__iff__simps_I12_J,axiom,
    ! [Ma: num,Q: num,Nb: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Ma))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q))) != aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q))) ).

% cong_exp_iff_simps(12)
tff(fact_5371_cong__exp__iff__simps_I12_J,axiom,
    ! [Ma: num,Q: num,Nb: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,Ma)),numera6620942414471956472nteger(aa_num_num(bit0,Q))) != modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,Nb)),numera6620942414471956472nteger(aa_num_num(bit0,Q))) ).

% cong_exp_iff_simps(12)
tff(fact_5372_cong__exp__iff__simps_I13_J,axiom,
    ! [Ma: num,Q: num,Nb: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Ma))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Ma)),aa_num_nat(numeral_numeral_nat,Q)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,Q)) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_5373_cong__exp__iff__simps_I13_J,axiom,
    ! [Ma: num,Q: num,Nb: num] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Ma))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q))) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q))) )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Ma)),aa_num_int(numeral_numeral_int,Q)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,Q)) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_5374_cong__exp__iff__simps_I13_J,axiom,
    ! [Ma: num,Q: num,Nb: num] :
      ( ( modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,Ma)),numera6620942414471956472nteger(aa_num_num(bit0,Q))) = modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,Nb)),numera6620942414471956472nteger(aa_num_num(bit0,Q))) )
    <=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(Ma),numera6620942414471956472nteger(Q)) = modulo364778990260209775nteger(numera6620942414471956472nteger(Nb),numera6620942414471956472nteger(Q)) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_5375_power__minus__Bit0,axiom,
    ! [X: real,K2: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K2))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K2))) ).

% power_minus_Bit0
tff(fact_5376_power__minus__Bit0,axiom,
    ! [X: int,K2: num] : aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K2))) = aa_nat_int(aa_int_fun_nat_int(power_power_int,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K2))) ).

% power_minus_Bit0
tff(fact_5377_power__minus__Bit0,axiom,
    ! [X: complex,K2: num] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K2))) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K2))) ).

% power_minus_Bit0
tff(fact_5378_power__minus__Bit0,axiom,
    ! [X: rat,K2: num] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K2))) = aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K2))) ).

% power_minus_Bit0
tff(fact_5379_power__minus__Bit0,axiom,
    ! [X: code_integer,K2: num] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K2))) = aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K2))) ).

% power_minus_Bit0
tff(fact_5380_take__bit__Suc__bit1,axiom,
    ! [Nb: nat,K2: num] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(Nb),aa_num_int(numeral_numeral_int,K2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% take_bit_Suc_bit1
tff(fact_5381_take__bit__Suc__bit1,axiom,
    ! [Nb: nat,K2: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_num_nat(numeral_numeral_nat,K2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat) ).

% take_bit_Suc_bit1
tff(fact_5382_take__bit__Suc__minus__1__eq,axiom,
    ! [Nb: nat] : bit_se1745604003318907178nteger(aa_nat_nat(suc,Nb),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(suc,Nb)),one_one_Code_integer) ).

% take_bit_Suc_minus_1_eq
tff(fact_5383_take__bit__Suc__minus__1__eq,axiom,
    ! [Nb: nat] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,Nb))),one_one_int) ).

% take_bit_Suc_minus_1_eq
tff(fact_5384_take__bit__numeral__minus__1__eq,axiom,
    ! [K2: num] : bit_se1745604003318907178nteger(aa_num_nat(numeral_numeral_nat,K2),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,K2)),one_one_Code_integer) ).

% take_bit_numeral_minus_1_eq
tff(fact_5385_take__bit__numeral__minus__1__eq,axiom,
    ! [K2: num] : aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,K2)),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,K2))),one_one_int) ).

% take_bit_numeral_minus_1_eq
tff(fact_5386_real__0__less__add__iff,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(plus_plus_real(X),Y2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,X)),Y2) ) ).

% real_0_less_add_iff
tff(fact_5387_real__add__less__0__iff,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(X),Y2)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),aa_real_real(uminus_uminus_real,X)) ) ).

% real_add_less_0_iff
tff(fact_5388_real__add__le__0__iff,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(X),Y2)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),aa_real_real(uminus_uminus_real,X)) ) ).

% real_add_le_0_iff
tff(fact_5389_real__0__le__add__iff,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(plus_plus_real(X),Y2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,X)),Y2) ) ).

% real_0_le_add_iff
tff(fact_5390_zmod__zminus2__eq__if,axiom,
    ! [A: int,B: int] :
      aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(uminus_uminus_int,B)) = $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int,zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),B)) ).

% zmod_zminus2_eq_if
tff(fact_5391_zmod__zminus1__eq__if,axiom,
    ! [A: int,B: int] :
      aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A)),B) = $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int,zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))) ).

% zmod_zminus1_eq_if
tff(fact_5392_numeral__code_I3_J,axiom,
    ! [Nb: num] :
      numera6690914467698888265omplex(aa_num_num(bit1,Nb)) = $let(
        m2: complex,
        m2:= numera6690914467698888265omplex(Nb),
        aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(m2),m2)),one_one_complex) ) ).

% numeral_code(3)
tff(fact_5393_numeral__code_I3_J,axiom,
    ! [Nb: num] :
      aa_num_real(numeral_numeral_real,aa_num_num(bit1,Nb)) = $let(
        m2: real,
        m2:= aa_num_real(numeral_numeral_real,Nb),
        aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(m2),m2)),one_one_real) ) ).

% numeral_code(3)
tff(fact_5394_numeral__code_I3_J,axiom,
    ! [Nb: num] :
      aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,Nb)) = $let(
        m2: rat,
        m2:= aa_num_rat(numeral_numeral_rat,Nb),
        aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(m2),m2)),one_one_rat) ) ).

% numeral_code(3)
tff(fact_5395_numeral__code_I3_J,axiom,
    ! [Nb: num] :
      aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb)) = $let(
        m2: nat,
        m2:= aa_num_nat(numeral_numeral_nat,Nb),
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,m2),m2)),one_one_nat) ) ).

% numeral_code(3)
tff(fact_5396_numeral__code_I3_J,axiom,
    ! [Nb: num] :
      aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)) = $let(
        m2: int,
        m2:= aa_num_int(numeral_numeral_int,Nb),
        aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,m2),m2)),one_one_int) ) ).

% numeral_code(3)
tff(fact_5397_power__numeral__odd,axiom,
    ! [Z2: complex,W: num] :
      aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Z2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = $let(
        w: complex,
        w:= aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Z2),aa_num_nat(numeral_numeral_nat,W)),
        aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(Z2),w)),w) ) ).

% power_numeral_odd
tff(fact_5398_power__numeral__odd,axiom,
    ! [Z2: real,W: num] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,Z2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = $let(
        w: real,
        w:= aa_nat_real(aa_real_fun_nat_real(power_power_real,Z2),aa_num_nat(numeral_numeral_nat,W)),
        aa_real_real(times_times_real(aa_real_real(times_times_real(Z2),w)),w) ) ).

% power_numeral_odd
tff(fact_5399_power__numeral__odd,axiom,
    ! [Z2: rat,W: num] :
      aa_nat_rat(power_power_rat(Z2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = $let(
        w: rat,
        w:= aa_nat_rat(power_power_rat(Z2),aa_num_nat(numeral_numeral_nat,W)),
        aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(Z2),w)),w) ) ).

% power_numeral_odd
tff(fact_5400_power__numeral__odd,axiom,
    ! [Z2: nat,W: num] :
      aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = $let(
        w: nat,
        w:= aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z2),aa_num_nat(numeral_numeral_nat,W)),
        aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Z2),w)),w) ) ).

% power_numeral_odd
tff(fact_5401_power__numeral__odd,axiom,
    ! [Z2: int,W: num] :
      aa_nat_int(aa_int_fun_nat_int(power_power_int,Z2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = $let(
        w: int,
        w:= aa_nat_int(aa_int_fun_nat_int(power_power_int,Z2),aa_num_nat(numeral_numeral_nat,W)),
        aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z2),w)),w) ) ).

% power_numeral_odd
tff(fact_5402_take__bit__minus__small__eq,axiom,
    ! [K2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))
       => ( aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(uminus_uminus_int,K2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),K2) ) ) ) ).

% take_bit_minus_small_eq
tff(fact_5403_numeral__Bit1__div__2,axiom,
    ! [Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_num_nat(numeral_numeral_nat,Nb) ).

% numeral_Bit1_div_2
tff(fact_5404_numeral__Bit1__div__2,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_num_int(numeral_numeral_int,Nb) ).

% numeral_Bit1_div_2
tff(fact_5405_pos__minus__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C))),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(times_times_real(A),C)) ) ) ).

% pos_minus_divide_less_eq
tff(fact_5406_pos__minus__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C)) ) ) ).

% pos_minus_divide_less_eq
tff(fact_5407_pos__less__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),C)),aa_real_real(uminus_uminus_real,B)) ) ) ).

% pos_less_minus_divide_eq
tff(fact_5408_pos__less__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% pos_less_minus_divide_eq
tff(fact_5409_neg__minus__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C))),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),C)),aa_real_real(uminus_uminus_real,B)) ) ) ).

% neg_minus_divide_less_eq
tff(fact_5410_neg__minus__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% neg_minus_divide_less_eq
tff(fact_5411_neg__less__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(times_times_real(A),C)) ) ) ).

% neg_less_minus_divide_eq
tff(fact_5412_neg__less__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C)) ) ) ).

% neg_less_minus_divide_eq
tff(fact_5413_minus__divide__less__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C))),A)
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(times_times_real(A),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),C)),aa_real_real(uminus_uminus_real,B)),aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)) ) ) ).

% minus_divide_less_eq
tff(fact_5414_minus__divide__less__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))),A)
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(uminus_uminus_rat,B)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)) ) ) ).

% minus_divide_less_eq
tff(fact_5415_less__minus__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C)))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),C)),aa_real_real(uminus_uminus_real,B)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(times_times_real(A),C)),aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)) ) ) ).

% less_minus_divide_eq
tff(fact_5416_less__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(uminus_uminus_rat,B)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)) ) ) ).

% less_minus_divide_eq
tff(fact_5417_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: real,C: real] :
      ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) = aa_real_real(divide_divide_real(B),C) )
    <=> $ite(C != zero_zero_real,aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C) = B,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) = zero_zero_real) ) ).

% eq_divide_eq_numeral(2)
tff(fact_5418_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: complex,C: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) = aa_complex_complex(divide1717551699836669952omplex(B),C) )
    <=> $ite(C != zero_zero_complex,aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),C) = B,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) = zero_zero_complex) ) ).

% eq_divide_eq_numeral(2)
tff(fact_5419_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) = aa_rat_rat(divide_divide_rat(B),C) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C) = B,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) = zero_zero_rat) ) ).

% eq_divide_eq_numeral(2)
tff(fact_5420_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W: num] :
      ( ( aa_real_real(divide_divide_real(B),C) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) )
    <=> $ite(C != zero_zero_real,B = aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) = zero_zero_real) ) ).

% divide_eq_eq_numeral(2)
tff(fact_5421_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: complex,C: complex,W: num] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B),C) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) )
    <=> $ite(C != zero_zero_complex,B = aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),C),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) = zero_zero_complex) ) ).

% divide_eq_eq_numeral(2)
tff(fact_5422_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( aa_rat_rat(divide_divide_rat(B),C) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) )
    <=> $ite(C != zero_zero_rat,B = aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) = zero_zero_rat) ) ).

% divide_eq_eq_numeral(2)
tff(fact_5423_odd__numeral,axiom,
    ! [Nb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),numera6620942414471956472nteger(aa_num_num(bit1,Nb))) ).

% odd_numeral
tff(fact_5424_odd__numeral,axiom,
    ! [Nb: num] : ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb))) ).

% odd_numeral
tff(fact_5425_odd__numeral,axiom,
    ! [Nb: num] : ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))) ).

% odd_numeral
tff(fact_5426_minus__divide__add__eq__iff,axiom,
    ! [Z2: real,X: real,Y2: real] :
      ( ( Z2 != zero_zero_real )
     => ( aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(X),Z2))),Y2) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,X)),aa_real_real(times_times_real(Y2),Z2))),Z2) ) ) ).

% minus_divide_add_eq_iff
tff(fact_5427_minus__divide__add__eq__iff,axiom,
    ! [Z2: complex,X: complex,Y2: complex] :
      ( ( Z2 != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(X),Z2))),Y2) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,X)),aa_complex_complex(times_times_complex(Y2),Z2))),Z2) ) ) ).

% minus_divide_add_eq_iff
tff(fact_5428_minus__divide__add__eq__iff,axiom,
    ! [Z2: rat,X: rat,Y2: rat] :
      ( ( Z2 != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(X),Z2))),Y2) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,X)),aa_rat_rat(times_times_rat(Y2),Z2))),Z2) ) ) ).

% minus_divide_add_eq_iff
tff(fact_5429_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: real,Z2: real,B: real] :
      aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(A),Z2))),B) = $ite(Z2 = zero_zero_real,B,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,A)),aa_real_real(times_times_real(B),Z2))),Z2)) ).

% add_divide_eq_if_simps(3)
tff(fact_5430_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: complex,Z2: complex,B: complex] :
      aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),Z2))),B) = $ite(Z2 = zero_zero_complex,B,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(times_times_complex(B),Z2))),Z2)) ).

% add_divide_eq_if_simps(3)
tff(fact_5431_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: rat,Z2: rat,B: rat] :
      aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),Z2))),B) = $ite(Z2 = zero_zero_rat,B,aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(times_times_rat(B),Z2))),Z2)) ).

% add_divide_eq_if_simps(3)
tff(fact_5432_cong__exp__iff__simps_I3_J,axiom,
    ! [Nb: num,Q: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q))) != zero_zero_nat ).

% cong_exp_iff_simps(3)
tff(fact_5433_cong__exp__iff__simps_I3_J,axiom,
    ! [Nb: num,Q: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q))) != zero_zero_int ).

% cong_exp_iff_simps(3)
tff(fact_5434_cong__exp__iff__simps_I3_J,axiom,
    ! [Nb: num,Q: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,Nb)),numera6620942414471956472nteger(aa_num_num(bit0,Q))) != zero_z3403309356797280102nteger ).

% cong_exp_iff_simps(3)
tff(fact_5435_minus__divide__diff__eq__iff,axiom,
    ! [Z2: real,X: real,Y2: real] :
      ( ( Z2 != zero_zero_real )
     => ( aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(X),Z2))),Y2) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,X)),aa_real_real(times_times_real(Y2),Z2))),Z2) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_5436_minus__divide__diff__eq__iff,axiom,
    ! [Z2: complex,X: complex,Y2: complex] :
      ( ( Z2 != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(X),Z2))),Y2) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,X)),aa_complex_complex(times_times_complex(Y2),Z2))),Z2) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_5437_minus__divide__diff__eq__iff,axiom,
    ! [Z2: rat,X: rat,Y2: rat] :
      ( ( Z2 != zero_zero_rat )
     => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(X),Z2))),Y2) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,X)),aa_rat_rat(times_times_rat(Y2),Z2))),Z2) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_5438_add__divide__eq__if__simps_I5_J,axiom,
    ! [A: real,Z2: real,B: real] :
      aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(A),Z2)),B) = $ite(Z2 = zero_zero_real,aa_real_real(uminus_uminus_real,B),aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(A),aa_real_real(times_times_real(B),Z2))),Z2)) ).

% add_divide_eq_if_simps(5)
tff(fact_5439_add__divide__eq__if__simps_I5_J,axiom,
    ! [A: complex,Z2: complex,B: complex] :
      aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(A),Z2)),B) = $ite(Z2 = zero_zero_complex,aa_complex_complex(uminus1482373934393186551omplex,B),aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(A),aa_complex_complex(times_times_complex(B),Z2))),Z2)) ).

% add_divide_eq_if_simps(5)
tff(fact_5440_add__divide__eq__if__simps_I5_J,axiom,
    ! [A: rat,Z2: rat,B: rat] :
      aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(A),Z2)),B) = $ite(Z2 = zero_zero_rat,aa_rat_rat(uminus_uminus_rat,B),aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(A),aa_rat_rat(times_times_rat(B),Z2))),Z2)) ).

% add_divide_eq_if_simps(5)
tff(fact_5441_add__divide__eq__if__simps_I6_J,axiom,
    ! [A: real,Z2: real,B: real] :
      aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(A),Z2))),B) = $ite(Z2 = zero_zero_real,aa_real_real(uminus_uminus_real,B),aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,A)),aa_real_real(times_times_real(B),Z2))),Z2)) ).

% add_divide_eq_if_simps(6)
tff(fact_5442_add__divide__eq__if__simps_I6_J,axiom,
    ! [A: complex,Z2: complex,B: complex] :
      aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),Z2))),B) = $ite(Z2 = zero_zero_complex,aa_complex_complex(uminus1482373934393186551omplex,B),aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(times_times_complex(B),Z2))),Z2)) ).

% add_divide_eq_if_simps(6)
tff(fact_5443_add__divide__eq__if__simps_I6_J,axiom,
    ! [A: rat,Z2: rat,B: rat] :
      aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),Z2))),B) = $ite(Z2 = zero_zero_rat,aa_rat_rat(uminus_uminus_rat,B),aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(times_times_rat(B),Z2))),Z2)) ).

% add_divide_eq_if_simps(6)
tff(fact_5444_power3__eq__cube,axiom,
    ! [A: complex] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(A),A)),A) ).

% power3_eq_cube
tff(fact_5445_power3__eq__cube,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_real_real(times_times_real(aa_real_real(times_times_real(A),A)),A) ).

% power3_eq_cube
tff(fact_5446_power3__eq__cube,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(A),A)),A) ).

% power3_eq_cube
tff(fact_5447_power3__eq__cube,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),A)),A) ).

% power3_eq_cube
tff(fact_5448_power3__eq__cube,axiom,
    ! [A: int] : aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),A)),A) ).

% power3_eq_cube
tff(fact_5449_even__minus,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) ) ).

% even_minus
tff(fact_5450_even__minus,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A) ) ).

% even_minus
tff(fact_5451_numeral__3__eq__3,axiom,
    aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) = aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat))) ).

% numeral_3_eq_3
tff(fact_5452_power2__eq__iff,axiom,
    ! [X: real,Y2: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
    <=> ( ( X = Y2 )
        | ( X = aa_real_real(uminus_uminus_real,Y2) ) ) ) ).

% power2_eq_iff
tff(fact_5453_power2__eq__iff,axiom,
    ! [X: int,Y2: int] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
    <=> ( ( X = Y2 )
        | ( X = aa_int_int(uminus_uminus_int,Y2) ) ) ) ).

% power2_eq_iff
tff(fact_5454_power2__eq__iff,axiom,
    ! [X: complex,Y2: complex] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
    <=> ( ( X = Y2 )
        | ( X = aa_complex_complex(uminus1482373934393186551omplex,Y2) ) ) ) ).

% power2_eq_iff
tff(fact_5455_power2__eq__iff,axiom,
    ! [X: rat,Y2: rat] :
      ( ( aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
    <=> ( ( X = Y2 )
        | ( X = aa_rat_rat(uminus_uminus_rat,Y2) ) ) ) ).

% power2_eq_iff
tff(fact_5456_power2__eq__iff,axiom,
    ! [X: code_integer,Y2: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_Code_integer(power_8256067586552552935nteger(Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
    <=> ( ( X = Y2 )
        | ( X = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Y2) ) ) ) ).

% power2_eq_iff
tff(fact_5457_Suc3__eq__add__3,axiom,
    ! [Nb: nat] : aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Nb) ).

% Suc3_eq_add_3
tff(fact_5458_verit__less__mono__div__int2,axiom,
    ! [A2: int,B3: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,Nb))
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B3),Nb)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),Nb)) ) ) ).

% verit_less_mono_div_int2
tff(fact_5459_div__eq__minus1,axiom,
    ! [B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),B) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ).

% div_eq_minus1
tff(fact_5460_take__bit__Suc__bit0,axiom,
    ! [Nb: nat,K2: num] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,K2))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(Nb),aa_num_int(numeral_numeral_int,K2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_Suc_bit0
tff(fact_5461_take__bit__Suc__bit0,axiom,
    ! [Nb: nat,K2: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K2))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_num_nat(numeral_numeral_nat,K2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% take_bit_Suc_bit0
tff(fact_5462_take__bit__eq__mod,axiom,
    ! [Nb: nat,A: code_integer] : bit_se1745604003318907178nteger(Nb,A) = modulo364778990260209775nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)) ).

% take_bit_eq_mod
tff(fact_5463_take__bit__eq__mod,axiom,
    ! [Nb: nat,A: int] : aa_int_int(bit_se2923211474154528505it_int(Nb),A) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% take_bit_eq_mod
tff(fact_5464_take__bit__eq__mod,axiom,
    ! [Nb: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% take_bit_eq_mod
tff(fact_5465_mult__commute__abs,axiom,
    ! [C: real] : aTP_Lamm_ca(C) = times_times_real(C) ).

% mult_commute_abs
tff(fact_5466_mult__commute__abs,axiom,
    ! [C: rat] : aTP_Lamm_cb(C) = times_times_rat(C) ).

% mult_commute_abs
tff(fact_5467_mult__commute__abs,axiom,
    ! [C: nat] : aa_nat_fun_nat_nat(aTP_Lamm_cc,C) = aa_nat_fun_nat_nat(times_times_nat,C) ).

% mult_commute_abs
tff(fact_5468_mult__commute__abs,axiom,
    ! [C: int] : aa_int_fun_int_int(aTP_Lamm_cd,C) = aa_int_fun_int_int(times_times_int,C) ).

% mult_commute_abs
tff(fact_5469_take__bit__nat__eq__self__iff,axiom,
    ! [Nb: nat,Ma: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Ma) = Ma )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ) ).

% take_bit_nat_eq_self_iff
tff(fact_5470_take__bit__nat__less__exp,axiom,
    ! [Nb: nat,Ma: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% take_bit_nat_less_exp
tff(fact_5471_take__bit__nat__eq__self,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Ma) = Ma ) ) ).

% take_bit_nat_eq_self
tff(fact_5472_num_Osize_I6_J,axiom,
    ! [X32: num] : aa_num_nat(size_size_num,aa_num_num(bit1,X32)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_size_num,X32)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size(6)
tff(fact_5473_take__bit__nat__def,axiom,
    ! [Nb: nat,Ma: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Ma) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% take_bit_nat_def
tff(fact_5474_take__bit__int__less__exp,axiom,
    ! [Nb: nat,K2: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se2923211474154528505it_int(Nb),K2)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% take_bit_int_less_exp
tff(fact_5475_pos__minus__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C))),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(times_times_real(A),C)) ) ) ).

% pos_minus_divide_le_eq
tff(fact_5476_pos__minus__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C)) ) ) ).

% pos_minus_divide_le_eq
tff(fact_5477_pos__le__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),C)),aa_real_real(uminus_uminus_real,B)) ) ) ).

% pos_le_minus_divide_eq
tff(fact_5478_pos__le__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% pos_le_minus_divide_eq
tff(fact_5479_neg__minus__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C))),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),C)),aa_real_real(uminus_uminus_real,B)) ) ) ).

% neg_minus_divide_le_eq
tff(fact_5480_neg__minus__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% neg_minus_divide_le_eq
tff(fact_5481_neg__le__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(times_times_real(A),C)) ) ) ).

% neg_le_minus_divide_eq
tff(fact_5482_neg__le__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C)) ) ) ).

% neg_le_minus_divide_eq
tff(fact_5483_minus__divide__le__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C))),A)
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(times_times_real(A),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),C)),aa_real_real(uminus_uminus_real,B)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)) ) ) ).

% minus_divide_le_eq
tff(fact_5484_minus__divide__le__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))),A)
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(uminus_uminus_rat,B)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)) ) ) ).

% minus_divide_le_eq
tff(fact_5485_le__minus__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C)))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),C)),aa_real_real(uminus_uminus_real,B)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(times_times_real(A),C)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)) ) ) ).

% le_minus_divide_eq
tff(fact_5486_le__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(uminus_uminus_rat,B)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)) ) ) ).

% le_minus_divide_eq
tff(fact_5487_less__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),aa_real_real(divide_divide_real(B),C))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C)),B),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C)),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),zero_zero_real)) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_5488_less__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),aa_rat_rat(divide_divide_rat(B),C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),zero_zero_rat)) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_5489_divide__less__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(B),C)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_5490_divide__less__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B),C)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_5491_take__bit__int__def,axiom,
    ! [Nb: nat,K2: int] : aa_int_int(bit_se2923211474154528505it_int(Nb),K2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% take_bit_int_def
tff(fact_5492_cong__exp__iff__simps_I7_J,axiom,
    ! [Q: num,Nb: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,Q)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(7)
tff(fact_5493_cong__exp__iff__simps_I7_J,axiom,
    ! [Q: num,Nb: num] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q))) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q))) )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,Q)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(7)
tff(fact_5494_cong__exp__iff__simps_I7_J,axiom,
    ! [Q: num,Nb: num] :
      ( ( modulo364778990260209775nteger(numera6620942414471956472nteger(one),numera6620942414471956472nteger(aa_num_num(bit0,Q))) = modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,Nb)),numera6620942414471956472nteger(aa_num_num(bit0,Q))) )
    <=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(Nb),numera6620942414471956472nteger(Q)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(7)
tff(fact_5495_cong__exp__iff__simps_I11_J,axiom,
    ! [Ma: num,Q: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Ma))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Ma)),aa_num_nat(numeral_numeral_nat,Q)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(11)
tff(fact_5496_cong__exp__iff__simps_I11_J,axiom,
    ! [Ma: num,Q: num] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Ma))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q))) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q))) )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Ma)),aa_num_int(numeral_numeral_int,Q)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(11)
tff(fact_5497_cong__exp__iff__simps_I11_J,axiom,
    ! [Ma: num,Q: num] :
      ( ( modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,Ma)),numera6620942414471956472nteger(aa_num_num(bit0,Q))) = modulo364778990260209775nteger(numera6620942414471956472nteger(one),numera6620942414471956472nteger(aa_num_num(bit0,Q))) )
    <=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(Ma),numera6620942414471956472nteger(Q)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(11)
tff(fact_5498_power2__eq__1__iff,axiom,
    ! [A: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_real )
    <=> ( ( A = one_one_real )
        | ( A = aa_real_real(uminus_uminus_real,one_one_real) ) ) ) ).

% power2_eq_1_iff
tff(fact_5499_power2__eq__1__iff,axiom,
    ! [A: int] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_int )
    <=> ( ( A = one_one_int )
        | ( A = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% power2_eq_1_iff
tff(fact_5500_power2__eq__1__iff,axiom,
    ! [A: complex] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_complex )
    <=> ( ( A = one_one_complex )
        | ( A = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ) ).

% power2_eq_1_iff
tff(fact_5501_power2__eq__1__iff,axiom,
    ! [A: rat] :
      ( ( aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_rat )
    <=> ( ( A = one_one_rat )
        | ( A = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ) ).

% power2_eq_1_iff
tff(fact_5502_power2__eq__1__iff,axiom,
    ! [A: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_Code_integer )
    <=> ( ( A = one_one_Code_integer )
        | ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ) ).

% power2_eq_1_iff
tff(fact_5503_uminus__power__if,axiom,
    ! [A: real,Nb: nat] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb),aa_real_real(uminus_uminus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))) ).

% uminus_power_if
tff(fact_5504_uminus__power__if,axiom,
    ! [A: int,Nb: nat] :
      aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,A)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb),aa_int_int(uminus_uminus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb))) ).

% uminus_power_if
tff(fact_5505_uminus__power__if,axiom,
    ! [A: complex,Nb: nat] :
      aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,A)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),Nb),aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),Nb))) ).

% uminus_power_if
tff(fact_5506_uminus__power__if,axiom,
    ! [A: rat,Nb: nat] :
      aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),aa_nat_rat(power_power_rat(A),Nb),aa_rat_rat(uminus_uminus_rat,aa_nat_rat(power_power_rat(A),Nb))) ).

% uminus_power_if
tff(fact_5507_uminus__power__if,axiom,
    ! [A: code_integer,Nb: nat] :
      aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb))) ).

% uminus_power_if
tff(fact_5508_Suc__div__eq__add3__div,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Ma)))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Ma)),Nb) ).

% Suc_div_eq_add3_div
tff(fact_5509_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K2)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_5510_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K2)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_5511_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K2)) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_5512_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K2)) = aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_5513_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K2)) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_5514_Suc__mod__eq__add3__mod,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Ma)))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Ma)),Nb) ).

% Suc_mod_eq_add3_mod
tff(fact_5515_realpow__square__minus__le,axiom,
    ! [U: real,X: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,U),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% realpow_square_minus_le
tff(fact_5516_take__bit__eq__0__iff,axiom,
    ! [Nb: nat,A: code_integer] :
      ( ( bit_se1745604003318907178nteger(Nb,A) = zero_z3403309356797280102nteger )
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)),A) ) ).

% take_bit_eq_0_iff
tff(fact_5517_take__bit__eq__0__iff,axiom,
    ! [Nb: nat,A: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Nb),A) = zero_zero_int )
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),A) ) ).

% take_bit_eq_0_iff
tff(fact_5518_take__bit__eq__0__iff,axiom,
    ! [Nb: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A) = zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),A) ) ).

% take_bit_eq_0_iff
tff(fact_5519_minus__mod__int__eq,axiom,
    ! [L: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,K2)),L) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,L),one_one_int)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,K2),one_one_int)),L)) ) ) ).

% minus_mod_int_eq
tff(fact_5520_zmod__minus1,axiom,
    ! [B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,one_one_int)),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,B),one_one_int) ) ) ).

% zmod_minus1
tff(fact_5521_zdiv__zminus2__eq__if,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(uminus_uminus_int,B)) = $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))),one_one_int)) ) ) ).

% zdiv_zminus2_eq_if
tff(fact_5522_zdiv__zminus1__eq__if,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A)),B) = $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))),one_one_int)) ) ) ).

% zdiv_zminus1_eq_if
tff(fact_5523_take__bit__nat__less__self__iff,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Ma)),Ma)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),Ma) ) ).

% take_bit_nat_less_self_iff
tff(fact_5524_zminus1__lemma,axiom,
    ! [A: int,B: int,Q: int,R: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q),R))
     => ( ( B != zero_zero_int )
       => aa_Pro4592068669816069595_int_o(eucl_rel_int(aa_int_int(uminus_uminus_int,A),B),
            aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,
                $ite(R = zero_zero_int,aa_int_int(uminus_uminus_int,Q),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,Q)),one_one_int))),
              $ite(R = zero_zero_int,zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),R)))) ) ) ).

% zminus1_lemma
tff(fact_5525_take__bit__int__less__self__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se2923211474154528505it_int(Nb),K2)),K2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),K2) ) ).

% take_bit_int_less_self_iff
tff(fact_5526_take__bit__int__greater__eq__self__iff,axiom,
    ! [K2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),aa_int_int(bit_se2923211474154528505it_int(Nb),K2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K2),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ) ).

% take_bit_int_greater_eq_self_iff
tff(fact_5527_le__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),aa_real_real(divide_divide_real(B),C))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C)),B),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),zero_zero_real)) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_5528_le__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),aa_rat_rat(divide_divide_rat(B),C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C)),B),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),zero_zero_rat)) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_5529_divide__le__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(B),C)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_5530_divide__le__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B),C)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_5531_square__le__1,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real) ) ) ).

% square_le_1
tff(fact_5532_square__le__1,axiom,
    ! [X: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),X)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_Code_integer) ) ) ).

% square_le_1
tff(fact_5533_square__le__1,axiom,
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_rat) ) ) ).

% square_le_1
tff(fact_5534_square__le__1,axiom,
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_int) ) ) ).

% square_le_1
tff(fact_5535_minus__power__mult__self,axiom,
    ! [A: real,Nb: nat] : aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% minus_power_mult_self
tff(fact_5536_minus__power__mult__self,axiom,
    ! [A: int,Nb: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,A)),Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,A)),Nb)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% minus_power_mult_self
tff(fact_5537_minus__power__mult__self,axiom,
    ! [A: complex,Nb: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,A)),Nb)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,A)),Nb)) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% minus_power_mult_self
tff(fact_5538_minus__power__mult__self,axiom,
    ! [A: rat,Nb: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),Nb)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),Nb)) = aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% minus_power_mult_self
tff(fact_5539_minus__power__mult__self,axiom,
    ! [A: code_integer,Nb: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),Nb)) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% minus_power_mult_self
tff(fact_5540_minus__one__power__iff,axiom,
    ! [Nb: nat] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),one_one_real,aa_real_real(uminus_uminus_real,one_one_real)) ).

% minus_one_power_iff
tff(fact_5541_minus__one__power__iff,axiom,
    ! [Nb: nat] :
      aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),one_one_int,aa_int_int(uminus_uminus_int,one_one_int)) ).

% minus_one_power_iff
tff(fact_5542_minus__one__power__iff,axiom,
    ! [Nb: nat] :
      aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),one_one_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) ).

% minus_one_power_iff
tff(fact_5543_minus__one__power__iff,axiom,
    ! [Nb: nat] :
      aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),one_one_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% minus_one_power_iff
tff(fact_5544_minus__one__power__iff,axiom,
    ! [Nb: nat] :
      aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% minus_one_power_iff
tff(fact_5545_minus__1__div__exp__eq__int,axiom,
    ! [Nb: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% minus_1_div_exp_eq_int
tff(fact_5546_div__pos__neg__trivial,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),L)),zero_zero_int)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),L) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% div_pos_neg_trivial
tff(fact_5547_signed__take__bit__int__less__eq__self__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_ri631733984087533419it_int(Nb),K2)),K2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))),K2) ) ).

% signed_take_bit_int_less_eq_self_iff
tff(fact_5548_signed__take__bit__int__greater__eq__minus__exp,axiom,
    ! [Nb: nat,K2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))),aa_int_int(bit_ri631733984087533419it_int(Nb),K2)) ).

% signed_take_bit_int_greater_eq_minus_exp
tff(fact_5549_signed__take__bit__int__greater__self__iff,axiom,
    ! [K2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),aa_int_int(bit_ri631733984087533419it_int(Nb),K2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K2),aa_int_int(uminus_uminus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))) ) ).

% signed_take_bit_int_greater_self_iff
tff(fact_5550_take__bit__int__eq__self,axiom,
    ! [K2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))
       => ( aa_int_int(bit_se2923211474154528505it_int(Nb),K2) = K2 ) ) ) ).

% take_bit_int_eq_self
tff(fact_5551_take__bit__int__eq__self__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Nb),K2) = K2 )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K2),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ) ) ).

% take_bit_int_eq_self_iff
tff(fact_5552_take__bit__incr__eq,axiom,
    ! [Nb: nat,K2: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Nb),K2) != aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),one_one_int) )
     => ( aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),one_one_int)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(bit_se2923211474154528505it_int(Nb),K2)) ) ) ).

% take_bit_incr_eq
tff(fact_5553_power__minus1__odd,axiom,
    ! [Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_real_real(uminus_uminus_real,one_one_real) ).

% power_minus1_odd
tff(fact_5554_power__minus1__odd,axiom,
    ! [Nb: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_int_int(uminus_uminus_int,one_one_int) ).

% power_minus1_odd
tff(fact_5555_power__minus1__odd,axiom,
    ! [Nb: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% power_minus1_odd
tff(fact_5556_power__minus1__odd,axiom,
    ! [Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% power_minus1_odd
tff(fact_5557_power__minus1__odd,axiom,
    ! [Nb: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% power_minus1_odd
tff(fact_5558_take__bit__Suc,axiom,
    ! [Nb: nat,A: code_integer] : bit_se1745604003318907178nteger(aa_nat_nat(suc,Nb),A) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,bit_se1745604003318907178nteger(Nb,divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))))),numera6620942414471956472nteger(aa_num_num(bit0,one))),modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))) ).

% take_bit_Suc
tff(fact_5559_take__bit__Suc,axiom,
    ! [Nb: nat,A: int] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% take_bit_Suc
tff(fact_5560_take__bit__Suc,axiom,
    ! [Nb: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,Nb)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% take_bit_Suc
tff(fact_5561_int__bit__induct,axiom,
    ! [P: fun_int_o,K2: int] :
      ( aa_int_o(P,zero_zero_int)
     => ( aa_int_o(P,aa_int_int(uminus_uminus_int,one_one_int))
       => ( ! [K: int] :
              ( aa_int_o(P,K)
             => ( ( K != zero_zero_int )
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(times_times_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ) )
         => ( ! [K: int] :
                ( aa_int_o(P,K)
               => ( ( K != aa_int_int(uminus_uminus_int,one_one_int) )
                 => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ) )
           => aa_int_o(P,K2) ) ) ) ) ).

% int_bit_induct
tff(fact_5562_signed__take__bit__int__eq__self,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))),K2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))
       => ( aa_int_int(bit_ri631733984087533419it_int(Nb),K2) = K2 ) ) ) ).

% signed_take_bit_int_eq_self
tff(fact_5563_signed__take__bit__int__eq__self__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( ( aa_int_int(bit_ri631733984087533419it_int(Nb),K2) = K2 )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))),K2)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K2),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ) ) ).

% signed_take_bit_int_eq_self_iff
tff(fact_5564_take__bit__int__less__eq,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),K2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se2923211474154528505it_int(Nb),K2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,K2),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))) ) ) ).

% take_bit_int_less_eq
tff(fact_5565_take__bit__int__greater__eq,axiom,
    ! [K2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))),aa_int_int(bit_se2923211474154528505it_int(Nb),K2)) ) ).

% take_bit_int_greater_eq
tff(fact_5566_signed__take__bit__eq__take__bit__shift,axiom,
    ! [Nb: nat,K2: int] : aa_int_int(bit_ri631733984087533419it_int(Nb),K2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% signed_take_bit_eq_take_bit_shift
tff(fact_5567_stable__imp__take__bit__eq,axiom,
    ! [A: code_integer,Nb: nat] :
      ( ( divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = A )
     => ( bit_se1745604003318907178nteger(Nb,A) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A),zero_z3403309356797280102nteger,minus_8373710615458151222nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb),one_one_Code_integer)) ) ) ).

% stable_imp_take_bit_eq
tff(fact_5568_stable__imp__take__bit__eq,axiom,
    ! [A: int,Nb: nat] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A )
     => ( aa_int_int(bit_se2923211474154528505it_int(Nb),A) = $ite(aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A),zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),one_one_int)) ) ) ).

% stable_imp_take_bit_eq
tff(fact_5569_stable__imp__take__bit__eq,axiom,
    ! [A: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = A )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),one_one_nat)) ) ) ).

% stable_imp_take_bit_eq
tff(fact_5570_divmod__step__nat__def,axiom,
    ! [L: num,Qr: product_prod_nat_nat] : unique5026877609467782581ep_nat(L,Qr) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_br(L)),Qr) ).

% divmod_step_nat_def
tff(fact_5571_divmod__step__int__def,axiom,
    ! [L: num,Qr: product_prod_int_int] : aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(L),Qr) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_bs(L)),Qr) ).

% divmod_step_int_def
tff(fact_5572_odd__mod__4__div__2,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
     => ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% odd_mod_4_div_2
tff(fact_5573_signed__take__bit__int__greater__eq,axiom,
    ! [K2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),aa_int_int(uminus_uminus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,Nb)))),aa_int_int(bit_ri631733984087533419it_int(Nb),K2)) ) ).

% signed_take_bit_int_greater_eq
tff(fact_5574_mod__exhaust__less__4,axiom,
    ! [Ma: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = zero_zero_nat )
      | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = one_one_nat )
      | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
      | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ) ) ).

% mod_exhaust_less_4
tff(fact_5575_case__prod__conv,axiom,
    ! [F: fun_na1668100129766719005at_nat,A: nat,B: nat] : aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(F),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B)) = aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,A),B) ).

% case_prod_conv
tff(fact_5576_case__prod__conv,axiom,
    ! [F: fun_na5914473681949881652_nat_o,A: nat,B: nat] : aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(F),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B)) = aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(F,A),B) ).

% case_prod_conv
tff(fact_5577_case__prod__conv,axiom,
    ! [F: fun_in1696471567231710611nt_int,A: int,B: int] : aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,A),B) ).

% case_prod_conv
tff(fact_5578_case__prod__conv,axiom,
    ! [F: fun_int_fun_int_o,A: int,B: int] :
      ( aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(F,A),B) ) ).

% case_prod_conv
tff(fact_5579_case__prod__conv,axiom,
    ! [F: fun_int_fun_int_int,A: int,B: int] : aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)) = aa_int_int(aa_int_fun_int_int(F,A),B) ).

% case_prod_conv
tff(fact_5580_compl__le__compl__iff,axiom,
    ! [X: set_int,Y2: set_int] :
      ( ord_less_eq_set_int(uminus1532241313380277803et_int(X),uminus1532241313380277803et_int(Y2))
    <=> ord_less_eq_set_int(Y2,X) ) ).

% compl_le_compl_iff
tff(fact_5581_divmod__algorithm__code_I6_J,axiom,
    ! [Ma: num,Nb: num] : unique5052692396658037445od_int(aa_num_num(bit1,Ma),aa_num_num(bit0,Nb)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_ce),unique5052692396658037445od_int(Ma,Nb)) ).

% divmod_algorithm_code(6)
tff(fact_5582_divmod__algorithm__code_I6_J,axiom,
    ! [Ma: num,Nb: num] : unique5055182867167087721od_nat(aa_num_num(bit1,Ma),aa_num_num(bit0,Nb)) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_cf),unique5055182867167087721od_nat(Ma,Nb)) ).

% divmod_algorithm_code(6)
tff(fact_5583_divmod__algorithm__code_I6_J,axiom,
    ! [Ma: num,Nb: num] : unique3479559517661332726nteger(aa_num_num(bit1,Ma),aa_num_num(bit0,Nb)) = produc6916734918728496179nteger(aTP_Lamm_cg,unique3479559517661332726nteger(Ma,Nb)) ).

% divmod_algorithm_code(6)
tff(fact_5584_dbl__dec__simps_I4_J,axiom,
    aa_real_real(neg_nu6075765906172075777c_real,aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_5585_dbl__dec__simps_I4_J,axiom,
    aa_int_int(neg_nu3811975205180677377ec_int,aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_5586_dbl__dec__simps_I4_J,axiom,
    aa_complex_complex(neg_nu6511756317524482435omplex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_5587_dbl__dec__simps_I4_J,axiom,
    aa_rat_rat(neg_nu3179335615603231917ec_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_5588_dbl__dec__simps_I4_J,axiom,
    neg_nu7757733837767384882nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_5589_divmod__algorithm__code_I7_J,axiom,
    ! [Ma: num,Nb: num] :
      unique5055182867167087721od_nat(aa_num_num(bit0,Ma),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ma),Nb),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Ma))),unique5026877609467782581ep_nat(aa_num_num(bit1,Nb),unique5055182867167087721od_nat(aa_num_num(bit0,Ma),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(7)
tff(fact_5590_divmod__algorithm__code_I7_J,axiom,
    ! [Ma: num,Nb: num] :
      unique5052692396658037445od_int(aa_num_num(bit0,Ma),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ma),Nb),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Ma))),aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(aa_num_num(bit1,Nb)),unique5052692396658037445od_int(aa_num_num(bit0,Ma),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(7)
tff(fact_5591_divmod__algorithm__code_I7_J,axiom,
    ! [Ma: num,Nb: num] :
      unique3479559517661332726nteger(aa_num_num(bit0,Ma),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ma),Nb),produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(aa_num_num(bit0,Ma))),unique4921790084139445826nteger(aa_num_num(bit1,Nb),unique3479559517661332726nteger(aa_num_num(bit0,Ma),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(7)
tff(fact_5592_case__prodI2,axiom,
    ! [P2: produc6271795597528267376eger_o,C: fun_Co480733329897759773un_o_o] :
      ( ! [A3: code_integer,B2: $o] :
          ( ( P2 = aa_o_P4192039665577170892eger_o(aa_Cod3473573201699832512eger_o(produc6677183202524767010eger_o,A3),(B2)) )
         => aa_o_o(aa_Cod6379977994291289310un_o_o(C,A3),(B2)) )
     => aa_Pro5286387174248033258er_o_o(produc7828578312038201481er_o_o(C),P2) ) ).

% case_prodI2
tff(fact_5593_case__prodI2,axiom,
    ! [P2: product_prod_num_num,C: fun_num_fun_num_o] :
      ( ! [A3: num,B2: num] :
          ( ( P2 = aa_num3156681643174609981um_num(aa_num8681986212680877308um_num(product_Pair_num_num,A3),B2) )
         => aa_num_o(aa_num_fun_num_o(C,A3),B2) )
     => aa_Pro7685007914727650815_num_o(produc5703948589228662326_num_o(C),P2) ) ).

% case_prodI2
tff(fact_5594_case__prodI2,axiom,
    ! [P2: product_prod_nat_num,C: fun_nat_fun_num_o] :
      ( ! [A3: nat,B2: num] :
          ( ( P2 = aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,A3),B2) )
         => aa_num_o(aa_nat_fun_num_o(C,A3),B2) )
     => aa_Pro6908818167415475913_num_o(produc4927758841916487424_num_o(C),P2) ) ).

% case_prodI2
tff(fact_5595_case__prodI2,axiom,
    ! [P2: product_prod_nat_nat,C: fun_nat_fun_nat_o] :
      ( ! [A3: nat,B2: nat] :
          ( ( P2 = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A3),B2) )
         => aa_nat_o(aa_nat_fun_nat_o(C,A3),B2) )
     => aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(C),P2) ) ).

% case_prodI2
tff(fact_5596_case__prodI2,axiom,
    ! [P2: product_prod_int_int,C: fun_int_fun_int_o] :
      ( ! [A3: int,B2: int] :
          ( ( P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A3),B2) )
         => aa_int_o(aa_int_fun_int_o(C,A3),B2) )
     => aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(C),P2) ) ).

% case_prodI2
tff(fact_5597_case__prodI,axiom,
    ! [F: fun_Co480733329897759773un_o_o,A: code_integer,B: $o] :
      ( aa_o_o(aa_Cod6379977994291289310un_o_o(F,A),(B))
     => aa_Pro5286387174248033258er_o_o(produc7828578312038201481er_o_o(F),aa_o_P4192039665577170892eger_o(aa_Cod3473573201699832512eger_o(produc6677183202524767010eger_o,A),(B))) ) ).

% case_prodI
tff(fact_5598_case__prodI,axiom,
    ! [F: fun_num_fun_num_o,A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(F,A),B)
     => aa_Pro7685007914727650815_num_o(produc5703948589228662326_num_o(F),aa_num3156681643174609981um_num(aa_num8681986212680877308um_num(product_Pair_num_num,A),B)) ) ).

% case_prodI
tff(fact_5599_case__prodI,axiom,
    ! [F: fun_nat_fun_num_o,A: nat,B: num] :
      ( aa_num_o(aa_nat_fun_num_o(F,A),B)
     => aa_Pro6908818167415475913_num_o(produc4927758841916487424_num_o(F),aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,A),B)) ) ).

% case_prodI
tff(fact_5600_case__prodI,axiom,
    ! [F: fun_nat_fun_nat_o,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(F,A),B)
     => aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(F),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B)) ) ).

% case_prodI
tff(fact_5601_case__prodI,axiom,
    ! [F: fun_int_fun_int_o,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(F,A),B)
     => aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)) ) ).

% case_prodI
tff(fact_5602_mem__case__prodI2,axiom,
    ! [P2: produc6271795597528267376eger_o,Z2: complex,C: fun_Co5015990272074005067omplex] :
      ( ! [A3: code_integer,B2: $o] :
          ( ( P2 = aa_o_P4192039665577170892eger_o(aa_Cod3473573201699832512eger_o(produc6677183202524767010eger_o,A3),(B2)) )
         => member_complex(Z2,aa_o_set_complex(aa_Cod2779864889637189172omplex(C,A3),(B2))) )
     => member_complex(Z2,produc1043322548047392435omplex(C,P2)) ) ).

% mem_case_prodI2
tff(fact_5603_mem__case__prodI2,axiom,
    ! [P2: produc6271795597528267376eger_o,Z2: real,C: fun_Co5368275265270938953t_real] :
      ( ! [A3: code_integer,B2: $o] :
          ( ( P2 = aa_o_P4192039665577170892eger_o(aa_Cod3473573201699832512eger_o(produc6677183202524767010eger_o,A3),(B2)) )
         => member_real(Z2,aa_o_set_real(aa_Cod7471715548787160626t_real(C,A3),(B2))) )
     => member_real(Z2,produc242741666403216561t_real(C,P2)) ) ).

% mem_case_prodI2
tff(fact_5604_mem__case__prodI2,axiom,
    ! [P2: produc6271795597528267376eger_o,Z2: nat,C: fun_Co7401227239307276909et_nat] :
      ( ! [A3: code_integer,B2: $o] :
          ( ( P2 = aa_o_P4192039665577170892eger_o(aa_Cod3473573201699832512eger_o(produc6677183202524767010eger_o,A3),(B2)) )
         => aa_set_nat_o(member_nat(Z2),aa_o_set_nat(aa_Cod3851761462451373910et_nat(C,A3),(B2))) )
     => aa_set_nat_o(member_nat(Z2),produc5431169771168744661et_nat(C,P2)) ) ).

% mem_case_prodI2
tff(fact_5605_mem__case__prodI2,axiom,
    ! [P2: produc6271795597528267376eger_o,Z2: int,C: fun_Co2587678820519358153et_int] :
      ( ! [A3: code_integer,B2: $o] :
          ( ( P2 = aa_o_P4192039665577170892eger_o(aa_Cod3473573201699832512eger_o(produc6677183202524767010eger_o,A3),(B2)) )
         => member_int(Z2,aa_o_set_int(aa_Cod1361678317519647154et_int(C,A3),(B2))) )
     => member_int(Z2,produc1253318751659547953et_int(C,P2)) ) ).

% mem_case_prodI2
tff(fact_5606_mem__case__prodI2,axiom,
    ! [P2: product_prod_num_num,Z2: complex,C: fun_nu9041852840018610706omplex] :
      ( ! [A3: num,B2: num] :
          ( ( P2 = aa_num3156681643174609981um_num(aa_num8681986212680877308um_num(product_Pair_num_num,A3),B2) )
         => member_complex(Z2,aa_num_set_complex(aa_num241437687621652691omplex(C,A3),B2)) )
     => member_complex(Z2,produc2866383454006189126omplex(C,P2)) ) ).

% mem_case_prodI2
tff(fact_5607_mem__case__prodI2,axiom,
    ! [P2: product_prod_num_num,Z2: real,C: fun_nu4698968662094489488t_real] :
      ( ! [A3: num,B2: num] :
          ( ( P2 = aa_num3156681643174609981um_num(aa_num8681986212680877308um_num(product_Pair_num_num,A3),B2) )
         => member_real(Z2,aa_num_set_real(aa_num2469771427700027217t_real(C,A3),B2)) )
     => member_real(Z2,produc8296048397933160132t_real(C,P2)) ) ).

% mem_case_prodI2
tff(fact_5608_mem__case__prodI2,axiom,
    ! [P2: product_prod_num_num,Z2: nat,C: fun_nu530747996811530292et_nat] :
      ( ! [A3: num,B2: num] :
          ( ( P2 = aa_num3156681643174609981um_num(aa_num8681986212680877308um_num(product_Pair_num_num,A3),B2) )
         => aa_set_nat_o(member_nat(Z2),aa_num_set_nat(aa_num5617506412915843061et_nat(C,A3),B2)) )
     => aa_set_nat_o(member_nat(Z2),produc1361121860356118632et_nat(C,P2)) ) ).

% mem_case_prodI2
tff(fact_5609_mem__case__prodI2,axiom,
    ! [P2: product_prod_num_num,Z2: int,C: fun_nu4940571614878387344et_int] :
      ( ! [A3: num,B2: num] :
          ( ( P2 = aa_num3156681643174609981um_num(aa_num8681986212680877308um_num(product_Pair_num_num,A3),B2) )
         => member_int(Z2,aa_num_set_int(aa_num3127423267984116305et_int(C,A3),B2)) )
     => member_int(Z2,produc6406642877701697732et_int(C,P2)) ) ).

% mem_case_prodI2
tff(fact_5610_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_num,Z2: complex,C: fun_na3925014923085203400omplex] :
      ( ! [A3: nat,B2: num] :
          ( ( P2 = aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,A3),B2) )
         => member_complex(Z2,aa_num_set_complex(aa_nat4347971807543021193omplex(C,A3),B2)) )
     => member_complex(Z2,produc6231982587499038204omplex(C,P2)) ) ).

% mem_case_prodI2
tff(fact_5611_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_num,Z2: real,C: fun_na3666575854997362246t_real] :
      ( ! [A3: nat,B2: num] :
          ( ( P2 = aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,A3),B2) )
         => member_real(Z2,aa_num_set_real(aa_nat1437378620602899975t_real(C,A3),B2)) )
     => member_real(Z2,produc1435849484188172666t_real(C,P2)) ) ).

% mem_case_prodI2
tff(fact_5612_mem__case__prodI,axiom,
    ! [Z2: complex,C: fun_Co5015990272074005067omplex,A: code_integer,B: $o] :
      ( member_complex(Z2,aa_o_set_complex(aa_Cod2779864889637189172omplex(C,A),(B)))
     => member_complex(Z2,produc1043322548047392435omplex(C,aa_o_P4192039665577170892eger_o(aa_Cod3473573201699832512eger_o(produc6677183202524767010eger_o,A),(B)))) ) ).

% mem_case_prodI
tff(fact_5613_mem__case__prodI,axiom,
    ! [Z2: real,C: fun_Co5368275265270938953t_real,A: code_integer,B: $o] :
      ( member_real(Z2,aa_o_set_real(aa_Cod7471715548787160626t_real(C,A),(B)))
     => member_real(Z2,produc242741666403216561t_real(C,aa_o_P4192039665577170892eger_o(aa_Cod3473573201699832512eger_o(produc6677183202524767010eger_o,A),(B)))) ) ).

% mem_case_prodI
tff(fact_5614_mem__case__prodI,axiom,
    ! [Z2: nat,C: fun_Co7401227239307276909et_nat,A: code_integer,B: $o] :
      ( aa_set_nat_o(member_nat(Z2),aa_o_set_nat(aa_Cod3851761462451373910et_nat(C,A),(B)))
     => aa_set_nat_o(member_nat(Z2),produc5431169771168744661et_nat(C,aa_o_P4192039665577170892eger_o(aa_Cod3473573201699832512eger_o(produc6677183202524767010eger_o,A),(B)))) ) ).

% mem_case_prodI
tff(fact_5615_mem__case__prodI,axiom,
    ! [Z2: int,C: fun_Co2587678820519358153et_int,A: code_integer,B: $o] :
      ( member_int(Z2,aa_o_set_int(aa_Cod1361678317519647154et_int(C,A),(B)))
     => member_int(Z2,produc1253318751659547953et_int(C,aa_o_P4192039665577170892eger_o(aa_Cod3473573201699832512eger_o(produc6677183202524767010eger_o,A),(B)))) ) ).

% mem_case_prodI
tff(fact_5616_mem__case__prodI,axiom,
    ! [Z2: complex,C: fun_nu9041852840018610706omplex,A: num,B: num] :
      ( member_complex(Z2,aa_num_set_complex(aa_num241437687621652691omplex(C,A),B))
     => member_complex(Z2,produc2866383454006189126omplex(C,aa_num3156681643174609981um_num(aa_num8681986212680877308um_num(product_Pair_num_num,A),B))) ) ).

% mem_case_prodI
tff(fact_5617_mem__case__prodI,axiom,
    ! [Z2: real,C: fun_nu4698968662094489488t_real,A: num,B: num] :
      ( member_real(Z2,aa_num_set_real(aa_num2469771427700027217t_real(C,A),B))
     => member_real(Z2,produc8296048397933160132t_real(C,aa_num3156681643174609981um_num(aa_num8681986212680877308um_num(product_Pair_num_num,A),B))) ) ).

% mem_case_prodI
tff(fact_5618_mem__case__prodI,axiom,
    ! [Z2: nat,C: fun_nu530747996811530292et_nat,A: num,B: num] :
      ( aa_set_nat_o(member_nat(Z2),aa_num_set_nat(aa_num5617506412915843061et_nat(C,A),B))
     => aa_set_nat_o(member_nat(Z2),produc1361121860356118632et_nat(C,aa_num3156681643174609981um_num(aa_num8681986212680877308um_num(product_Pair_num_num,A),B))) ) ).

% mem_case_prodI
tff(fact_5619_mem__case__prodI,axiom,
    ! [Z2: int,C: fun_nu4940571614878387344et_int,A: num,B: num] :
      ( member_int(Z2,aa_num_set_int(aa_num3127423267984116305et_int(C,A),B))
     => member_int(Z2,produc6406642877701697732et_int(C,aa_num3156681643174609981um_num(aa_num8681986212680877308um_num(product_Pair_num_num,A),B))) ) ).

% mem_case_prodI
tff(fact_5620_mem__case__prodI,axiom,
    ! [Z2: complex,C: fun_na3925014923085203400omplex,A: nat,B: num] :
      ( member_complex(Z2,aa_num_set_complex(aa_nat4347971807543021193omplex(C,A),B))
     => member_complex(Z2,produc6231982587499038204omplex(C,aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,A),B))) ) ).

% mem_case_prodI
tff(fact_5621_mem__case__prodI,axiom,
    ! [Z2: real,C: fun_na3666575854997362246t_real,A: nat,B: num] :
      ( member_real(Z2,aa_num_set_real(aa_nat1437378620602899975t_real(C,A),B))
     => member_real(Z2,produc1435849484188172666t_real(C,aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,A),B))) ) ).

% mem_case_prodI
tff(fact_5622_case__prodI2_H,axiom,
    ! [P2: product_prod_nat_nat,C: fun_na5914473681949881652_nat_o,X: product_prod_nat_nat] :
      ( ! [A3: nat,B2: nat] :
          ( ( aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A3),B2) = P2 )
         => aa_Pro6150637604445964947_nat_o(aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(C,A3),B2),X) )
     => aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(C),P2),X) ) ).

% case_prodI2'
tff(fact_5623_dbl__dec__simps_I3_J,axiom,
    aa_complex_complex(neg_nu6511756317524482435omplex,one_one_complex) = one_one_complex ).

% dbl_dec_simps(3)
tff(fact_5624_dbl__dec__simps_I3_J,axiom,
    aa_real_real(neg_nu6075765906172075777c_real,one_one_real) = one_one_real ).

% dbl_dec_simps(3)
tff(fact_5625_dbl__dec__simps_I3_J,axiom,
    aa_rat_rat(neg_nu3179335615603231917ec_rat,one_one_rat) = one_one_rat ).

% dbl_dec_simps(3)
tff(fact_5626_dbl__dec__simps_I3_J,axiom,
    aa_int_int(neg_nu3811975205180677377ec_int,one_one_int) = one_one_int ).

% dbl_dec_simps(3)
tff(fact_5627_dbl__dec__simps_I2_J,axiom,
    aa_real_real(neg_nu6075765906172075777c_real,zero_zero_real) = aa_real_real(uminus_uminus_real,one_one_real) ).

% dbl_dec_simps(2)
tff(fact_5628_dbl__dec__simps_I2_J,axiom,
    aa_int_int(neg_nu3811975205180677377ec_int,zero_zero_int) = aa_int_int(uminus_uminus_int,one_one_int) ).

% dbl_dec_simps(2)
tff(fact_5629_dbl__dec__simps_I2_J,axiom,
    aa_complex_complex(neg_nu6511756317524482435omplex,zero_zero_complex) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% dbl_dec_simps(2)
tff(fact_5630_dbl__dec__simps_I2_J,axiom,
    aa_rat_rat(neg_nu3179335615603231917ec_rat,zero_zero_rat) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% dbl_dec_simps(2)
tff(fact_5631_dbl__dec__simps_I2_J,axiom,
    neg_nu7757733837767384882nteger(zero_z3403309356797280102nteger) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% dbl_dec_simps(2)
tff(fact_5632_divmod__algorithm__code_I2_J,axiom,
    ! [Ma: num] : unique5052692396658037445od_int(Ma,one) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_num_int(numeral_numeral_int,Ma)),zero_zero_int) ).

% divmod_algorithm_code(2)
tff(fact_5633_divmod__algorithm__code_I2_J,axiom,
    ! [Ma: num] : unique5055182867167087721od_nat(Ma,one) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_num_nat(numeral_numeral_nat,Ma)),zero_zero_nat) ).

% divmod_algorithm_code(2)
tff(fact_5634_divmod__algorithm__code_I2_J,axiom,
    ! [Ma: num] : unique3479559517661332726nteger(Ma,one) = produc1086072967326762835nteger(numera6620942414471956472nteger(Ma),zero_z3403309356797280102nteger) ).

% divmod_algorithm_code(2)
tff(fact_5635_divmod__algorithm__code_I3_J,axiom,
    ! [Nb: num] : unique5052692396658037445od_int(one,aa_num_num(bit0,Nb)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),aa_num_int(numeral_numeral_int,one)) ).

% divmod_algorithm_code(3)
tff(fact_5636_divmod__algorithm__code_I3_J,axiom,
    ! [Nb: num] : unique5055182867167087721od_nat(one,aa_num_num(bit0,Nb)) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,one)) ).

% divmod_algorithm_code(3)
tff(fact_5637_divmod__algorithm__code_I3_J,axiom,
    ! [Nb: num] : unique3479559517661332726nteger(one,aa_num_num(bit0,Nb)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(one)) ).

% divmod_algorithm_code(3)
tff(fact_5638_divmod__algorithm__code_I4_J,axiom,
    ! [Nb: num] : unique5052692396658037445od_int(one,aa_num_num(bit1,Nb)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),aa_num_int(numeral_numeral_int,one)) ).

% divmod_algorithm_code(4)
tff(fact_5639_divmod__algorithm__code_I4_J,axiom,
    ! [Nb: num] : unique5055182867167087721od_nat(one,aa_num_num(bit1,Nb)) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,one)) ).

% divmod_algorithm_code(4)
tff(fact_5640_divmod__algorithm__code_I4_J,axiom,
    ! [Nb: num] : unique3479559517661332726nteger(one,aa_num_num(bit1,Nb)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(one)) ).

% divmod_algorithm_code(4)
tff(fact_5641_divmod__algorithm__code_I5_J,axiom,
    ! [Ma: num,Nb: num] : unique5052692396658037445od_int(aa_num_num(bit0,Ma),aa_num_num(bit0,Nb)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_ch),unique5052692396658037445od_int(Ma,Nb)) ).

% divmod_algorithm_code(5)
tff(fact_5642_divmod__algorithm__code_I5_J,axiom,
    ! [Ma: num,Nb: num] : unique5055182867167087721od_nat(aa_num_num(bit0,Ma),aa_num_num(bit0,Nb)) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_ci),unique5055182867167087721od_nat(Ma,Nb)) ).

% divmod_algorithm_code(5)
tff(fact_5643_divmod__algorithm__code_I5_J,axiom,
    ! [Ma: num,Nb: num] : unique3479559517661332726nteger(aa_num_num(bit0,Ma),aa_num_num(bit0,Nb)) = produc6916734918728496179nteger(aTP_Lamm_cj,unique3479559517661332726nteger(Ma,Nb)) ).

% divmod_algorithm_code(5)
tff(fact_5644_divmod__algorithm__code_I8_J,axiom,
    ! [Ma: num,Nb: num] :
      unique5055182867167087721od_nat(aa_num_num(bit1,Ma),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Ma),Nb),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Ma))),unique5026877609467782581ep_nat(aa_num_num(bit1,Nb),unique5055182867167087721od_nat(aa_num_num(bit1,Ma),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(8)
tff(fact_5645_divmod__algorithm__code_I8_J,axiom,
    ! [Ma: num,Nb: num] :
      unique5052692396658037445od_int(aa_num_num(bit1,Ma),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Ma),Nb),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Ma))),aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(aa_num_num(bit1,Nb)),unique5052692396658037445od_int(aa_num_num(bit1,Ma),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(8)
tff(fact_5646_divmod__algorithm__code_I8_J,axiom,
    ! [Ma: num,Nb: num] :
      unique3479559517661332726nteger(aa_num_num(bit1,Ma),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Ma),Nb),produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(aa_num_num(bit1,Ma))),unique4921790084139445826nteger(aa_num_num(bit1,Nb),unique3479559517661332726nteger(aa_num_num(bit1,Ma),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(8)
tff(fact_5647_uminus__set__def,axiom,
    ! [A2: set_real] : uminus612125837232591019t_real(A2) = collect_real(uminus2572480474538279996real_o(aTP_Lamm_ab(A2))) ).

% uminus_set_def
tff(fact_5648_uminus__set__def,axiom,
    ! [A2: set_complex] : uminus8566677241136511917omplex(A2) = collect_complex(uminus4035047602201070266plex_o(aTP_Lamm_ac(A2))) ).

% uminus_set_def
tff(fact_5649_uminus__set__def,axiom,
    ! [A2: set_list_nat] : uminus3195874150345416415st_nat(A2) = collect_list_nat(uminus8676031017987022600_nat_o(aTP_Lamm_ad(A2))) ).

% uminus_set_def
tff(fact_5650_uminus__set__def,axiom,
    ! [A2: set_set_nat] : uminus613421341184616069et_nat(A2) = collect_set_nat(uminus1495274395075166690_nat_o(aTP_Lamm_ae(A2))) ).

% uminus_set_def
tff(fact_5651_uminus__set__def,axiom,
    ! [A2: set_nat] : aa_set_nat_set_nat(uminus5710092332889474511et_nat,A2) = collect_nat(uminus6183504819137795864_nat_o(aTP_Lamm_af(A2))) ).

% uminus_set_def
tff(fact_5652_uminus__set__def,axiom,
    ! [A2: set_int] : uminus1532241313380277803et_int(A2) = collect_int(uminus4362462473705107900_int_o(aTP_Lamm_ag(A2))) ).

% uminus_set_def
tff(fact_5653_take__bit__minus,axiom,
    ! [Nb: nat,K2: int] : aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(uminus_uminus_int,aa_int_int(bit_se2923211474154528505it_int(Nb),K2))) = aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(uminus_uminus_int,K2)) ).

% take_bit_minus
tff(fact_5654_Collect__neg__eq,axiom,
    ! [P: fun_complex_o] : collect_complex(aTP_Lamm_ck(P)) = uminus8566677241136511917omplex(collect_complex(P)) ).

% Collect_neg_eq
tff(fact_5655_Collect__neg__eq,axiom,
    ! [P: fun_list_nat_o] : collect_list_nat(aTP_Lamm_cl(P)) = uminus3195874150345416415st_nat(collect_list_nat(P)) ).

% Collect_neg_eq
tff(fact_5656_Collect__neg__eq,axiom,
    ! [P: fun_set_nat_o] : collect_set_nat(aTP_Lamm_cm(P)) = uminus613421341184616069et_nat(collect_set_nat(P)) ).

% Collect_neg_eq
tff(fact_5657_Collect__neg__eq,axiom,
    ! [P: fun_nat_o] : collect_nat(aTP_Lamm_cn(P)) = aa_set_nat_set_nat(uminus5710092332889474511et_nat,collect_nat(P)) ).

% Collect_neg_eq
tff(fact_5658_Collect__neg__eq,axiom,
    ! [P: fun_int_o] : collect_int(aTP_Lamm_co(P)) = uminus1532241313380277803et_int(collect_int(P)) ).

% Collect_neg_eq
tff(fact_5659_Compl__eq,axiom,
    ! [A2: set_real] : uminus612125837232591019t_real(A2) = collect_real(aTP_Lamm_cp(A2)) ).

% Compl_eq
tff(fact_5660_Compl__eq,axiom,
    ! [A2: set_complex] : uminus8566677241136511917omplex(A2) = collect_complex(aTP_Lamm_cq(A2)) ).

% Compl_eq
tff(fact_5661_Compl__eq,axiom,
    ! [A2: set_list_nat] : uminus3195874150345416415st_nat(A2) = collect_list_nat(aTP_Lamm_cr(A2)) ).

% Compl_eq
tff(fact_5662_Compl__eq,axiom,
    ! [A2: set_set_nat] : uminus613421341184616069et_nat(A2) = collect_set_nat(aTP_Lamm_cs(A2)) ).

% Compl_eq
tff(fact_5663_Compl__eq,axiom,
    ! [A2: set_nat] : aa_set_nat_set_nat(uminus5710092332889474511et_nat,A2) = collect_nat(aTP_Lamm_ct(A2)) ).

% Compl_eq
tff(fact_5664_Compl__eq,axiom,
    ! [A2: set_int] : uminus1532241313380277803et_int(A2) = collect_int(aTP_Lamm_cu(A2)) ).

% Compl_eq
tff(fact_5665_case__prodD,axiom,
    ! [F: fun_Co480733329897759773un_o_o,A: code_integer,B: $o] :
      ( aa_Pro5286387174248033258er_o_o(produc7828578312038201481er_o_o(F),aa_o_P4192039665577170892eger_o(aa_Cod3473573201699832512eger_o(produc6677183202524767010eger_o,A),(B)))
     => aa_o_o(aa_Cod6379977994291289310un_o_o(F,A),(B)) ) ).

% case_prodD
tff(fact_5666_case__prodD,axiom,
    ! [F: fun_num_fun_num_o,A: num,B: num] :
      ( aa_Pro7685007914727650815_num_o(produc5703948589228662326_num_o(F),aa_num3156681643174609981um_num(aa_num8681986212680877308um_num(product_Pair_num_num,A),B))
     => aa_num_o(aa_num_fun_num_o(F,A),B) ) ).

% case_prodD
tff(fact_5667_case__prodD,axiom,
    ! [F: fun_nat_fun_num_o,A: nat,B: num] :
      ( aa_Pro6908818167415475913_num_o(produc4927758841916487424_num_o(F),aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,A),B))
     => aa_num_o(aa_nat_fun_num_o(F,A),B) ) ).

% case_prodD
tff(fact_5668_case__prodD,axiom,
    ! [F: fun_nat_fun_nat_o,A: nat,B: nat] :
      ( aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(F),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B))
     => aa_nat_o(aa_nat_fun_nat_o(F,A),B) ) ).

% case_prodD
tff(fact_5669_case__prodD,axiom,
    ! [F: fun_int_fun_int_o,A: int,B: int] :
      ( aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B))
     => aa_int_o(aa_int_fun_int_o(F,A),B) ) ).

% case_prodD
tff(fact_5670_case__prodE,axiom,
    ! [C: fun_Co480733329897759773un_o_o,P2: produc6271795597528267376eger_o] :
      ( aa_Pro5286387174248033258er_o_o(produc7828578312038201481er_o_o(C),P2)
     => ~ ! [X4: code_integer,Y4: $o] :
            ( ( P2 = aa_o_P4192039665577170892eger_o(aa_Cod3473573201699832512eger_o(produc6677183202524767010eger_o,X4),(Y4)) )
           => ~ aa_o_o(aa_Cod6379977994291289310un_o_o(C,X4),(Y4)) ) ) ).

% case_prodE
tff(fact_5671_case__prodE,axiom,
    ! [C: fun_num_fun_num_o,P2: product_prod_num_num] :
      ( aa_Pro7685007914727650815_num_o(produc5703948589228662326_num_o(C),P2)
     => ~ ! [X4: num,Y4: num] :
            ( ( P2 = aa_num3156681643174609981um_num(aa_num8681986212680877308um_num(product_Pair_num_num,X4),Y4) )
           => ~ aa_num_o(aa_num_fun_num_o(C,X4),Y4) ) ) ).

% case_prodE
tff(fact_5672_case__prodE,axiom,
    ! [C: fun_nat_fun_num_o,P2: product_prod_nat_num] :
      ( aa_Pro6908818167415475913_num_o(produc4927758841916487424_num_o(C),P2)
     => ~ ! [X4: nat,Y4: num] :
            ( ( P2 = aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,X4),Y4) )
           => ~ aa_num_o(aa_nat_fun_num_o(C,X4),Y4) ) ) ).

% case_prodE
tff(fact_5673_case__prodE,axiom,
    ! [C: fun_nat_fun_nat_o,P2: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(C),P2)
     => ~ ! [X4: nat,Y4: nat] :
            ( ( P2 = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X4),Y4) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(C,X4),Y4) ) ) ).

% case_prodE
tff(fact_5674_case__prodE,axiom,
    ! [C: fun_int_fun_int_o,P2: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(C),P2)
     => ~ ! [X4: int,Y4: int] :
            ( ( P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Y4) )
           => ~ aa_int_o(aa_int_fun_int_o(C,X4),Y4) ) ) ).

% case_prodE
tff(fact_5675_case__prodE_H,axiom,
    ! [C: fun_na5914473681949881652_nat_o,P2: product_prod_nat_nat,Z2: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(C),P2),Z2)
     => ~ ! [X4: nat,Y4: nat] :
            ( ( P2 = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X4),Y4) )
           => ~ aa_Pro6150637604445964947_nat_o(aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(C,X4),Y4),Z2) ) ) ).

% case_prodE'
tff(fact_5676_case__prodD_H,axiom,
    ! [R4: fun_na5914473681949881652_nat_o,A: nat,B: nat,C: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(R4),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A),B)),C)
     => aa_Pro6150637604445964947_nat_o(aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(R4,A),B),C) ) ).

% case_prodD'
tff(fact_5677_prod_Ocase__distrib,axiom,
    ! [H: fun_o_o,F: fun_int_fun_int_o,Prod: product_prod_int_int] :
      ( aa_o_o(H,aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(F),Prod))
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_cv(H),F)),Prod) ) ).

% prod.case_distrib
tff(fact_5678_prod_Ocase__distrib,axiom,
    ! [H: fun_o_int,F: fun_int_fun_int_o,Prod: product_prod_int_int] : aa_o_int(H,aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(F),Prod)) = aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(aa_fun7323606162024495668nt_int(aTP_Lamm_cw(H),F)),Prod) ).

% prod.case_distrib
tff(fact_5679_prod_Ocase__distrib,axiom,
    ! [H: fun_int_o,F: fun_int_fun_int_int,Prod: product_prod_int_int] :
      ( aa_int_o(H,aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(F),Prod))
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_fun82482788810740534_int_o(aTP_Lamm_cx(H),F)),Prod) ) ).

% prod.case_distrib
tff(fact_5680_prod_Ocase__distrib,axiom,
    ! [H: fun_int_int,F: fun_int_fun_int_int,Prod: product_prod_int_int] : aa_int_int(H,aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(F),Prod)) = aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(aa_fun6119123525526402106nt_int(aTP_Lamm_cy(H),F)),Prod) ).

% prod.case_distrib
tff(fact_5681_prod_Ocase__distrib,axiom,
    ! [H: fun_Pr3238374608701176162_int_o,F: fun_in1696471567231710611nt_int,Prod: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(H,aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(F),Prod))
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_fun4545072666309863515_int_o(aTP_Lamm_cz(H),F)),Prod) ) ).

% prod.case_distrib
tff(fact_5682_prod_Ocase__distrib,axiom,
    ! [H: fun_Pr982841511892225912nt_int,F: fun_in1696471567231710611nt_int,Prod: product_prod_int_int] : aa_Pro6799258434465214121nt_int(H,aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(F),Prod)) = aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(aa_fun7461304428735181397nt_int(aTP_Lamm_da(H),F)),Prod) ).

% prod.case_distrib
tff(fact_5683_prod_Ocase__distrib,axiom,
    ! [H: fun_o_8988054265482332260nt_int,F: fun_int_fun_int_o,Prod: product_prod_int_int] : aa_o_P3134438092330336605nt_int(H,aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(F),Prod)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_fun4023131440832735197nt_int(aTP_Lamm_db(H),F)),Prod) ).

% prod.case_distrib
tff(fact_5684_prod_Ocase__distrib,axiom,
    ! [H: fun_in4211310926729683942nt_int,F: fun_int_fun_int_int,Prod: product_prod_int_int] : aa_int7462565939943104719nt_int(H,aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(F),Prod)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_fun1034830265760468567nt_int(aTP_Lamm_dc(H),F)),Prod) ).

% prod.case_distrib
tff(fact_5685_prod_Ocase__distrib,axiom,
    ! [H: fun_Pr2478283820680522243nt_int,F: fun_in1696471567231710611nt_int,Prod: product_prod_int_int] : aa_Pro6970504127699319548nt_int(H,aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(F),Prod)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_fun4152872853588681340nt_int(aTP_Lamm_dd(H),F)),Prod) ).

% prod.case_distrib
tff(fact_5686_prod_Ocase__distrib,axiom,
    ! [H: fun_fu3275390434952296369_nat_o,F: fun_na5914473681949881652_nat_o,Prod: product_prod_nat_nat] : aa_fun359341089595923442_nat_o(H,aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(F),Prod)) = aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(aa_fun824319262034445606_nat_o(aTP_Lamm_de(H),F)),Prod) ).

% prod.case_distrib
tff(fact_5687_divmod__int__def,axiom,
    ! [Ma: num,Nb: num] : unique5052692396658037445od_int(Ma,Nb) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,Ma)),aa_num_int(numeral_numeral_int,Nb))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Ma)),aa_num_int(numeral_numeral_int,Nb))) ).

% divmod_int_def
tff(fact_5688_divmod__def,axiom,
    ! [Ma: num,Nb: num] : unique5052692396658037445od_int(Ma,Nb) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,Ma)),aa_num_int(numeral_numeral_int,Nb))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Ma)),aa_num_int(numeral_numeral_int,Nb))) ).

% divmod_def
tff(fact_5689_divmod__def,axiom,
    ! [Ma: num,Nb: num] : unique5055182867167087721od_nat(Ma,Nb) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,Ma)),aa_num_nat(numeral_numeral_nat,Nb))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Ma)),aa_num_nat(numeral_numeral_nat,Nb))) ).

% divmod_def
tff(fact_5690_divmod__def,axiom,
    ! [Ma: num,Nb: num] : unique3479559517661332726nteger(Ma,Nb) = produc1086072967326762835nteger(divide6298287555418463151nteger(numera6620942414471956472nteger(Ma),numera6620942414471956472nteger(Nb)),modulo364778990260209775nteger(numera6620942414471956472nteger(Ma),numera6620942414471956472nteger(Nb))) ).

% divmod_def
tff(fact_5691_divmod_H__nat__def,axiom,
    ! [Ma: num,Nb: num] : unique5055182867167087721od_nat(Ma,Nb) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,Ma)),aa_num_nat(numeral_numeral_nat,Nb))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Ma)),aa_num_nat(numeral_numeral_nat,Nb))) ).

% divmod'_nat_def
tff(fact_5692_dbl__dec__def,axiom,
    ! [X: complex] : aa_complex_complex(neg_nu6511756317524482435omplex,X) = aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(X),X)),one_one_complex) ).

% dbl_dec_def
tff(fact_5693_dbl__dec__def,axiom,
    ! [X: real] : aa_real_real(neg_nu6075765906172075777c_real,X) = aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(X),X)),one_one_real) ).

% dbl_dec_def
tff(fact_5694_dbl__dec__def,axiom,
    ! [X: rat] : aa_rat_rat(neg_nu3179335615603231917ec_rat,X) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(X),X)),one_one_rat) ).

% dbl_dec_def
tff(fact_5695_dbl__dec__def,axiom,
    ! [X: int] : aa_int_int(neg_nu3811975205180677377ec_int,X) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),X)),one_one_int) ).

% dbl_dec_def
tff(fact_5696_compl__mono,axiom,
    ! [X: set_int,Y2: set_int] :
      ( ord_less_eq_set_int(X,Y2)
     => ord_less_eq_set_int(uminus1532241313380277803et_int(Y2),uminus1532241313380277803et_int(X)) ) ).

% compl_mono
tff(fact_5697_compl__le__swap1,axiom,
    ! [Y2: set_int,X: set_int] :
      ( ord_less_eq_set_int(Y2,uminus1532241313380277803et_int(X))
     => ord_less_eq_set_int(X,uminus1532241313380277803et_int(Y2)) ) ).

% compl_le_swap1
tff(fact_5698_compl__le__swap2,axiom,
    ! [Y2: set_int,X: set_int] :
      ( ord_less_eq_set_int(uminus1532241313380277803et_int(Y2),X)
     => ord_less_eq_set_int(uminus1532241313380277803et_int(X),Y2) ) ).

% compl_le_swap2
tff(fact_5699_cond__case__prod__eta,axiom,
    ! [F: fun_na1668100129766719005at_nat,G: fun_Pr2477239776880934807at_nat] :
      ( ! [X4: nat,Y4: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,X4),Y4) = aa_Pro8023792847356852296at_nat(G,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X4),Y4))
     => ( produc4528391901023682385at_nat(F) = G ) ) ).

% cond_case_prod_eta
tff(fact_5700_cond__case__prod__eta,axiom,
    ! [F: fun_na5914473681949881652_nat_o,G: fun_Pr2028404973800837934_nat_o] :
      ( ! [X4: nat,Y4: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(F,X4),Y4) = aa_Pro989595756709598047_nat_o(G,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X4),Y4))
     => ( produc6395135316500328296_nat_o(F) = G ) ) ).

% cond_case_prod_eta
tff(fact_5701_cond__case__prod__eta,axiom,
    ! [F: fun_in1696471567231710611nt_int,G: fun_Pr2478283820680522243nt_int] :
      ( ! [X4: int,Y4: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,X4),Y4) = aa_Pro6970504127699319548nt_int(G,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Y4))
     => ( produc4245557441103728435nt_int(F) = G ) ) ).

% cond_case_prod_eta
tff(fact_5702_cond__case__prod__eta,axiom,
    ! [F: fun_int_fun_int_o,G: fun_Pr3238374608701176162_int_o] :
      ( ! [X4: int,Y4: int] :
          ( aa_int_o(aa_int_fun_int_o(F,X4),Y4)
        <=> aa_Pro4592068669816069595_int_o(G,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Y4)) )
     => ( produc4947309494688390418_int_o(F) = G ) ) ).

% cond_case_prod_eta
tff(fact_5703_cond__case__prod__eta,axiom,
    ! [F: fun_int_fun_int_int,G: fun_Pr982841511892225912nt_int] :
      ( ! [X4: int,Y4: int] : aa_int_int(aa_int_fun_int_int(F,X4),Y4) = aa_Pro6799258434465214121nt_int(G,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Y4))
     => ( produc8211389475949308722nt_int(F) = G ) ) ).

% cond_case_prod_eta
tff(fact_5704_case__prod__eta,axiom,
    ! [F: fun_Pr2477239776880934807at_nat] : produc4528391901023682385at_nat(aTP_Lamm_df(F)) = F ).

% case_prod_eta
tff(fact_5705_case__prod__eta,axiom,
    ! [F: fun_Pr2028404973800837934_nat_o] : produc6395135316500328296_nat_o(aTP_Lamm_dg(F)) = F ).

% case_prod_eta
tff(fact_5706_case__prod__eta,axiom,
    ! [F: fun_Pr2478283820680522243nt_int] : produc4245557441103728435nt_int(aTP_Lamm_dh(F)) = F ).

% case_prod_eta
tff(fact_5707_case__prod__eta,axiom,
    ! [F: fun_Pr3238374608701176162_int_o] : produc4947309494688390418_int_o(aTP_Lamm_di(F)) = F ).

% case_prod_eta
tff(fact_5708_case__prod__eta,axiom,
    ! [F: fun_Pr982841511892225912nt_int] : produc8211389475949308722nt_int(aTP_Lamm_dj(F)) = F ).

% case_prod_eta
tff(fact_5709_case__prodE2,axiom,
    ! [Q2: fun_fu8391177546919232046_nat_o,P: fun_na1668100129766719005at_nat,Z2: product_prod_nat_nat] :
      ( aa_fun3538214411680374423_nat_o(Q2,aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(P),Z2))
     => ~ ! [X4: nat,Y4: nat] :
            ( ( Z2 = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X4),Y4) )
           => ~ aa_fun3538214411680374423_nat_o(Q2,aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(P,X4),Y4)) ) ) ).

% case_prodE2
tff(fact_5710_case__prodE2,axiom,
    ! [Q2: fun_fu6891447142836604759at_o_o,P: fun_na5914473681949881652_nat_o,Z2: product_prod_nat_nat] :
      ( aa_fun8408151826566693824at_o_o(Q2,aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(P),Z2))
     => ~ ! [X4: nat,Y4: nat] :
            ( ( Z2 = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X4),Y4) )
           => ~ aa_fun8408151826566693824at_o_o(Q2,aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(P,X4),Y4)) ) ) ).

% case_prodE2
tff(fact_5711_case__prodE2,axiom,
    ! [Q2: fun_Pr3238374608701176162_int_o,P: fun_in1696471567231710611nt_int,Z2: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(Q2,aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(P),Z2))
     => ~ ! [X4: int,Y4: int] :
            ( ( Z2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Y4) )
           => ~ aa_Pro4592068669816069595_int_o(Q2,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(P,X4),Y4)) ) ) ).

% case_prodE2
tff(fact_5712_case__prodE2,axiom,
    ! [Q2: fun_o_o,P: fun_int_fun_int_o,Z2: product_prod_int_int] :
      ( aa_o_o(Q2,aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(P),Z2))
     => ~ ! [X4: int,Y4: int] :
            ( ( Z2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Y4) )
           => ~ aa_o_o(Q2,aa_int_o(aa_int_fun_int_o(P,X4),Y4)) ) ) ).

% case_prodE2
tff(fact_5713_case__prodE2,axiom,
    ! [Q2: fun_int_o,P: fun_int_fun_int_int,Z2: product_prod_int_int] :
      ( aa_int_o(Q2,aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(P),Z2))
     => ~ ! [X4: int,Y4: int] :
            ( ( Z2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Y4) )
           => ~ aa_int_o(Q2,aa_int_int(aa_int_fun_int_int(P,X4),Y4)) ) ) ).

% case_prodE2
tff(fact_5714_divmod__divmod__step,axiom,
    ! [Ma: num,Nb: num] :
      unique5055182867167087721od_nat(Ma,Nb) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Ma),Nb),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,Ma)),unique5026877609467782581ep_nat(Nb,unique5055182867167087721od_nat(Ma,aa_num_num(bit0,Nb)))) ).

% divmod_divmod_step
tff(fact_5715_divmod__divmod__step,axiom,
    ! [Ma: num,Nb: num] :
      unique5052692396658037445od_int(Ma,Nb) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Ma),Nb),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),aa_num_int(numeral_numeral_int,Ma)),aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(Nb),unique5052692396658037445od_int(Ma,aa_num_num(bit0,Nb)))) ).

% divmod_divmod_step
tff(fact_5716_divmod__divmod__step,axiom,
    ! [Ma: num,Nb: num] :
      unique3479559517661332726nteger(Ma,Nb) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Ma),Nb),produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(Ma)),unique4921790084139445826nteger(Nb,unique3479559517661332726nteger(Ma,aa_num_num(bit0,Nb)))) ).

% divmod_divmod_step
tff(fact_5717_dvd__numeral__simp,axiom,
    ! [Ma: num,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,Ma)),aa_num_int(numeral_numeral_int,Nb))
    <=> aa_Pro4592068669816069595_int_o(unique6319869463603278526ux_int,unique5052692396658037445od_int(Nb,Ma)) ) ).

% dvd_numeral_simp
tff(fact_5718_dvd__numeral__simp,axiom,
    ! [Ma: num,Nb: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,Ma)),aa_num_nat(numeral_numeral_nat,Nb))
    <=> aa_Pro6150637604445964947_nat_o(unique6322359934112328802ux_nat,unique5055182867167087721od_nat(Nb,Ma)) ) ).

% dvd_numeral_simp
tff(fact_5719_dvd__numeral__simp,axiom,
    ! [Ma: num,Nb: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(Ma)),numera6620942414471956472nteger(Nb))
    <=> unique5706413561485394159nteger(unique3479559517661332726nteger(Nb,Ma)) ) ).

% dvd_numeral_simp
tff(fact_5720_one__div__minus__numeral,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,aa_Pro6799258434465214121nt_int(adjust_div,unique5052692396658037445od_int(one,Nb))) ).

% one_div_minus_numeral
tff(fact_5721_minus__one__div__numeral,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,aa_Pro6799258434465214121nt_int(adjust_div,unique5052692396658037445od_int(one,Nb))) ).

% minus_one_div_numeral
tff(fact_5722_signed__take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K2: num] : aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K2)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(pred_numeral,L)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% signed_take_bit_numeral_minus_bit1
tff(fact_5723_divmod__nat__if,axiom,
    ! [Ma: nat,Nb: nat] :
      divmod_nat(Ma,Nb) = $ite(
        ( ( Nb = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb) ),
        aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,zero_zero_nat),Ma),
        aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_dk),divmod_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb),Nb)) ) ).

% divmod_nat_if
tff(fact_5724_take__bit__Suc__minus__bit1,axiom,
    ! [Nb: nat,K2: num] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K2)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,K2))))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% take_bit_Suc_minus_bit1
tff(fact_5725_signed__take__bit__numeral__bit1,axiom,
    ! [L: num,K2: num] : aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(numeral_numeral_nat,L)),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(pred_numeral,L)),aa_num_int(numeral_numeral_int,K2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% signed_take_bit_numeral_bit1
tff(fact_5726_split__part,axiom,
    ! [P: $o,Q2: fun_int_fun_int_o,X2: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_dl((P)),Q2)),X2)
    <=> ( (P)
        & aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(Q2),X2) ) ) ).

% split_part
tff(fact_5727_pred__numeral__simps_I1_J,axiom,
    aa_num_nat(pred_numeral,one) = zero_zero_nat ).

% pred_numeral_simps(1)
tff(fact_5728_eq__numeral__Suc,axiom,
    ! [K2: num,Nb: nat] :
      ( ( aa_num_nat(numeral_numeral_nat,K2) = aa_nat_nat(suc,Nb) )
    <=> ( aa_num_nat(pred_numeral,K2) = Nb ) ) ).

% eq_numeral_Suc
tff(fact_5729_Suc__eq__numeral,axiom,
    ! [Nb: nat,K2: num] :
      ( ( aa_nat_nat(suc,Nb) = aa_num_nat(numeral_numeral_nat,K2) )
    <=> ( Nb = aa_num_nat(pred_numeral,K2) ) ) ).

% Suc_eq_numeral
tff(fact_5730_pred__numeral__inc,axiom,
    ! [K2: num] : aa_num_nat(pred_numeral,aa_num_num(inc,K2)) = aa_num_nat(numeral_numeral_nat,K2) ).

% pred_numeral_inc
tff(fact_5731_pred__numeral__simps_I3_J,axiom,
    ! [K2: num] : aa_num_nat(pred_numeral,aa_num_num(bit1,K2)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K2)) ).

% pred_numeral_simps(3)
tff(fact_5732_less__numeral__Suc,axiom,
    ! [K2: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,K2)),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(pred_numeral,K2)),Nb) ) ).

% less_numeral_Suc
tff(fact_5733_less__Suc__numeral,axiom,
    ! [Nb: nat,K2: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,K2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_num_nat(pred_numeral,K2)) ) ).

% less_Suc_numeral
tff(fact_5734_le__numeral__Suc,axiom,
    ! [K2: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,K2)),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(pred_numeral,K2)),Nb) ) ).

% le_numeral_Suc
tff(fact_5735_le__Suc__numeral,axiom,
    ! [Nb: nat,K2: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,K2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_num_nat(pred_numeral,K2)) ) ).

% le_Suc_numeral
tff(fact_5736_diff__numeral__Suc,axiom,
    ! [K2: num,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,K2)),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(pred_numeral,K2)),Nb) ).

% diff_numeral_Suc
tff(fact_5737_diff__Suc__numeral,axiom,
    ! [Nb: nat,K2: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,K2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_num_nat(pred_numeral,K2)) ).

% diff_Suc_numeral
tff(fact_5738_add__neg__numeral__special_I5_J,axiom,
    ! [Nb: num] : aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(inc,Nb))) ).

% add_neg_numeral_special(5)
tff(fact_5739_add__neg__numeral__special_I5_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,Nb))) ).

% add_neg_numeral_special(5)
tff(fact_5740_add__neg__numeral__special_I5_J,axiom,
    ! [Nb: num] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb))) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(inc,Nb))) ).

% add_neg_numeral_special(5)
tff(fact_5741_add__neg__numeral__special_I5_J,axiom,
    ! [Nb: num] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(inc,Nb))) ).

% add_neg_numeral_special(5)
tff(fact_5742_add__neg__numeral__special_I5_J,axiom,
    ! [Nb: num] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(inc,Nb))) ).

% add_neg_numeral_special(5)
tff(fact_5743_add__neg__numeral__special_I6_J,axiom,
    ! [Ma: num] : aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Ma))),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(inc,Ma))) ).

% add_neg_numeral_special(6)
tff(fact_5744_add__neg__numeral__special_I6_J,axiom,
    ! [Ma: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Ma))),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,Ma))) ).

% add_neg_numeral_special(6)
tff(fact_5745_add__neg__numeral__special_I6_J,axiom,
    ! [Ma: num] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Ma))),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(inc,Ma))) ).

% add_neg_numeral_special(6)
tff(fact_5746_add__neg__numeral__special_I6_J,axiom,
    ! [Ma: num] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Ma))),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(inc,Ma))) ).

% add_neg_numeral_special(6)
tff(fact_5747_add__neg__numeral__special_I6_J,axiom,
    ! [Ma: num] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Ma)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(inc,Ma))) ).

% add_neg_numeral_special(6)
tff(fact_5748_diff__numeral__special_I6_J,axiom,
    ! [Ma: num] : aa_real_real(minus_minus_real(aa_num_real(numeral_numeral_real,Ma)),aa_real_real(uminus_uminus_real,one_one_real)) = aa_num_real(numeral_numeral_real,aa_num_num(inc,Ma)) ).

% diff_numeral_special(6)
tff(fact_5749_diff__numeral__special_I6_J,axiom,
    ! [Ma: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Ma)),aa_int_int(uminus_uminus_int,one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(inc,Ma)) ).

% diff_numeral_special(6)
tff(fact_5750_diff__numeral__special_I6_J,axiom,
    ! [Ma: num] : aa_complex_complex(minus_minus_complex(numera6690914467698888265omplex(Ma)),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = numera6690914467698888265omplex(aa_num_num(inc,Ma)) ).

% diff_numeral_special(6)
tff(fact_5751_diff__numeral__special_I6_J,axiom,
    ! [Ma: num] : aa_rat_rat(minus_minus_rat(aa_num_rat(numeral_numeral_rat,Ma)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_num_rat(numeral_numeral_rat,aa_num_num(inc,Ma)) ).

% diff_numeral_special(6)
tff(fact_5752_diff__numeral__special_I6_J,axiom,
    ! [Ma: num] : minus_8373710615458151222nteger(numera6620942414471956472nteger(Ma),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = numera6620942414471956472nteger(aa_num_num(inc,Ma)) ).

% diff_numeral_special(6)
tff(fact_5753_diff__numeral__special_I5_J,axiom,
    ! [Nb: num] : aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,one_one_real)),aa_num_real(numeral_numeral_real,Nb)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(inc,Nb))) ).

% diff_numeral_special(5)
tff(fact_5754_diff__numeral__special_I5_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,Nb))) ).

% diff_numeral_special(5)
tff(fact_5755_diff__numeral__special_I5_J,axiom,
    ! [Nb: num] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),numera6690914467698888265omplex(Nb)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(inc,Nb))) ).

% diff_numeral_special(5)
tff(fact_5756_diff__numeral__special_I5_J,axiom,
    ! [Nb: num] : aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_num_rat(numeral_numeral_rat,Nb)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(inc,Nb))) ).

% diff_numeral_special(5)
tff(fact_5757_diff__numeral__special_I5_J,axiom,
    ! [Nb: num] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),numera6620942414471956472nteger(Nb)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(inc,Nb))) ).

% diff_numeral_special(5)
tff(fact_5758_signed__take__bit__numeral__bit0,axiom,
    ! [L: num,K2: num] : aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(numeral_numeral_nat,L)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,K2))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(pred_numeral,L)),aa_num_int(numeral_numeral_int,K2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% signed_take_bit_numeral_bit0
tff(fact_5759_signed__take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K2: num] : aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K2)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% signed_take_bit_numeral_minus_bit0
tff(fact_5760_Collect__case__prod__mono,axiom,
    ! [A2: fun_int_fun_int_o,B3: fun_int_fun_int_o] :
      ( ord_le77980682512674322_int_o(A2,B3)
     => ord_le2843351958646193337nt_int(collec213857154873943460nt_int(produc4947309494688390418_int_o(A2)),collec213857154873943460nt_int(produc4947309494688390418_int_o(B3))) ) ).

% Collect_case_prod_mono
tff(fact_5761_prod_Odisc__eq__case,axiom,
    ! [Prod: product_prod_int_int] : aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aTP_Lamm_dm),Prod) ).

% prod.disc_eq_case
tff(fact_5762_num__induct,axiom,
    ! [P: fun_num_o,X: num] :
      ( aa_num_o(P,one)
     => ( ! [X4: num] :
            ( aa_num_o(P,X4)
           => aa_num_o(P,aa_num_num(inc,X4)) )
       => aa_num_o(P,X) ) ) ).

% num_induct
tff(fact_5763_add__inc,axiom,
    ! [X: num,Y2: num] : aa_num_num(plus_plus_num(X),aa_num_num(inc,Y2)) = aa_num_num(inc,aa_num_num(plus_plus_num(X),Y2)) ).

% add_inc
tff(fact_5764_numeral__eq__Suc,axiom,
    ! [K2: num] : aa_num_nat(numeral_numeral_nat,K2) = aa_nat_nat(suc,aa_num_nat(pred_numeral,K2)) ).

% numeral_eq_Suc
tff(fact_5765_inc_Osimps_I1_J,axiom,
    aa_num_num(inc,one) = aa_num_num(bit0,one) ).

% inc.simps(1)
tff(fact_5766_inc_Osimps_I2_J,axiom,
    ! [X: num] : aa_num_num(inc,aa_num_num(bit0,X)) = aa_num_num(bit1,X) ).

% inc.simps(2)
tff(fact_5767_inc_Osimps_I3_J,axiom,
    ! [X: num] : aa_num_num(inc,aa_num_num(bit1,X)) = aa_num_num(bit0,aa_num_num(inc,X)) ).

% inc.simps(3)
tff(fact_5768_add__One,axiom,
    ! [X: num] : aa_num_num(plus_plus_num(X),one) = aa_num_num(inc,X) ).

% add_One
tff(fact_5769_mult__inc,axiom,
    ! [X: num,Y2: num] : aa_num_num(times_times_num(X),aa_num_num(inc,Y2)) = aa_num_num(plus_plus_num(aa_num_num(times_times_num(X),Y2)),X) ).

% mult_inc
tff(fact_5770_pred__numeral__def,axiom,
    ! [K2: num] : aa_num_nat(pred_numeral,K2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,K2)),one_one_nat) ).

% pred_numeral_def
tff(fact_5771_numeral__inc,axiom,
    ! [X: num] : numera6690914467698888265omplex(aa_num_num(inc,X)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(X)),one_one_complex) ).

% numeral_inc
tff(fact_5772_numeral__inc,axiom,
    ! [X: num] : aa_num_real(numeral_numeral_real,aa_num_num(inc,X)) = aa_real_real(plus_plus_real(aa_num_real(numeral_numeral_real,X)),one_one_real) ).

% numeral_inc
tff(fact_5773_numeral__inc,axiom,
    ! [X: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(inc,X)) = aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,X)),one_one_rat) ).

% numeral_inc
tff(fact_5774_numeral__inc,axiom,
    ! [X: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(inc,X)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,X)),one_one_nat) ).

% numeral_inc
tff(fact_5775_numeral__inc,axiom,
    ! [X: num] : aa_num_int(numeral_numeral_int,aa_num_num(inc,X)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,X)),one_one_int) ).

% numeral_inc
tff(fact_5776_Divides_Oadjust__div__def,axiom,
    ! [Qr: product_prod_int_int] : aa_Pro6799258434465214121nt_int(adjust_div,Qr) = aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(aTP_Lamm_dn),Qr) ).

% Divides.adjust_div_def
tff(fact_5777_take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K2: num] : aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K2)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,K2))))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% take_bit_numeral_minus_bit1
tff(fact_5778_take__bit__numeral__bit0,axiom,
    ! [L: num,K2: num] : aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,L)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,K2))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(pred_numeral,L)),aa_num_int(numeral_numeral_int,K2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_numeral_bit0
tff(fact_5779_take__bit__numeral__bit0,axiom,
    ! [L: num,K2: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_num_nat(numeral_numeral_nat,L)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K2))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_num_nat(pred_numeral,L)),aa_num_nat(numeral_numeral_nat,K2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% take_bit_numeral_bit0
tff(fact_5780_divmod__nat__def,axiom,
    ! [Ma: nat,Nb: nat] : divmod_nat(Ma,Nb) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),Nb)) ).

% divmod_nat_def
tff(fact_5781_take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K2: num] : aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K2)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_numeral_minus_bit0
tff(fact_5782_take__bit__numeral__bit1,axiom,
    ! [L: num,K2: num] : aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,L)),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(pred_numeral,L)),aa_num_int(numeral_numeral_int,K2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% take_bit_numeral_bit1
tff(fact_5783_take__bit__numeral__bit1,axiom,
    ! [L: num,K2: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_num_nat(numeral_numeral_nat,L)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_num_nat(pred_numeral,L)),aa_num_nat(numeral_numeral_nat,K2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat) ).

% take_bit_numeral_bit1
tff(fact_5784_dbl__inc__simps_I3_J,axiom,
    aa_complex_complex(neg_nu8557863876264182079omplex,one_one_complex) = numera6690914467698888265omplex(aa_num_num(bit1,one)) ).

% dbl_inc_simps(3)
tff(fact_5785_dbl__inc__simps_I3_J,axiom,
    aa_real_real(neg_nu8295874005876285629c_real,one_one_real) = aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)) ).

% dbl_inc_simps(3)
tff(fact_5786_dbl__inc__simps_I3_J,axiom,
    aa_rat_rat(neg_nu5219082963157363817nc_rat,one_one_rat) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,one)) ).

% dbl_inc_simps(3)
tff(fact_5787_dbl__inc__simps_I3_J,axiom,
    aa_int_int(neg_nu5851722552734809277nc_int,one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,one)) ).

% dbl_inc_simps(3)
tff(fact_5788_divmod__BitM__2__eq,axiom,
    ! [Ma: num] : unique5052692396658037445od_int(aa_num_num(bitM,Ma),aa_num_num(bit0,one)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Ma)),one_one_int)),one_one_int) ).

% divmod_BitM_2_eq
tff(fact_5789_of__int__code__if,axiom,
    ! [K2: int] :
      aa_int_real(ring_1_of_int_real,K2) = $ite(
        K2 = zero_zero_int,
        zero_zero_real,
        $ite(
          aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int),
          aa_real_real(uminus_uminus_real,aa_int_real(ring_1_of_int_real,aa_int_int(uminus_uminus_int,K2))),
          $let(
            l: real,
            l:= aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,aa_real_real(plus_plus_real(l),one_one_real)) ) ) ) ).

% of_int_code_if
tff(fact_5790_of__int__code__if,axiom,
    ! [K2: int] :
      aa_int_int(ring_1_of_int_int,K2) = $ite(
        K2 = zero_zero_int,
        zero_zero_int,
        $ite(
          aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int),
          aa_int_int(uminus_uminus_int,aa_int_int(ring_1_of_int_int,aa_int_int(uminus_uminus_int,K2))),
          $let(
            l: int,
            l:= aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,aa_int_int(aa_int_fun_int_int(plus_plus_int,l),one_one_int)) ) ) ) ).

% of_int_code_if
tff(fact_5791_of__int__code__if,axiom,
    ! [K2: int] :
      aa_int_complex(ring_17405671764205052669omplex,K2) = $ite(
        K2 = zero_zero_int,
        zero_zero_complex,
        $ite(
          aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int),
          aa_complex_complex(uminus1482373934393186551omplex,aa_int_complex(ring_17405671764205052669omplex,aa_int_int(uminus_uminus_int,K2))),
          $let(
            l: complex,
            l:= aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_int_complex(ring_17405671764205052669omplex,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,aa_complex_complex(plus_plus_complex(l),one_one_complex)) ) ) ) ).

% of_int_code_if
tff(fact_5792_of__int__code__if,axiom,
    ! [K2: int] :
      aa_int_rat(ring_1_of_int_rat,K2) = $ite(
        K2 = zero_zero_int,
        zero_zero_rat,
        $ite(
          aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int),
          aa_rat_rat(uminus_uminus_rat,aa_int_rat(ring_1_of_int_rat,aa_int_int(uminus_uminus_int,K2))),
          $let(
            l: rat,
            l:= aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,aa_rat_rat(plus_plus_rat(l),one_one_rat)) ) ) ) ).

% of_int_code_if
tff(fact_5793_of__int__code__if,axiom,
    ! [K2: int] :
      ring_18347121197199848620nteger(K2) = $ite(
        K2 = zero_zero_int,
        zero_z3403309356797280102nteger,
        $ite(
          aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int),
          aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,ring_18347121197199848620nteger(aa_int_int(uminus_uminus_int,K2))),
          $let(
            l: code_integer,
            l:= aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),ring_18347121197199848620nteger(aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,plus_p5714425477246183910nteger(l,one_one_Code_integer)) ) ) ) ).

% of_int_code_if
tff(fact_5794_int__ge__less__than2__def,axiom,
    ! [D: int] : int_ge_less_than2(D) = collec213857154873943460nt_int(produc4947309494688390418_int_o(aTP_Lamm_do(D))) ).

% int_ge_less_than2_def
tff(fact_5795_int__ge__less__than__def,axiom,
    ! [D: int] : int_ge_less_than(D) = collec213857154873943460nt_int(produc4947309494688390418_int_o(aTP_Lamm_dp(D))) ).

% int_ge_less_than_def
tff(fact_5796_dbl__dec__simps_I1_J,axiom,
    ! [K2: num] : aa_real_real(neg_nu6075765906172075777c_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,K2))) = aa_real_real(uminus_uminus_real,aa_real_real(neg_nu8295874005876285629c_real,aa_num_real(numeral_numeral_real,K2))) ).

% dbl_dec_simps(1)
tff(fact_5797_dbl__dec__simps_I1_J,axiom,
    ! [K2: num] : aa_int_int(neg_nu3811975205180677377ec_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))) = aa_int_int(uminus_uminus_int,aa_int_int(neg_nu5851722552734809277nc_int,aa_num_int(numeral_numeral_int,K2))) ).

% dbl_dec_simps(1)
tff(fact_5798_dbl__dec__simps_I1_J,axiom,
    ! [K2: num] : aa_complex_complex(neg_nu6511756317524482435omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(K2))) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(neg_nu8557863876264182079omplex,numera6690914467698888265omplex(K2))) ).

% dbl_dec_simps(1)
tff(fact_5799_dbl__dec__simps_I1_J,axiom,
    ! [K2: num] : aa_rat_rat(neg_nu3179335615603231917ec_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,K2))) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(neg_nu5219082963157363817nc_rat,aa_num_rat(numeral_numeral_rat,K2))) ).

% dbl_dec_simps(1)
tff(fact_5800_dbl__dec__simps_I1_J,axiom,
    ! [K2: num] : neg_nu7757733837767384882nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(K2))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,neg_nu5831290666863070958nteger(numera6620942414471956472nteger(K2))) ).

% dbl_dec_simps(1)
tff(fact_5801_dbl__inc__simps_I1_J,axiom,
    ! [K2: num] : aa_real_real(neg_nu8295874005876285629c_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,K2))) = aa_real_real(uminus_uminus_real,aa_real_real(neg_nu6075765906172075777c_real,aa_num_real(numeral_numeral_real,K2))) ).

% dbl_inc_simps(1)
tff(fact_5802_dbl__inc__simps_I1_J,axiom,
    ! [K2: num] : aa_int_int(neg_nu5851722552734809277nc_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))) = aa_int_int(uminus_uminus_int,aa_int_int(neg_nu3811975205180677377ec_int,aa_num_int(numeral_numeral_int,K2))) ).

% dbl_inc_simps(1)
tff(fact_5803_dbl__inc__simps_I1_J,axiom,
    ! [K2: num] : aa_complex_complex(neg_nu8557863876264182079omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(K2))) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(neg_nu6511756317524482435omplex,numera6690914467698888265omplex(K2))) ).

% dbl_inc_simps(1)
tff(fact_5804_dbl__inc__simps_I1_J,axiom,
    ! [K2: num] : aa_rat_rat(neg_nu5219082963157363817nc_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,K2))) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(neg_nu3179335615603231917ec_rat,aa_num_rat(numeral_numeral_rat,K2))) ).

% dbl_inc_simps(1)
tff(fact_5805_dbl__inc__simps_I1_J,axiom,
    ! [K2: num] : neg_nu5831290666863070958nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(K2))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,neg_nu7757733837767384882nteger(numera6620942414471956472nteger(K2))) ).

% dbl_inc_simps(1)
tff(fact_5806_of__int__le__iff,axiom,
    ! [W: int,Z2: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,W)),aa_int_real(ring_1_of_int_real,Z2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Z2) ) ).

% of_int_le_iff
tff(fact_5807_of__int__le__iff,axiom,
    ! [W: int,Z2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,W)),aa_int_rat(ring_1_of_int_rat,Z2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Z2) ) ).

% of_int_le_iff
tff(fact_5808_of__int__le__iff,axiom,
    ! [W: int,Z2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Z2) ) ).

% of_int_le_iff
tff(fact_5809_of__int__numeral,axiom,
    ! [K2: num] : aa_int_complex(ring_17405671764205052669omplex,aa_num_int(numeral_numeral_int,K2)) = numera6690914467698888265omplex(K2) ).

% of_int_numeral
tff(fact_5810_of__int__numeral,axiom,
    ! [K2: num] : aa_int_real(ring_1_of_int_real,aa_num_int(numeral_numeral_int,K2)) = aa_num_real(numeral_numeral_real,K2) ).

% of_int_numeral
tff(fact_5811_of__int__numeral,axiom,
    ! [K2: num] : aa_int_rat(ring_1_of_int_rat,aa_num_int(numeral_numeral_int,K2)) = aa_num_rat(numeral_numeral_rat,K2) ).

% of_int_numeral
tff(fact_5812_of__int__numeral,axiom,
    ! [K2: num] : aa_int_int(ring_1_of_int_int,aa_num_int(numeral_numeral_int,K2)) = aa_num_int(numeral_numeral_int,K2) ).

% of_int_numeral
tff(fact_5813_of__int__eq__numeral__iff,axiom,
    ! [Z2: int,Nb: num] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Z2) = numera6690914467698888265omplex(Nb) )
    <=> ( Z2 = aa_num_int(numeral_numeral_int,Nb) ) ) ).

% of_int_eq_numeral_iff
tff(fact_5814_of__int__eq__numeral__iff,axiom,
    ! [Z2: int,Nb: num] :
      ( ( aa_int_real(ring_1_of_int_real,Z2) = aa_num_real(numeral_numeral_real,Nb) )
    <=> ( Z2 = aa_num_int(numeral_numeral_int,Nb) ) ) ).

% of_int_eq_numeral_iff
tff(fact_5815_of__int__eq__numeral__iff,axiom,
    ! [Z2: int,Nb: num] :
      ( ( aa_int_rat(ring_1_of_int_rat,Z2) = aa_num_rat(numeral_numeral_rat,Nb) )
    <=> ( Z2 = aa_num_int(numeral_numeral_int,Nb) ) ) ).

% of_int_eq_numeral_iff
tff(fact_5816_of__int__eq__numeral__iff,axiom,
    ! [Z2: int,Nb: num] :
      ( ( aa_int_int(ring_1_of_int_int,Z2) = aa_num_int(numeral_numeral_int,Nb) )
    <=> ( Z2 = aa_num_int(numeral_numeral_int,Nb) ) ) ).

% of_int_eq_numeral_iff
tff(fact_5817_of__int__less__iff,axiom,
    ! [W: int,Z2: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,W)),aa_int_real(ring_1_of_int_real,Z2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z2) ) ).

% of_int_less_iff
tff(fact_5818_of__int__less__iff,axiom,
    ! [W: int,Z2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,W)),aa_int_rat(ring_1_of_int_rat,Z2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z2) ) ).

% of_int_less_iff
tff(fact_5819_of__int__less__iff,axiom,
    ! [W: int,Z2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z2) ) ).

% of_int_less_iff
tff(fact_5820_of__int__1,axiom,
    aa_int_complex(ring_17405671764205052669omplex,one_one_int) = one_one_complex ).

% of_int_1
tff(fact_5821_of__int__1,axiom,
    aa_int_int(ring_1_of_int_int,one_one_int) = one_one_int ).

% of_int_1
tff(fact_5822_of__int__1,axiom,
    aa_int_real(ring_1_of_int_real,one_one_int) = one_one_real ).

% of_int_1
tff(fact_5823_of__int__1,axiom,
    aa_int_rat(ring_1_of_int_rat,one_one_int) = one_one_rat ).

% of_int_1
tff(fact_5824_of__int__eq__1__iff,axiom,
    ! [Z2: int] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Z2) = one_one_complex )
    <=> ( Z2 = one_one_int ) ) ).

% of_int_eq_1_iff
tff(fact_5825_of__int__eq__1__iff,axiom,
    ! [Z2: int] :
      ( ( aa_int_int(ring_1_of_int_int,Z2) = one_one_int )
    <=> ( Z2 = one_one_int ) ) ).

% of_int_eq_1_iff
tff(fact_5826_of__int__eq__1__iff,axiom,
    ! [Z2: int] :
      ( ( aa_int_real(ring_1_of_int_real,Z2) = one_one_real )
    <=> ( Z2 = one_one_int ) ) ).

% of_int_eq_1_iff
tff(fact_5827_of__int__eq__1__iff,axiom,
    ! [Z2: int] :
      ( ( aa_int_rat(ring_1_of_int_rat,Z2) = one_one_rat )
    <=> ( Z2 = one_one_int ) ) ).

% of_int_eq_1_iff
tff(fact_5828_of__int__mult,axiom,
    ! [W: int,Z2: int] : aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z2)) = aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,W)),aa_int_real(ring_1_of_int_real,Z2)) ).

% of_int_mult
tff(fact_5829_of__int__mult,axiom,
    ! [W: int,Z2: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z2)) = aa_rat_rat(times_times_rat(aa_int_rat(ring_1_of_int_rat,W)),aa_int_rat(ring_1_of_int_rat,Z2)) ).

% of_int_mult
tff(fact_5830_of__int__mult,axiom,
    ! [W: int,Z2: int] : aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z2)) ).

% of_int_mult
tff(fact_5831_of__int__add,axiom,
    ! [W: int,Z2: int] : aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,W),Z2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z2)) ).

% of_int_add
tff(fact_5832_of__int__add,axiom,
    ! [W: int,Z2: int] : aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,W),Z2)) = aa_real_real(plus_plus_real(aa_int_real(ring_1_of_int_real,W)),aa_int_real(ring_1_of_int_real,Z2)) ).

% of_int_add
tff(fact_5833_of__int__add,axiom,
    ! [W: int,Z2: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,W),Z2)) = aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,W)),aa_int_rat(ring_1_of_int_rat,Z2)) ).

% of_int_add
tff(fact_5834_of__int__power,axiom,
    ! [Z2: int,Nb: nat] : aa_int_rat(ring_1_of_int_rat,aa_nat_int(aa_int_fun_nat_int(power_power_int,Z2),Nb)) = aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,Z2)),Nb) ).

% of_int_power
tff(fact_5835_of__int__power,axiom,
    ! [Z2: int,Nb: nat] : aa_int_real(ring_1_of_int_real,aa_nat_int(aa_int_fun_nat_int(power_power_int,Z2),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,Z2)),Nb) ).

% of_int_power
tff(fact_5836_of__int__power,axiom,
    ! [Z2: int,Nb: nat] : aa_int_int(ring_1_of_int_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,Z2),Nb)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(ring_1_of_int_int,Z2)),Nb) ).

% of_int_power
tff(fact_5837_of__int__power,axiom,
    ! [Z2: int,Nb: nat] : aa_int_complex(ring_17405671764205052669omplex,aa_nat_int(aa_int_fun_nat_int(power_power_int,Z2),Nb)) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_int_complex(ring_17405671764205052669omplex,Z2)),Nb) ).

% of_int_power
tff(fact_5838_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W) = aa_int_rat(ring_1_of_int_rat,X) )
    <=> ( aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W) = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_5839_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W) = aa_int_real(ring_1_of_int_real,X) )
    <=> ( aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W) = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_5840_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(ring_1_of_int_int,B)),W) = aa_int_int(ring_1_of_int_int,X) )
    <=> ( aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W) = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_5841_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_int_complex(ring_17405671764205052669omplex,B)),W) = aa_int_complex(ring_17405671764205052669omplex,X) )
    <=> ( aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W) = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_5842_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( aa_int_rat(ring_1_of_int_rat,X) = aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W) )
    <=> ( X = aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_5843_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( aa_int_real(ring_1_of_int_real,X) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W) )
    <=> ( X = aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_5844_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( aa_int_int(ring_1_of_int_int,X) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(ring_1_of_int_int,B)),W) )
    <=> ( X = aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_5845_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,X) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_int_complex(ring_17405671764205052669omplex,B)),W) )
    <=> ( X = aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_5846_dbl__inc__simps_I2_J,axiom,
    aa_complex_complex(neg_nu8557863876264182079omplex,zero_zero_complex) = one_one_complex ).

% dbl_inc_simps(2)
tff(fact_5847_dbl__inc__simps_I2_J,axiom,
    aa_real_real(neg_nu8295874005876285629c_real,zero_zero_real) = one_one_real ).

% dbl_inc_simps(2)
tff(fact_5848_dbl__inc__simps_I2_J,axiom,
    aa_rat_rat(neg_nu5219082963157363817nc_rat,zero_zero_rat) = one_one_rat ).

% dbl_inc_simps(2)
tff(fact_5849_dbl__inc__simps_I2_J,axiom,
    aa_int_int(neg_nu5851722552734809277nc_int,zero_zero_int) = one_one_int ).

% dbl_inc_simps(2)
tff(fact_5850_dbl__inc__simps_I4_J,axiom,
    aa_real_real(neg_nu8295874005876285629c_real,aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,one_one_real) ).

% dbl_inc_simps(4)
tff(fact_5851_dbl__inc__simps_I4_J,axiom,
    aa_int_int(neg_nu5851722552734809277nc_int,aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% dbl_inc_simps(4)
tff(fact_5852_dbl__inc__simps_I4_J,axiom,
    aa_complex_complex(neg_nu8557863876264182079omplex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% dbl_inc_simps(4)
tff(fact_5853_dbl__inc__simps_I4_J,axiom,
    aa_rat_rat(neg_nu5219082963157363817nc_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% dbl_inc_simps(4)
tff(fact_5854_dbl__inc__simps_I4_J,axiom,
    neg_nu5831290666863070958nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% dbl_inc_simps(4)
tff(fact_5855_dbl__inc__simps_I5_J,axiom,
    ! [K2: num] : aa_complex_complex(neg_nu8557863876264182079omplex,numera6690914467698888265omplex(K2)) = numera6690914467698888265omplex(aa_num_num(bit1,K2)) ).

% dbl_inc_simps(5)
tff(fact_5856_dbl__inc__simps_I5_J,axiom,
    ! [K2: num] : aa_real_real(neg_nu8295874005876285629c_real,aa_num_real(numeral_numeral_real,K2)) = aa_num_real(numeral_numeral_real,aa_num_num(bit1,K2)) ).

% dbl_inc_simps(5)
tff(fact_5857_dbl__inc__simps_I5_J,axiom,
    ! [K2: num] : aa_rat_rat(neg_nu5219082963157363817nc_rat,aa_num_rat(numeral_numeral_rat,K2)) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,K2)) ).

% dbl_inc_simps(5)
tff(fact_5858_dbl__inc__simps_I5_J,axiom,
    ! [K2: num] : aa_int_int(neg_nu5851722552734809277nc_int,aa_num_int(numeral_numeral_int,K2)) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,K2)) ).

% dbl_inc_simps(5)
tff(fact_5859_dbl__dec__simps_I5_J,axiom,
    ! [K2: num] : aa_complex_complex(neg_nu6511756317524482435omplex,numera6690914467698888265omplex(K2)) = numera6690914467698888265omplex(aa_num_num(bitM,K2)) ).

% dbl_dec_simps(5)
tff(fact_5860_dbl__dec__simps_I5_J,axiom,
    ! [K2: num] : aa_real_real(neg_nu6075765906172075777c_real,aa_num_real(numeral_numeral_real,K2)) = aa_num_real(numeral_numeral_real,aa_num_num(bitM,K2)) ).

% dbl_dec_simps(5)
tff(fact_5861_dbl__dec__simps_I5_J,axiom,
    ! [K2: num] : aa_rat_rat(neg_nu3179335615603231917ec_rat,aa_num_rat(numeral_numeral_rat,K2)) = aa_num_rat(numeral_numeral_rat,aa_num_num(bitM,K2)) ).

% dbl_dec_simps(5)
tff(fact_5862_dbl__dec__simps_I5_J,axiom,
    ! [K2: num] : aa_int_int(neg_nu3811975205180677377ec_int,aa_num_int(numeral_numeral_int,K2)) = aa_num_int(numeral_numeral_int,aa_num_num(bitM,K2)) ).

% dbl_dec_simps(5)
tff(fact_5863_pred__numeral__simps_I2_J,axiom,
    ! [K2: num] : aa_num_nat(pred_numeral,aa_num_num(bit0,K2)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bitM,K2)) ).

% pred_numeral_simps(2)
tff(fact_5864_of__int__0__le__iff,axiom,
    ! [Z2: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z2) ) ).

% of_int_0_le_iff
tff(fact_5865_of__int__0__le__iff,axiom,
    ! [Z2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z2) ) ).

% of_int_0_le_iff
tff(fact_5866_of__int__0__le__iff,axiom,
    ! [Z2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z2) ) ).

% of_int_0_le_iff
tff(fact_5867_of__int__le__0__iff,axiom,
    ! [Z2: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z2)),zero_zero_real)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z2),zero_zero_int) ) ).

% of_int_le_0_iff
tff(fact_5868_of__int__le__0__iff,axiom,
    ! [Z2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z2)),zero_zero_rat)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z2),zero_zero_int) ) ).

% of_int_le_0_iff
tff(fact_5869_of__int__le__0__iff,axiom,
    ! [Z2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z2)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z2),zero_zero_int) ) ).

% of_int_le_0_iff
tff(fact_5870_of__int__less__0__iff,axiom,
    ! [Z2: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Z2)),zero_zero_real)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),zero_zero_int) ) ).

% of_int_less_0_iff
tff(fact_5871_of__int__less__0__iff,axiom,
    ! [Z2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z2)),zero_zero_rat)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),zero_zero_int) ) ).

% of_int_less_0_iff
tff(fact_5872_of__int__less__0__iff,axiom,
    ! [Z2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,Z2)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),zero_zero_int) ) ).

% of_int_less_0_iff
tff(fact_5873_of__int__0__less__iff,axiom,
    ! [Z2: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z2) ) ).

% of_int_0_less_iff
tff(fact_5874_of__int__0__less__iff,axiom,
    ! [Z2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z2) ) ).

% of_int_0_less_iff
tff(fact_5875_of__int__0__less__iff,axiom,
    ! [Z2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z2) ) ).

% of_int_0_less_iff
tff(fact_5876_of__int__le__numeral__iff,axiom,
    ! [Z2: int,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z2)),aa_num_real(numeral_numeral_real,Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z2),aa_num_int(numeral_numeral_int,Nb)) ) ).

% of_int_le_numeral_iff
tff(fact_5877_of__int__le__numeral__iff,axiom,
    ! [Z2: int,Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z2)),aa_num_rat(numeral_numeral_rat,Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z2),aa_num_int(numeral_numeral_int,Nb)) ) ).

% of_int_le_numeral_iff
tff(fact_5878_of__int__le__numeral__iff,axiom,
    ! [Z2: int,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z2)),aa_num_int(numeral_numeral_int,Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z2),aa_num_int(numeral_numeral_int,Nb)) ) ).

% of_int_le_numeral_iff
tff(fact_5879_of__int__numeral__le__iff,axiom,
    ! [Nb: num,Z2: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Nb)),aa_int_real(ring_1_of_int_real,Z2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Nb)),Z2) ) ).

% of_int_numeral_le_iff
tff(fact_5880_of__int__numeral__le__iff,axiom,
    ! [Nb: num,Z2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Nb)),aa_int_rat(ring_1_of_int_rat,Z2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Nb)),Z2) ) ).

% of_int_numeral_le_iff
tff(fact_5881_of__int__numeral__le__iff,axiom,
    ! [Nb: num,Z2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Nb)),aa_int_int(ring_1_of_int_int,Z2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Nb)),Z2) ) ).

% of_int_numeral_le_iff
tff(fact_5882_of__int__less__numeral__iff,axiom,
    ! [Z2: int,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Z2)),aa_num_real(numeral_numeral_real,Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),aa_num_int(numeral_numeral_int,Nb)) ) ).

% of_int_less_numeral_iff
tff(fact_5883_of__int__less__numeral__iff,axiom,
    ! [Z2: int,Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z2)),aa_num_rat(numeral_numeral_rat,Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),aa_num_int(numeral_numeral_int,Nb)) ) ).

% of_int_less_numeral_iff
tff(fact_5884_of__int__less__numeral__iff,axiom,
    ! [Z2: int,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,Z2)),aa_num_int(numeral_numeral_int,Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),aa_num_int(numeral_numeral_int,Nb)) ) ).

% of_int_less_numeral_iff
tff(fact_5885_of__int__numeral__less__iff,axiom,
    ! [Nb: num,Z2: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,Nb)),aa_int_real(ring_1_of_int_real,Z2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Nb)),Z2) ) ).

% of_int_numeral_less_iff
tff(fact_5886_of__int__numeral__less__iff,axiom,
    ! [Nb: num,Z2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,Nb)),aa_int_rat(ring_1_of_int_rat,Z2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Nb)),Z2) ) ).

% of_int_numeral_less_iff
tff(fact_5887_of__int__numeral__less__iff,axiom,
    ! [Nb: num,Z2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Nb)),aa_int_int(ring_1_of_int_int,Z2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Nb)),Z2) ) ).

% of_int_numeral_less_iff
tff(fact_5888_of__int__1__le__iff,axiom,
    ! [Z2: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_int_real(ring_1_of_int_real,Z2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Z2) ) ).

% of_int_1_le_iff
tff(fact_5889_of__int__1__le__iff,axiom,
    ! [Z2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_int_rat(ring_1_of_int_rat,Z2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Z2) ) ).

% of_int_1_le_iff
tff(fact_5890_of__int__1__le__iff,axiom,
    ! [Z2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_int_int(ring_1_of_int_int,Z2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Z2) ) ).

% of_int_1_le_iff
tff(fact_5891_of__int__le__1__iff,axiom,
    ! [Z2: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z2)),one_one_real)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z2),one_one_int) ) ).

% of_int_le_1_iff
tff(fact_5892_of__int__le__1__iff,axiom,
    ! [Z2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z2)),one_one_rat)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z2),one_one_int) ) ).

% of_int_le_1_iff
tff(fact_5893_of__int__le__1__iff,axiom,
    ! [Z2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z2)),one_one_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z2),one_one_int) ) ).

% of_int_le_1_iff
tff(fact_5894_of__int__less__1__iff,axiom,
    ! [Z2: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Z2)),one_one_real)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),one_one_int) ) ).

% of_int_less_1_iff
tff(fact_5895_of__int__less__1__iff,axiom,
    ! [Z2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z2)),one_one_rat)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),one_one_int) ) ).

% of_int_less_1_iff
tff(fact_5896_of__int__less__1__iff,axiom,
    ! [Z2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,Z2)),one_one_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),one_one_int) ) ).

% of_int_less_1_iff
tff(fact_5897_of__int__1__less__iff,axiom,
    ! [Z2: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_int_real(ring_1_of_int_real,Z2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Z2) ) ).

% of_int_1_less_iff
tff(fact_5898_of__int__1__less__iff,axiom,
    ! [Z2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_int_rat(ring_1_of_int_rat,Z2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Z2) ) ).

% of_int_1_less_iff
tff(fact_5899_of__int__1__less__iff,axiom,
    ! [Z2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(ring_1_of_int_int,Z2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Z2) ) ).

% of_int_1_less_iff
tff(fact_5900_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W)),aa_int_real(ring_1_of_int_real,X))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W)),X) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_5901_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W)),aa_int_rat(ring_1_of_int_rat,X))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W)),X) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_5902_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(ring_1_of_int_int,B)),W)),aa_int_int(ring_1_of_int_int,X))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W)),X) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_5903_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W)) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_5904_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,X)),aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W)) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_5905_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,X)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(ring_1_of_int_int,B)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W)) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_5906_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,Nb: nat,Y2: int] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,numera6690914467698888265omplex(X)),Nb) = aa_int_complex(ring_17405671764205052669omplex,Y2) )
    <=> ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X)),Nb) = Y2 ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_5907_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,Nb: nat,Y2: int] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,X)),Nb) = aa_int_real(ring_1_of_int_real,Y2) )
    <=> ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X)),Nb) = Y2 ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_5908_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,Nb: nat,Y2: int] :
      ( ( aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X)),Nb) = aa_int_rat(ring_1_of_int_rat,Y2) )
    <=> ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X)),Nb) = Y2 ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_5909_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,Nb: nat,Y2: int] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X)),Nb) = aa_int_int(ring_1_of_int_int,Y2) )
    <=> ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X)),Nb) = Y2 ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_5910_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y2: int,X: num,Nb: nat] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Y2) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,numera6690914467698888265omplex(X)),Nb) )
    <=> ( Y2 = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X)),Nb) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_5911_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y2: int,X: num,Nb: nat] :
      ( ( aa_int_real(ring_1_of_int_real,Y2) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,X)),Nb) )
    <=> ( Y2 = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X)),Nb) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_5912_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y2: int,X: num,Nb: nat] :
      ( ( aa_int_rat(ring_1_of_int_rat,Y2) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X)),Nb) )
    <=> ( Y2 = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X)),Nb) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_5913_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y2: int,X: num,Nb: nat] :
      ( ( aa_int_int(ring_1_of_int_int,Y2) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X)),Nb) )
    <=> ( Y2 = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X)),Nb) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_5914_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W)),aa_int_real(ring_1_of_int_real,X))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W)),X) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_5915_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W)),aa_int_rat(ring_1_of_int_rat,X))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W)),X) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_5916_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(ring_1_of_int_int,B)),W)),aa_int_int(ring_1_of_int_int,X))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W)),X) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_5917_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,X),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W)) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_5918_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,X)),aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,X),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W)) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_5919_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,X)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(ring_1_of_int_int,B)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,X),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),W)) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_5920_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,Nb: nat,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,X)),Nb)),aa_int_real(ring_1_of_int_real,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X)),Nb)),A) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_5921_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,Nb: nat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X)),Nb)),aa_int_rat(ring_1_of_int_rat,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X)),Nb)),A) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_5922_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,Nb: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X)),Nb)),aa_int_int(ring_1_of_int_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X)),Nb)),A) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_5923_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,X)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X)),Nb)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_5924_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X)),Nb)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_5925_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X)),Nb)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_5926_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,Nb: nat,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,X)),Nb)),aa_int_real(ring_1_of_int_real,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X)),Nb)),A) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_5927_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,Nb: nat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X)),Nb)),aa_int_rat(ring_1_of_int_rat,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X)),Nb)),A) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_5928_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,Nb: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X)),Nb)),aa_int_int(ring_1_of_int_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X)),Nb)),A) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_5929_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,X)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X)),Nb)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_5930_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X)),Nb)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_5931_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X)),Nb)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_5932_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,Nb: nat,Y2: int] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,X))),Nb) = aa_int_real(ring_1_of_int_real,Y2) )
    <=> ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Nb) = Y2 ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_5933_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,Nb: nat,Y2: int] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Nb) = aa_int_int(ring_1_of_int_int,Y2) )
    <=> ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Nb) = Y2 ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_5934_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,Nb: nat,Y2: int] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(X))),Nb) = aa_int_complex(ring_17405671764205052669omplex,Y2) )
    <=> ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Nb) = Y2 ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_5935_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,Nb: nat,Y2: int] :
      ( ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,X))),Nb) = aa_int_rat(ring_1_of_int_rat,Y2) )
    <=> ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Nb) = Y2 ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_5936_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,Nb: nat,Y2: int] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(X))),Nb) = ring_18347121197199848620nteger(Y2) )
    <=> ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Nb) = Y2 ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_5937_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y2: int,X: num,Nb: nat] :
      ( ( aa_int_real(ring_1_of_int_real,Y2) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,X))),Nb) )
    <=> ( Y2 = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Nb) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_5938_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y2: int,X: num,Nb: nat] :
      ( ( aa_int_int(ring_1_of_int_int,Y2) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Nb) )
    <=> ( Y2 = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Nb) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_5939_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y2: int,X: num,Nb: nat] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Y2) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(X))),Nb) )
    <=> ( Y2 = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Nb) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_5940_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y2: int,X: num,Nb: nat] :
      ( ( aa_int_rat(ring_1_of_int_rat,Y2) = aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,X))),Nb) )
    <=> ( Y2 = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Nb) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_5941_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y2: int,X: num,Nb: nat] :
      ( ( ring_18347121197199848620nteger(Y2) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(X))),Nb) )
    <=> ( Y2 = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Nb) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_5942_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,Nb: nat,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,X))),Nb)),aa_int_real(ring_1_of_int_real,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Nb)),A) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_5943_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,Nb: nat,A: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(X))),Nb)),ring_18347121197199848620nteger(A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Nb)),A) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_5944_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,Nb: nat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,X))),Nb)),aa_int_rat(ring_1_of_int_rat,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Nb)),A) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_5945_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,Nb: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Nb)),aa_int_int(ring_1_of_int_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Nb)),A) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_5946_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,X))),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Nb)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_5947_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,ring_18347121197199848620nteger(A)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(X))),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Nb)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_5948_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,X))),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Nb)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_5949_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Nb)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_5950_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,Nb: nat,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,X))),Nb)),aa_int_real(ring_1_of_int_real,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Nb)),A) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_5951_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,Nb: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Nb)),aa_int_int(ring_1_of_int_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Nb)),A) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_5952_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,Nb: nat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,X))),Nb)),aa_int_rat(ring_1_of_int_rat,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Nb)),A) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_5953_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,Nb: nat,A: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(X))),Nb)),ring_18347121197199848620nteger(A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Nb)),A) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_5954_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,X))),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Nb)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_5955_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Nb)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_5956_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,X))),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Nb)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_5957_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,ring_18347121197199848620nteger(A)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(X))),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,X))),Nb)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_5958_mult__of__int__commute,axiom,
    ! [X: int,Y2: real] : aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,X)),Y2) = aa_real_real(times_times_real(Y2),aa_int_real(ring_1_of_int_real,X)) ).

% mult_of_int_commute
tff(fact_5959_mult__of__int__commute,axiom,
    ! [X: int,Y2: rat] : aa_rat_rat(times_times_rat(aa_int_rat(ring_1_of_int_rat,X)),Y2) = aa_rat_rat(times_times_rat(Y2),aa_int_rat(ring_1_of_int_rat,X)) ).

% mult_of_int_commute
tff(fact_5960_mult__of__int__commute,axiom,
    ! [X: int,Y2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(ring_1_of_int_int,X)),Y2) = aa_int_int(aa_int_fun_int_int(times_times_int,Y2),aa_int_int(ring_1_of_int_int,X)) ).

% mult_of_int_commute
tff(fact_5961_semiring__norm_I26_J,axiom,
    aa_num_num(bitM,one) = one ).

% semiring_norm(26)
tff(fact_5962_take__bit__of__int,axiom,
    ! [Nb: nat,K2: int] : aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(ring_1_of_int_int,K2)) = aa_int_int(ring_1_of_int_int,aa_int_int(bit_se2923211474154528505it_int(Nb),K2)) ).

% take_bit_of_int
tff(fact_5963_semiring__norm_I28_J,axiom,
    ! [Nb: num] : aa_num_num(bitM,aa_num_num(bit1,Nb)) = aa_num_num(bit1,aa_num_num(bit0,Nb)) ).

% semiring_norm(28)
tff(fact_5964_semiring__norm_I27_J,axiom,
    ! [Nb: num] : aa_num_num(bitM,aa_num_num(bit0,Nb)) = aa_num_num(bit1,aa_num_num(bitM,Nb)) ).

% semiring_norm(27)
tff(fact_5965_inc__BitM__eq,axiom,
    ! [Nb: num] : aa_num_num(inc,aa_num_num(bitM,Nb)) = aa_num_num(bit0,Nb) ).

% inc_BitM_eq
tff(fact_5966_BitM__inc__eq,axiom,
    ! [Nb: num] : aa_num_num(bitM,aa_num_num(inc,Nb)) = aa_num_num(bit1,Nb) ).

% BitM_inc_eq
tff(fact_5967_real__of__int__div4,axiom,
    ! [Nb: int,X: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,Nb),X))),aa_real_real(divide_divide_real(aa_int_real(ring_1_of_int_real,Nb)),aa_int_real(ring_1_of_int_real,X))) ).

% real_of_int_div4
tff(fact_5968_real__of__int__div,axiom,
    ! [D: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),Nb)
     => ( aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,Nb),D)) = aa_real_real(divide_divide_real(aa_int_real(ring_1_of_int_real,Nb)),aa_int_real(ring_1_of_int_real,D)) ) ) ).

% real_of_int_div
tff(fact_5969_eval__nat__numeral_I2_J,axiom,
    ! [Nb: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb)) = aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,aa_num_num(bitM,Nb))) ).

% eval_nat_numeral(2)
tff(fact_5970_BitM__plus__one,axiom,
    ! [Nb: num] : aa_num_num(plus_plus_num(aa_num_num(bitM,Nb)),one) = aa_num_num(bit0,Nb) ).

% BitM_plus_one
tff(fact_5971_one__plus__BitM,axiom,
    ! [Nb: num] : aa_num_num(plus_plus_num(one),aa_num_num(bitM,Nb)) = aa_num_num(bit0,Nb) ).

% one_plus_BitM
tff(fact_5972_of__int__nonneg,axiom,
    ! [Z2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z2)) ) ).

% of_int_nonneg
tff(fact_5973_of__int__nonneg,axiom,
    ! [Z2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z2)) ) ).

% of_int_nonneg
tff(fact_5974_of__int__nonneg,axiom,
    ! [Z2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z2)) ) ).

% of_int_nonneg
tff(fact_5975_of__int__pos,axiom,
    ! [Z2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z2)) ) ).

% of_int_pos
tff(fact_5976_of__int__pos,axiom,
    ! [Z2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z2)) ) ).

% of_int_pos
tff(fact_5977_of__int__pos,axiom,
    ! [Z2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z2)) ) ).

% of_int_pos
tff(fact_5978_of__int__neg__numeral,axiom,
    ! [K2: num] : aa_int_real(ring_1_of_int_real,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,K2)) ).

% of_int_neg_numeral
tff(fact_5979_of__int__neg__numeral,axiom,
    ! [K2: num] : aa_int_int(ring_1_of_int_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2)) ).

% of_int_neg_numeral
tff(fact_5980_of__int__neg__numeral,axiom,
    ! [K2: num] : aa_int_complex(ring_17405671764205052669omplex,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(K2)) ).

% of_int_neg_numeral
tff(fact_5981_of__int__neg__numeral,axiom,
    ! [K2: num] : aa_int_rat(ring_1_of_int_rat,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,K2)) ).

% of_int_neg_numeral
tff(fact_5982_of__int__neg__numeral,axiom,
    ! [K2: num] : ring_18347121197199848620nteger(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(K2)) ).

% of_int_neg_numeral
tff(fact_5983_int__le__real__less,axiom,
    ! [Nb: int,Ma: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Nb),Ma)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Nb)),aa_real_real(plus_plus_real(aa_int_real(ring_1_of_int_real,Ma)),one_one_real)) ) ).

% int_le_real_less
tff(fact_5984_int__less__real__le,axiom,
    ! [Nb: int,Ma: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Nb),Ma)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_int_real(ring_1_of_int_real,Nb)),one_one_real)),aa_int_real(ring_1_of_int_real,Ma)) ) ).

% int_less_real_le
tff(fact_5985_dbl__inc__def,axiom,
    ! [X: complex] : aa_complex_complex(neg_nu8557863876264182079omplex,X) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(X),X)),one_one_complex) ).

% dbl_inc_def
tff(fact_5986_dbl__inc__def,axiom,
    ! [X: real] : aa_real_real(neg_nu8295874005876285629c_real,X) = aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(X),X)),one_one_real) ).

% dbl_inc_def
tff(fact_5987_dbl__inc__def,axiom,
    ! [X: rat] : aa_rat_rat(neg_nu5219082963157363817nc_rat,X) = aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(X),X)),one_one_rat) ).

% dbl_inc_def
tff(fact_5988_dbl__inc__def,axiom,
    ! [X: int] : aa_int_int(neg_nu5851722552734809277nc_int,X) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),X)),one_one_int) ).

% dbl_inc_def
tff(fact_5989_real__of__int__div__aux,axiom,
    ! [X: int,D: int] : aa_real_real(divide_divide_real(aa_int_real(ring_1_of_int_real,X)),aa_int_real(ring_1_of_int_real,D)) = aa_real_real(plus_plus_real(aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),D))),aa_real_real(divide_divide_real(aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X),D))),aa_int_real(ring_1_of_int_real,D))) ).

% real_of_int_div_aux
tff(fact_5990_numeral__BitM,axiom,
    ! [Nb: num] : numera6690914467698888265omplex(aa_num_num(bitM,Nb)) = aa_complex_complex(minus_minus_complex(numera6690914467698888265omplex(aa_num_num(bit0,Nb))),one_one_complex) ).

% numeral_BitM
tff(fact_5991_numeral__BitM,axiom,
    ! [Nb: num] : aa_num_real(numeral_numeral_real,aa_num_num(bitM,Nb)) = aa_real_real(minus_minus_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,Nb))),one_one_real) ).

% numeral_BitM
tff(fact_5992_numeral__BitM,axiom,
    ! [Nb: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(bitM,Nb)) = aa_rat_rat(minus_minus_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,Nb))),one_one_rat) ).

% numeral_BitM
tff(fact_5993_numeral__BitM,axiom,
    ! [Nb: num] : aa_num_int(numeral_numeral_int,aa_num_num(bitM,Nb)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))),one_one_int) ).

% numeral_BitM
tff(fact_5994_odd__numeral__BitM,axiom,
    ! [W: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),numera6620942414471956472nteger(aa_num_num(bitM,W))) ).

% odd_numeral_BitM
tff(fact_5995_odd__numeral__BitM,axiom,
    ! [W: num] : ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,aa_num_num(bitM,W))) ).

% odd_numeral_BitM
tff(fact_5996_odd__numeral__BitM,axiom,
    ! [W: num] : ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_int(numeral_numeral_int,aa_num_num(bitM,W))) ).

% odd_numeral_BitM
tff(fact_5997_real__of__int__div2,axiom,
    ! [Nb: int,X: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_int_real(ring_1_of_int_real,Nb)),aa_int_real(ring_1_of_int_real,X))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,Nb),X)))) ).

% real_of_int_div2
tff(fact_5998_real__of__int__div3,axiom,
    ! [Nb: int,X: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_int_real(ring_1_of_int_real,Nb)),aa_int_real(ring_1_of_int_real,X))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,Nb),X)))),one_one_real) ).

% real_of_int_div3
tff(fact_5999_even__of__int__iff,axiom,
    ! [K2: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),ring_18347121197199848620nteger(K2))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K2) ) ).

% even_of_int_iff
tff(fact_6000_even__of__int__iff,axiom,
    ! [K2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(ring_1_of_int_int,K2))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K2) ) ).

% even_of_int_iff
tff(fact_6001_floor__exists,axiom,
    ! [X: real] :
    ? [Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),X)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),one_one_int))) ) ).

% floor_exists
tff(fact_6002_floor__exists,axiom,
    ! [X: rat] :
    ? [Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),X)
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),one_one_int))) ) ).

% floor_exists
tff(fact_6003_floor__exists1,axiom,
    ! [X: real] :
    ? [X4: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,X4)),X)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),one_one_int)))
      & ! [Y5: int] :
          ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Y5)),X)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,Y5),one_one_int))) )
         => ( Y5 = X4 ) ) ) ).

% floor_exists1
tff(fact_6004_floor__exists1,axiom,
    ! [X: rat] :
    ? [X4: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,X4)),X)
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),one_one_int)))
      & ! [Y5: int] :
          ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Y5)),X)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,Y5),one_one_int))) )
         => ( Y5 = X4 ) ) ) ).

% floor_exists1
tff(fact_6005_pred__subset__eq2,axiom,
    ! [R4: set_Pr448751882837621926eger_o,S3: set_Pr448751882837621926eger_o] :
      ( ord_le2909477399203599485un_o_o(aTP_Lamm_dq(R4),aTP_Lamm_dq(S3))
    <=> ord_le8980329558974975238eger_o(R4,S3) ) ).

% pred_subset_eq2
tff(fact_6006_pred__subset__eq2,axiom,
    ! [R4: set_Pr8218934625190621173um_num,S3: set_Pr8218934625190621173um_num] :
      ( ord_le3306795289226699318_num_o(aTP_Lamm_dr(R4),aTP_Lamm_dr(S3))
    <=> ord_le880128212290418581um_num(R4,S3) ) ).

% pred_subset_eq2
tff(fact_6007_pred__subset__eq2,axiom,
    ! [R4: set_Pr6200539531224447659at_num,S3: set_Pr6200539531224447659at_num] :
      ( ord_le1526520017675560704_num_o(aTP_Lamm_ds(R4),aTP_Lamm_ds(S3))
    <=> ord_le8085105155179020875at_num(R4,S3) ) ).

% pred_subset_eq2
tff(fact_6008_pred__subset__eq2,axiom,
    ! [R4: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ord_le768339454706049738_nat_o(aTP_Lamm_dt(R4),aTP_Lamm_dt(S3))
    <=> ord_le3146513528884898305at_nat(R4,S3) ) ).

% pred_subset_eq2
tff(fact_6009_pred__subset__eq2,axiom,
    ! [R4: set_Pr958786334691620121nt_int,S3: set_Pr958786334691620121nt_int] :
      ( ord_le77980682512674322_int_o(aTP_Lamm_du(R4),aTP_Lamm_du(S3))
    <=> ord_le2843351958646193337nt_int(R4,S3) ) ).

% pred_subset_eq2
tff(fact_6010_accp__subset,axiom,
    ! [R1: fun_Pr8877776288780692068_nat_o,R22: fun_Pr8877776288780692068_nat_o] :
      ( ord_le4038806025055212036_nat_o(R1,R22)
     => ord_le1769015658490769310_nat_o(accp_P2887432264394892906BT_nat(R22),accp_P2887432264394892906BT_nat(R1)) ) ).

% accp_subset
tff(fact_6011_accp__subset,axiom,
    ! [R1: fun_Pr4799464687642144302_num_o,R22: fun_Pr4799464687642144302_num_o] :
      ( ord_le6421466325430216142_num_o(R1,R22)
     => ord_le3107050023622517222_num_o(accp_P3113834385874906142um_num(R22),accp_P3113834385874906142um_num(R1)) ) ).

% accp_subset
tff(fact_6012_accp__subset,axiom,
    ! [R1: fun_Pr2028404973800837934_nat_o,R22: fun_Pr2028404973800837934_nat_o] :
      ( ord_le3650406611588909774_nat_o(R1,R22)
     => aa_fun8408151826566693824at_o_o(ord_le1572679713340831354_nat_o(accp_P4275260045618599050at_nat(R22)),accp_P4275260045618599050at_nat(R1)) ) ).

% accp_subset
tff(fact_6013_accp__subset,axiom,
    ! [R1: fun_Pr9018117161326028078_int_o,R22: fun_Pr9018117161326028078_int_o] :
      ( ord_le1416746762259324110_int_o(R1,R22)
     => ord_le14110778710936002_int_o(accp_P1096762738010456898nt_int(R22),accp_P1096762738010456898nt_int(R1)) ) ).

% accp_subset
tff(fact_6014_accp__subset,axiom,
    ! [R1: fun_nat_fun_nat_o,R22: fun_nat_fun_nat_o] :
      ( ord_le768339454706049738_nat_o(R1,R22)
     => ord_le7240017405068453697_nat_o(accp_nat(R22),accp_nat(R1)) ) ).

% accp_subset
tff(fact_6015_signed__take__bit__eq__take__bit__minus,axiom,
    ! [Nb: nat,K2: int] : aa_int_int(bit_ri631733984087533419it_int(Nb),K2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),K2)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,Nb))),aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(bit_se1146084159140164899it_int(K2),Nb)))) ).

% signed_take_bit_eq_take_bit_minus
tff(fact_6016_mask__numeral,axiom,
    ! [Nb: num] : aa_nat_nat(bit_se2002935070580805687sk_nat,aa_num_nat(numeral_numeral_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(bit_se2002935070580805687sk_nat,aa_num_nat(pred_numeral,Nb)))) ).

% mask_numeral
tff(fact_6017_mask__numeral,axiom,
    ! [Nb: num] : aa_nat_int(bit_se2000444600071755411sk_int,aa_num_nat(numeral_numeral_nat,Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(bit_se2000444600071755411sk_int,aa_num_nat(pred_numeral,Nb)))) ).

% mask_numeral
tff(fact_6018_mask__nat__positive__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(bit_se2002935070580805687sk_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% mask_nat_positive_iff
tff(fact_6019_mask__0,axiom,
    aa_nat_nat(bit_se2002935070580805687sk_nat,zero_zero_nat) = zero_zero_nat ).

% mask_0
tff(fact_6020_mask__0,axiom,
    aa_nat_int(bit_se2000444600071755411sk_int,zero_zero_nat) = zero_zero_int ).

% mask_0
tff(fact_6021_mask__eq__0__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_nat(bit_se2002935070580805687sk_nat,Nb) = zero_zero_nat )
    <=> ( Nb = zero_zero_nat ) ) ).

% mask_eq_0_iff
tff(fact_6022_mask__eq__0__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_int(bit_se2000444600071755411sk_int,Nb) = zero_zero_int )
    <=> ( Nb = zero_zero_nat ) ) ).

% mask_eq_0_iff
tff(fact_6023_bit__numeral__Bit0__Suc__iff,axiom,
    ! [Ma: num,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Ma))),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,Ma)),Nb) ) ).

% bit_numeral_Bit0_Suc_iff
tff(fact_6024_bit__numeral__Bit0__Suc__iff,axiom,
    ! [Ma: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Ma))),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,Ma)),Nb) ) ).

% bit_numeral_Bit0_Suc_iff
tff(fact_6025_bit__numeral__Bit1__Suc__iff,axiom,
    ! [Ma: num,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Ma))),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,Ma)),Nb) ) ).

% bit_numeral_Bit1_Suc_iff
tff(fact_6026_bit__numeral__Bit1__Suc__iff,axiom,
    ! [Ma: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Ma))),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,Ma)),Nb) ) ).

% bit_numeral_Bit1_Suc_iff
tff(fact_6027_mask__Suc__0,axiom,
    aa_nat_nat(bit_se2002935070580805687sk_nat,aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% mask_Suc_0
tff(fact_6028_mask__Suc__0,axiom,
    aa_nat_int(bit_se2000444600071755411sk_int,aa_nat_nat(suc,zero_zero_nat)) = one_one_int ).

% mask_Suc_0
tff(fact_6029_take__bit__minus__one__eq__mask,axiom,
    ! [Nb: nat] : bit_se1745604003318907178nteger(Nb,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_nat_Code_integer(bit_se2119862282449309892nteger,Nb) ).

% take_bit_minus_one_eq_mask
tff(fact_6030_take__bit__minus__one__eq__mask,axiom,
    ! [Nb: nat] : aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(uminus_uminus_int,one_one_int)) = aa_nat_int(bit_se2000444600071755411sk_int,Nb) ).

% take_bit_minus_one_eq_mask
tff(fact_6031_signed__take__bit__nonnegative__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_ri631733984087533419it_int(Nb),K2))
    <=> ~ aa_nat_o(bit_se1146084159140164899it_int(K2),Nb) ) ).

% signed_take_bit_nonnegative_iff
tff(fact_6032_signed__take__bit__negative__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_ri631733984087533419it_int(Nb),K2)),zero_zero_int)
    <=> aa_nat_o(bit_se1146084159140164899it_int(K2),Nb) ) ).

% signed_take_bit_negative_iff
tff(fact_6033_bit__numeral__simps_I2_J,axiom,
    ! [W: num,Nb: num] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))),aa_num_nat(numeral_numeral_nat,Nb))
    <=> aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,W)),aa_num_nat(pred_numeral,Nb)) ) ).

% bit_numeral_simps(2)
tff(fact_6034_bit__numeral__simps_I2_J,axiom,
    ! [W: num,Nb: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))),aa_num_nat(numeral_numeral_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,W)),aa_num_nat(pred_numeral,Nb)) ) ).

% bit_numeral_simps(2)
tff(fact_6035_bit__minus__numeral__Bit0__Suc__iff,axiom,
    ! [W: num,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,W)))),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),Nb) ) ).

% bit_minus_numeral_Bit0_Suc_iff
tff(fact_6036_bit__numeral__simps_I3_J,axiom,
    ! [W: num,Nb: num] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,W))),aa_num_nat(numeral_numeral_nat,Nb))
    <=> aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,W)),aa_num_nat(pred_numeral,Nb)) ) ).

% bit_numeral_simps(3)
tff(fact_6037_bit__numeral__simps_I3_J,axiom,
    ! [W: num,Nb: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))),aa_num_nat(numeral_numeral_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,W)),aa_num_nat(pred_numeral,Nb)) ) ).

% bit_numeral_simps(3)
tff(fact_6038_bit__minus__numeral__Bit1__Suc__iff,axiom,
    ! [W: num,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,W)))),aa_nat_nat(suc,Nb))
    <=> ~ aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,W)),Nb) ) ).

% bit_minus_numeral_Bit1_Suc_iff
tff(fact_6039_bit__0,axiom,
    ! [A: code_integer] :
      ( aa_nat_o(bit_se9216721137139052372nteger(A),zero_zero_nat)
    <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A) ) ).

% bit_0
tff(fact_6040_bit__0,axiom,
    ! [A: int] :
      ( aa_nat_o(bit_se1146084159140164899it_int(A),zero_zero_nat)
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) ) ).

% bit_0
tff(fact_6041_bit__0,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),zero_zero_nat)
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A) ) ).

% bit_0
tff(fact_6042_bit__minus__numeral__int_I1_J,axiom,
    ! [W: num,Nb: num] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,W)))),aa_num_nat(numeral_numeral_nat,Nb))
    <=> aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),aa_num_nat(pred_numeral,Nb)) ) ).

% bit_minus_numeral_int(1)
tff(fact_6043_bit__minus__numeral__int_I2_J,axiom,
    ! [W: num,Nb: num] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,W)))),aa_num_nat(numeral_numeral_nat,Nb))
    <=> ~ aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,W)),aa_num_nat(pred_numeral,Nb)) ) ).

% bit_minus_numeral_int(2)
tff(fact_6044_bit__mod__2__iff,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_nat_o(bit_se9216721137139052372nteger(modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))),Nb)
    <=> ( ( Nb = zero_zero_nat )
        & ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A) ) ) ).

% bit_mod_2_iff
tff(fact_6045_bit__mod__2__iff,axiom,
    ! [A: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),Nb)
    <=> ( ( Nb = zero_zero_nat )
        & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) ) ) ).

% bit_mod_2_iff
tff(fact_6046_bit__mod__2__iff,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Nb)
    <=> ( ( Nb = zero_zero_nat )
        & ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A) ) ) ).

% bit_mod_2_iff
tff(fact_6047_of__int__mask__eq,axiom,
    ! [Nb: nat] : aa_int_int(ring_1_of_int_int,aa_nat_int(bit_se2000444600071755411sk_int,Nb)) = aa_nat_int(bit_se2000444600071755411sk_int,Nb) ).

% of_int_mask_eq
tff(fact_6048_bit__numeral__iff,axiom,
    ! [Ma: num,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,Ma)),Nb)
    <=> aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,Ma)),Nb) ) ).

% bit_numeral_iff
tff(fact_6049_bit__numeral__iff,axiom,
    ! [Ma: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,Ma)),Nb)
    <=> aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,Ma)),Nb) ) ).

% bit_numeral_iff
tff(fact_6050_bit__disjunctive__add__iff,axiom,
    ! [A: int,B: int,Nb: nat] :
      ( ! [N: nat] :
          ( ~ aa_nat_o(bit_se1146084159140164899it_int(A),N)
          | ~ aa_nat_o(bit_se1146084159140164899it_int(B),N) )
     => ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),Nb)
      <=> ( aa_nat_o(bit_se1146084159140164899it_int(A),Nb)
          | aa_nat_o(bit_se1146084159140164899it_int(B),Nb) ) ) ) ).

% bit_disjunctive_add_iff
tff(fact_6051_bit__disjunctive__add__iff,axiom,
    ! [A: nat,B: nat,Nb: nat] :
      ( ! [N: nat] :
          ( ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),N)
          | ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,B),N) )
     => ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),Nb)
      <=> ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Nb)
          | aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,B),Nb) ) ) ) ).

% bit_disjunctive_add_iff
tff(fact_6052_bit__unset__bit__iff,axiom,
    ! [Ma: nat,A: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Ma),A)),Nb)
    <=> ( aa_nat_o(bit_se1146084159140164899it_int(A),Nb)
        & ( Ma != Nb ) ) ) ).

% bit_unset_bit_iff
tff(fact_6053_bit__unset__bit__iff,axiom,
    ! [Ma: nat,A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,Ma),A)),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Nb)
        & ( Ma != Nb ) ) ) ).

% bit_unset_bit_iff
tff(fact_6054_less__eq__mask,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(bit_se2002935070580805687sk_nat,Nb)) ).

% less_eq_mask
tff(fact_6055_not__bit__1__Suc,axiom,
    ! [Nb: nat] : ~ aa_nat_o(bit_se1146084159140164899it_int(one_one_int),aa_nat_nat(suc,Nb)) ).

% not_bit_1_Suc
tff(fact_6056_not__bit__1__Suc,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,one_one_nat),aa_nat_nat(suc,Nb)) ).

% not_bit_1_Suc
tff(fact_6057_bit__1__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(one_one_int),Nb)
    <=> ( Nb = zero_zero_nat ) ) ).

% bit_1_iff
tff(fact_6058_bit__1__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,one_one_nat),Nb)
    <=> ( Nb = zero_zero_nat ) ) ).

% bit_1_iff
tff(fact_6059_bit__numeral__simps_I1_J,axiom,
    ! [Nb: num] : ~ aa_nat_o(bit_se1146084159140164899it_int(one_one_int),aa_num_nat(numeral_numeral_nat,Nb)) ).

% bit_numeral_simps(1)
tff(fact_6060_bit__numeral__simps_I1_J,axiom,
    ! [Nb: num] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Nb)) ).

% bit_numeral_simps(1)
tff(fact_6061_bit__take__bit__iff,axiom,
    ! [Ma: nat,A: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(bit_se2923211474154528505it_int(Ma),A)),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Ma)
        & aa_nat_o(bit_se1146084159140164899it_int(A),Nb) ) ) ).

% bit_take_bit_iff
tff(fact_6062_bit__take__bit__iff,axiom,
    ! [Ma: nat,A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Ma),A)),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Ma)
        & aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Nb) ) ) ).

% bit_take_bit_iff
tff(fact_6063_bit__of__bool__iff,axiom,
    ! [B: $o,Nb: nat] :
      ( aa_nat_o(bit_se9216721137139052372nteger(zero_n356916108424825756nteger((B))),Nb)
    <=> ( (B)
        & ( Nb = zero_zero_nat ) ) ) ).

% bit_of_bool_iff
tff(fact_6064_bit__of__bool__iff,axiom,
    ! [B: $o,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_o_int(zero_n2684676970156552555ol_int,(B))),Nb)
    <=> ( (B)
        & ( Nb = zero_zero_nat ) ) ) ).

% bit_of_bool_iff
tff(fact_6065_bit__of__bool__iff,axiom,
    ! [B: $o,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(B))),Nb)
    <=> ( (B)
        & ( Nb = zero_zero_nat ) ) ) ).

% bit_of_bool_iff
tff(fact_6066_signed__take__bit__eq__if__positive,axiom,
    ! [A: int,Nb: nat] :
      ( ~ aa_nat_o(bit_se1146084159140164899it_int(A),Nb)
     => ( aa_int_int(bit_ri631733984087533419it_int(Nb),A) = aa_int_int(bit_se2923211474154528505it_int(Nb),A) ) ) ).

% signed_take_bit_eq_if_positive
tff(fact_6067_mask__nonnegative__int,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(bit_se2000444600071755411sk_int,Nb)) ).

% mask_nonnegative_int
tff(fact_6068_not__mask__negative__int,axiom,
    ! [Nb: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(bit_se2000444600071755411sk_int,Nb)),zero_zero_int) ).

% not_mask_negative_int
tff(fact_6069_bit__not__int__iff_H,axiom,
    ! [K2: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,K2)),one_one_int)),Nb)
    <=> ~ aa_nat_o(bit_se1146084159140164899it_int(K2),Nb) ) ).

% bit_not_int_iff'
tff(fact_6070_flip__bit__eq__if,axiom,
    ! [Nb: nat,A: int] :
      aa_int_int(bit_se2159334234014336723it_int(Nb),A) = aa_int_int(aa_nat_fun_int_int($ite(aa_nat_o(bit_se1146084159140164899it_int(A),Nb),bit_se4203085406695923979it_int,bit_se7879613467334960850it_int),Nb),A) ).

% flip_bit_eq_if
tff(fact_6071_flip__bit__eq__if,axiom,
    ! [Nb: nat,A: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,Nb),A) = aa_nat_nat(aa_nat_fun_nat_nat($ite(aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Nb),bit_se4205575877204974255it_nat,bit_se7882103937844011126it_nat),Nb),A) ).

% flip_bit_eq_if
tff(fact_6072_less__mask,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(bit_se2002935070580805687sk_nat,Nb)) ) ).

% less_mask
tff(fact_6073_bit__imp__take__bit__positive,axiom,
    ! [Nb: nat,Ma: nat,K2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Ma)
     => ( aa_nat_o(bit_se1146084159140164899it_int(K2),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(bit_se2923211474154528505it_int(Ma),K2)) ) ) ).

% bit_imp_take_bit_positive
tff(fact_6074_bit__concat__bit__iff,axiom,
    ! [Ma: nat,K2: int,L: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(bit_concat_bit(Ma),K2),L)),Nb)
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Ma)
          & aa_nat_o(bit_se1146084159140164899it_int(K2),Nb) )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
          & aa_nat_o(bit_se1146084159140164899it_int(L),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Ma)) ) ) ) ).

% bit_concat_bit_iff
tff(fact_6075_signed__take__bit__eq__concat__bit,axiom,
    ! [Nb: nat,K2: int] : aa_int_int(bit_ri631733984087533419it_int(Nb),K2) = aa_int_int(aa_int_fun_int_int(bit_concat_bit(Nb),K2),aa_int_int(uminus_uminus_int,aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(bit_se1146084159140164899it_int(K2),Nb)))) ).

% signed_take_bit_eq_concat_bit
tff(fact_6076_exp__eq__0__imp__not__bit,axiom,
    ! [Nb: nat,A: int] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb) = zero_zero_int )
     => ~ aa_nat_o(bit_se1146084159140164899it_int(A),Nb) ) ).

% exp_eq_0_imp_not_bit
tff(fact_6077_exp__eq__0__imp__not__bit,axiom,
    ! [Nb: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) = zero_zero_nat )
     => ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Nb) ) ).

% exp_eq_0_imp_not_bit
tff(fact_6078_bit__Suc,axiom,
    ! [A: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(A),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),Nb) ) ).

% bit_Suc
tff(fact_6079_bit__Suc,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Nb) ) ).

% bit_Suc
tff(fact_6080_stable__imp__bit__iff__odd,axiom,
    ! [A: code_integer,Nb: nat] :
      ( ( divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = A )
     => ( aa_nat_o(bit_se9216721137139052372nteger(A),Nb)
      <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A) ) ) ).

% stable_imp_bit_iff_odd
tff(fact_6081_stable__imp__bit__iff__odd,axiom,
    ! [A: int,Nb: nat] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A )
     => ( aa_nat_o(bit_se1146084159140164899it_int(A),Nb)
      <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) ) ) ).

% stable_imp_bit_iff_odd
tff(fact_6082_stable__imp__bit__iff__odd,axiom,
    ! [A: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = A )
     => ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Nb)
      <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A) ) ) ).

% stable_imp_bit_iff_odd
tff(fact_6083_bit__iff__idd__imp__stable,axiom,
    ! [A: code_integer] :
      ( ! [N: nat] :
          ( aa_nat_o(bit_se9216721137139052372nteger(A),N)
        <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A) )
     => ( divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = A ) ) ).

% bit_iff_idd_imp_stable
tff(fact_6084_bit__iff__idd__imp__stable,axiom,
    ! [A: int] :
      ( ! [N: nat] :
          ( aa_nat_o(bit_se1146084159140164899it_int(A),N)
        <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A ) ) ).

% bit_iff_idd_imp_stable
tff(fact_6085_bit__iff__idd__imp__stable,axiom,
    ! [A: nat] :
      ( ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),N)
        <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = A ) ) ).

% bit_iff_idd_imp_stable
tff(fact_6086_pred__equals__eq2,axiom,
    ! [R4: set_Pr448751882837621926eger_o,S3: set_Pr448751882837621926eger_o] :
      ( ! [X3: code_integer,Xa2: $o] :
          ( member1379723562493234055eger_o(aa_o_P4192039665577170892eger_o(aa_Cod3473573201699832512eger_o(produc6677183202524767010eger_o,X3),(Xa2)),R4)
        <=> member1379723562493234055eger_o(aa_o_P4192039665577170892eger_o(aa_Cod3473573201699832512eger_o(produc6677183202524767010eger_o,X3),(Xa2)),S3) )
    <=> ( R4 = S3 ) ) ).

% pred_equals_eq2
tff(fact_6087_pred__equals__eq2,axiom,
    ! [R4: set_Pr8218934625190621173um_num,S3: set_Pr8218934625190621173um_num] :
      ( ! [X3: num,Xa2: num] :
          ( member7279096912039735102um_num(aa_num3156681643174609981um_num(aa_num8681986212680877308um_num(product_Pair_num_num,X3),Xa2),R4)
        <=> member7279096912039735102um_num(aa_num3156681643174609981um_num(aa_num8681986212680877308um_num(product_Pair_num_num,X3),Xa2),S3) )
    <=> ( R4 = S3 ) ) ).

% pred_equals_eq2
tff(fact_6088_pred__equals__eq2,axiom,
    ! [R4: set_Pr6200539531224447659at_num,S3: set_Pr6200539531224447659at_num] :
      ( ! [X3: nat,Xa2: num] :
          ( member9148766508732265716at_num(aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,X3),Xa2),R4)
        <=> member9148766508732265716at_num(aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,X3),Xa2),S3) )
    <=> ( R4 = S3 ) ) ).

% pred_equals_eq2
tff(fact_6089_pred__equals__eq2,axiom,
    ! [R4: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ! [X3: nat,Xa2: nat] :
          ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X3),Xa2),R4)
        <=> member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X3),Xa2),S3) )
    <=> ( R4 = S3 ) ) ).

% pred_equals_eq2
tff(fact_6090_pred__equals__eq2,axiom,
    ! [R4: set_Pr958786334691620121nt_int,S3: set_Pr958786334691620121nt_int] :
      ( ! [X3: int,Xa2: int] :
          ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Xa2),R4)
        <=> member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Xa2),S3) )
    <=> ( R4 = S3 ) ) ).

% pred_equals_eq2
tff(fact_6091_take__bit__eq__mask__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Nb),K2) = aa_nat_int(bit_se2000444600071755411sk_int,Nb) )
    <=> ( aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),one_one_int)) = zero_zero_int ) ) ).

% take_bit_eq_mask_iff
tff(fact_6092_int__bit__bound,axiom,
    ! [K2: int] :
      ~ ! [N: nat] :
          ( ! [M: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M)
             => ( aa_nat_o(bit_se1146084159140164899it_int(K2),M)
              <=> aa_nat_o(bit_se1146084159140164899it_int(K2),N) ) )
         => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
             => ( aa_nat_o(bit_se1146084159140164899it_int(K2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))
              <=> ~ aa_nat_o(bit_se1146084159140164899it_int(K2),N) ) ) ) ).

% int_bit_bound
tff(fact_6093_bit__iff__odd,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_nat_o(bit_se9216721137139052372nteger(A),Nb)
    <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb))) ) ).

% bit_iff_odd
tff(fact_6094_bit__iff__odd,axiom,
    ! [A: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(A),Nb)
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))) ) ).

% bit_iff_odd
tff(fact_6095_bit__iff__odd,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Nb)
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ) ).

% bit_iff_odd
tff(fact_6096_Suc__mask__eq__exp,axiom,
    ! [Nb: nat] : aa_nat_nat(suc,aa_nat_nat(bit_se2002935070580805687sk_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ).

% Suc_mask_eq_exp
tff(fact_6097_mask__nat__less__exp,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(bit_se2002935070580805687sk_nat,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% mask_nat_less_exp
tff(fact_6098_bit__int__def,axiom,
    ! [K2: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(K2),Nb)
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))) ) ).

% bit_int_def
tff(fact_6099_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_Code_integer(bit_se2119862282449309892nteger,Nb))
    <=> ( Nb = zero_zero_nat ) ) ).

% semiring_bit_operations_class.even_mask_iff
tff(fact_6100_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(bit_se2002935070580805687sk_nat,Nb))
    <=> ( Nb = zero_zero_nat ) ) ).

% semiring_bit_operations_class.even_mask_iff
tff(fact_6101_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(bit_se2000444600071755411sk_int,Nb))
    <=> ( Nb = zero_zero_nat ) ) ).

% semiring_bit_operations_class.even_mask_iff
tff(fact_6102_even__bit__succ__iff,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
     => ( aa_nat_o(bit_se9216721137139052372nteger(plus_p5714425477246183910nteger(one_one_Code_integer,A)),Nb)
      <=> ( aa_nat_o(bit_se9216721137139052372nteger(A),Nb)
          | ( Nb = zero_zero_nat ) ) ) ) ).

% even_bit_succ_iff
tff(fact_6103_even__bit__succ__iff,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),Nb)
      <=> ( aa_nat_o(bit_se1146084159140164899it_int(A),Nb)
          | ( Nb = zero_zero_nat ) ) ) ) ).

% even_bit_succ_iff
tff(fact_6104_even__bit__succ__iff,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),Nb)
      <=> ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Nb)
          | ( Nb = zero_zero_nat ) ) ) ) ).

% even_bit_succ_iff
tff(fact_6105_odd__bit__iff__bit__pred,axiom,
    ! [A: code_integer,Nb: nat] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
     => ( aa_nat_o(bit_se9216721137139052372nteger(A),Nb)
      <=> ( aa_nat_o(bit_se9216721137139052372nteger(minus_8373710615458151222nteger(A,one_one_Code_integer)),Nb)
          | ( Nb = zero_zero_nat ) ) ) ) ).

% odd_bit_iff_bit_pred
tff(fact_6106_odd__bit__iff__bit__pred,axiom,
    ! [A: int,Nb: nat] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(bit_se1146084159140164899it_int(A),Nb)
      <=> ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,A),one_one_int)),Nb)
          | ( Nb = zero_zero_nat ) ) ) ) ).

% odd_bit_iff_bit_pred
tff(fact_6107_odd__bit__iff__bit__pred,axiom,
    ! [A: nat,Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Nb)
      <=> ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),one_one_nat)),Nb)
          | ( Nb = zero_zero_nat ) ) ) ) ).

% odd_bit_iff_bit_pred
tff(fact_6108_mask__nat__def,axiom,
    ! [Nb: nat] : aa_nat_nat(bit_se2002935070580805687sk_nat,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),one_one_nat) ).

% mask_nat_def
tff(fact_6109_mask__half__int,axiom,
    ! [Nb: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(bit_se2000444600071755411sk_int,Nb)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_nat_int(bit_se2000444600071755411sk_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)) ).

% mask_half_int
tff(fact_6110_mask__int__def,axiom,
    ! [Nb: nat] : aa_nat_int(bit_se2000444600071755411sk_int,Nb) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),one_one_int) ).

% mask_int_def
tff(fact_6111_ex__le__of__int,axiom,
    ! [X: real] :
    ? [Z: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,Z)) ).

% ex_le_of_int
tff(fact_6112_ex__le__of__int,axiom,
    ! [X: rat] :
    ? [Z: int] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,Z)) ).

% ex_le_of_int
tff(fact_6113_ex__of__int__less,axiom,
    ! [X: real] :
    ? [Z: int] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),X) ).

% ex_of_int_less
tff(fact_6114_ex__of__int__less,axiom,
    ! [X: rat] :
    ? [Z: int] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),X) ).

% ex_of_int_less
tff(fact_6115_ex__less__of__int,axiom,
    ! [X: real] :
    ? [Z: int] : aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_int_real(ring_1_of_int_real,Z)) ).

% ex_less_of_int
tff(fact_6116_ex__less__of__int,axiom,
    ! [X: rat] :
    ? [Z: int] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,Z)) ).

% ex_less_of_int
tff(fact_6117_subrelI,axiom,
    ! [R: set_Pr448751882837621926eger_o,S: set_Pr448751882837621926eger_o] :
      ( ! [X4: code_integer,Y4: $o] :
          ( member1379723562493234055eger_o(aa_o_P4192039665577170892eger_o(aa_Cod3473573201699832512eger_o(produc6677183202524767010eger_o,X4),(Y4)),R)
         => member1379723562493234055eger_o(aa_o_P4192039665577170892eger_o(aa_Cod3473573201699832512eger_o(produc6677183202524767010eger_o,X4),(Y4)),S) )
     => ord_le8980329558974975238eger_o(R,S) ) ).

% subrelI
tff(fact_6118_subrelI,axiom,
    ! [R: set_Pr8218934625190621173um_num,S: set_Pr8218934625190621173um_num] :
      ( ! [X4: num,Y4: num] :
          ( member7279096912039735102um_num(aa_num3156681643174609981um_num(aa_num8681986212680877308um_num(product_Pair_num_num,X4),Y4),R)
         => member7279096912039735102um_num(aa_num3156681643174609981um_num(aa_num8681986212680877308um_num(product_Pair_num_num,X4),Y4),S) )
     => ord_le880128212290418581um_num(R,S) ) ).

% subrelI
tff(fact_6119_subrelI,axiom,
    ! [R: set_Pr6200539531224447659at_num,S: set_Pr6200539531224447659at_num] :
      ( ! [X4: nat,Y4: num] :
          ( member9148766508732265716at_num(aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,X4),Y4),R)
         => member9148766508732265716at_num(aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,X4),Y4),S) )
     => ord_le8085105155179020875at_num(R,S) ) ).

% subrelI
tff(fact_6120_subrelI,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ! [X4: nat,Y4: nat] :
          ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X4),Y4),R)
         => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X4),Y4),S) )
     => ord_le3146513528884898305at_nat(R,S) ) ).

% subrelI
tff(fact_6121_subrelI,axiom,
    ! [R: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
      ( ! [X4: int,Y4: int] :
          ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Y4),R)
         => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Y4),S) )
     => ord_le2843351958646193337nt_int(R,S) ) ).

% subrelI
tff(fact_6122_bit__sum__mult__2__cases,axiom,
    ! [A: code_integer,B: code_integer,Nb: nat] :
      ( ! [J2: nat] : ~ aa_nat_o(bit_se9216721137139052372nteger(A),aa_nat_nat(suc,J2))
     => ( aa_nat_o(bit_se9216721137139052372nteger(plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B))),Nb)
      <=> $ite(Nb = zero_zero_nat,~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A),aa_nat_o(bit_se9216721137139052372nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B)),Nb)) ) ) ).

% bit_sum_mult_2_cases
tff(fact_6123_bit__sum__mult__2__cases,axiom,
    ! [A: int,B: int,Nb: nat] :
      ( ! [J2: nat] : ~ aa_nat_o(bit_se1146084159140164899it_int(A),aa_nat_nat(suc,J2))
     => ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B))),Nb)
      <=> $ite(Nb = zero_zero_nat,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A),aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B)),Nb)) ) ) ).

% bit_sum_mult_2_cases
tff(fact_6124_bit__sum__mult__2__cases,axiom,
    ! [A: nat,B: nat,Nb: nat] :
      ( ! [J2: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),aa_nat_nat(suc,J2))
     => ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B))),Nb)
      <=> $ite(Nb = zero_zero_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A),aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)),Nb)) ) ) ).

% bit_sum_mult_2_cases
tff(fact_6125_bit__rec,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_nat_o(bit_se9216721137139052372nteger(A),Nb)
    <=> $ite(Nb = zero_zero_nat,~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A),aa_nat_o(bit_se9216721137139052372nteger(divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ) ).

% bit_rec
tff(fact_6126_bit__rec,axiom,
    ! [A: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(A),Nb)
    <=> $ite(Nb = zero_zero_nat,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A),aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ) ).

% bit_rec
tff(fact_6127_bit__rec,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Nb)
    <=> $ite(Nb = zero_zero_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A),aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ) ).

% bit_rec
tff(fact_6128_mask__eq__exp__minus__1,axiom,
    ! [Nb: nat] : aa_nat_nat(bit_se2002935070580805687sk_nat,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),one_one_nat) ).

% mask_eq_exp_minus_1
tff(fact_6129_mask__eq__exp__minus__1,axiom,
    ! [Nb: nat] : aa_nat_int(bit_se2000444600071755411sk_int,Nb) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),one_one_int) ).

% mask_eq_exp_minus_1
tff(fact_6130_accp__subset__induct,axiom,
    ! [D3: fun_Pr7546709192626087934_nat_o,R4: fun_Pr8877776288780692068_nat_o,X: produc9072475918466114483BT_nat,P: fun_Pr7546709192626087934_nat_o] :
      ( ord_le1769015658490769310_nat_o(D3,accp_P2887432264394892906BT_nat(R4))
     => ( ! [X4: produc9072475918466114483BT_nat,Z: produc9072475918466114483BT_nat] :
            ( aa_Pro632845113475883111_nat_o(D3,X4)
           => ( aa_Pro632845113475883111_nat_o(aa_Pro1578556086376436173_nat_o(R4,Z),X4)
             => aa_Pro632845113475883111_nat_o(D3,Z) ) )
       => ( aa_Pro632845113475883111_nat_o(D3,X)
         => ( ! [X4: produc9072475918466114483BT_nat] :
                ( aa_Pro632845113475883111_nat_o(D3,X4)
               => ( ! [Z3: produc9072475918466114483BT_nat] :
                      ( aa_Pro632845113475883111_nat_o(aa_Pro1578556086376436173_nat_o(R4,Z3),X4)
                     => aa_Pro632845113475883111_nat_o(P,Z3) )
                 => aa_Pro632845113475883111_nat_o(P,X4) ) )
           => aa_Pro632845113475883111_nat_o(P,X) ) ) ) ) ).

% accp_subset_induct
tff(fact_6131_accp__subset__induct,axiom,
    ! [D3: fun_Pr6331313853612757382_num_o,R4: fun_Pr4799464687642144302_num_o,X: product_prod_num_num,P: fun_Pr6331313853612757382_num_o] :
      ( ord_le3107050023622517222_num_o(D3,accp_P3113834385874906142um_num(R4))
     => ( ! [X4: product_prod_num_num,Z: product_prod_num_num] :
            ( aa_Pro7685007914727650815_num_o(D3,X4)
           => ( aa_Pro7685007914727650815_num_o(aa_Pro4218333719626769503_num_o(R4,Z),X4)
             => aa_Pro7685007914727650815_num_o(D3,Z) ) )
       => ( aa_Pro7685007914727650815_num_o(D3,X)
         => ( ! [X4: product_prod_num_num] :
                ( aa_Pro7685007914727650815_num_o(D3,X4)
               => ( ! [Z3: product_prod_num_num] :
                      ( aa_Pro7685007914727650815_num_o(aa_Pro4218333719626769503_num_o(R4,Z3),X4)
                     => aa_Pro7685007914727650815_num_o(P,Z3) )
                 => aa_Pro7685007914727650815_num_o(P,X4) ) )
           => aa_Pro7685007914727650815_num_o(P,X) ) ) ) ) ).

% accp_subset_induct
tff(fact_6132_accp__subset__induct,axiom,
    ! [D3: fun_Pr4796943543331071514_nat_o,R4: fun_Pr2028404973800837934_nat_o,X: product_prod_nat_nat,P: fun_Pr4796943543331071514_nat_o] :
      ( aa_fun8408151826566693824at_o_o(ord_le1572679713340831354_nat_o(D3),accp_P4275260045618599050at_nat(R4))
     => ( ! [X4: product_prod_nat_nat,Z: product_prod_nat_nat] :
            ( aa_Pro6150637604445964947_nat_o(D3,X4)
           => ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(R4,Z),X4)
             => aa_Pro6150637604445964947_nat_o(D3,Z) ) )
       => ( aa_Pro6150637604445964947_nat_o(D3,X)
         => ( ! [X4: product_prod_nat_nat] :
                ( aa_Pro6150637604445964947_nat_o(D3,X4)
               => ( ! [Z3: product_prod_nat_nat] :
                      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(R4,Z3),X4)
                     => aa_Pro6150637604445964947_nat_o(P,Z3) )
                 => aa_Pro6150637604445964947_nat_o(P,X4) ) )
           => aa_Pro6150637604445964947_nat_o(P,X) ) ) ) ) ).

% accp_subset_induct
tff(fact_6133_accp__subset__induct,axiom,
    ! [D3: fun_Pr3238374608701176162_int_o,R4: fun_Pr9018117161326028078_int_o,X: product_prod_int_int,P: fun_Pr3238374608701176162_int_o] :
      ( ord_le14110778710936002_int_o(D3,accp_P1096762738010456898nt_int(R4))
     => ( ! [X4: product_prod_int_int,Z: product_prod_int_int] :
            ( aa_Pro4592068669816069595_int_o(D3,X4)
           => ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(R4,Z),X4)
             => aa_Pro4592068669816069595_int_o(D3,Z) ) )
       => ( aa_Pro4592068669816069595_int_o(D3,X)
         => ( ! [X4: product_prod_int_int] :
                ( aa_Pro4592068669816069595_int_o(D3,X4)
               => ( ! [Z3: product_prod_int_int] :
                      ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(R4,Z3),X4)
                     => aa_Pro4592068669816069595_int_o(P,Z3) )
                 => aa_Pro4592068669816069595_int_o(P,X4) ) )
           => aa_Pro4592068669816069595_int_o(P,X) ) ) ) ) ).

% accp_subset_induct
tff(fact_6134_accp__subset__induct,axiom,
    ! [D3: fun_nat_o,R4: fun_nat_fun_nat_o,X: nat,P: fun_nat_o] :
      ( ord_le7240017405068453697_nat_o(D3,accp_nat(R4))
     => ( ! [X4: nat,Z: nat] :
            ( aa_nat_o(D3,X4)
           => ( aa_nat_o(aa_nat_fun_nat_o(R4,Z),X4)
             => aa_nat_o(D3,Z) ) )
       => ( aa_nat_o(D3,X)
         => ( ! [X4: nat] :
                ( aa_nat_o(D3,X4)
               => ( ! [Z3: nat] :
                      ( aa_nat_o(aa_nat_fun_nat_o(R4,Z3),X4)
                     => aa_nat_o(P,Z3) )
                 => aa_nat_o(P,X4) ) )
           => aa_nat_o(P,X) ) ) ) ) ).

% accp_subset_induct
tff(fact_6135_set__bit__eq,axiom,
    ! [Nb: nat,K2: int] : aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,Nb),K2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,~ aa_nat_o(bit_se1146084159140164899it_int(K2),Nb))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))) ).

% set_bit_eq
tff(fact_6136_unset__bit__eq,axiom,
    ! [Nb: nat,K2: int] : aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Nb),K2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,K2),aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(bit_se1146084159140164899it_int(K2),Nb))),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))) ).

% unset_bit_eq
tff(fact_6137_take__bit__Suc__from__most,axiom,
    ! [Nb: nat,K2: int] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),K2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(bit_se1146084159140164899it_int(K2),Nb)))),aa_int_int(bit_se2923211474154528505it_int(Nb),K2)) ).

% take_bit_Suc_from_most
tff(fact_6138_pred__subset__eq,axiom,
    ! [R4: set_complex,S3: set_complex] :
      ( ord_le7644322562001619171plex_o(aTP_Lamm_ac(R4),aTP_Lamm_ac(S3))
    <=> ord_le211207098394363844omplex(R4,S3) ) ).

% pred_subset_eq
tff(fact_6139_pred__subset__eq,axiom,
    ! [R4: set_real,S3: set_real] :
      ( ord_le3605727316730466277real_o(aTP_Lamm_ab(R4),aTP_Lamm_ab(S3))
    <=> ord_less_eq_set_real(R4,S3) ) ).

% pred_subset_eq
tff(fact_6140_pred__subset__eq,axiom,
    ! [R4: set_set_nat,S3: set_set_nat] :
      ( ord_le6468474429079055883_nat_o(aTP_Lamm_ae(R4),aTP_Lamm_ae(S3))
    <=> ord_le6893508408891458716et_nat(R4,S3) ) ).

% pred_subset_eq
tff(fact_6141_pred__subset__eq,axiom,
    ! [R4: set_nat,S3: set_nat] :
      ( ord_le7240017405068453697_nat_o(aTP_Lamm_af(R4),aTP_Lamm_af(S3))
    <=> aa_set_nat_o(ord_less_eq_set_nat(R4),S3) ) ).

% pred_subset_eq
tff(fact_6142_pred__subset__eq,axiom,
    ! [R4: set_int,S3: set_int] :
      ( ord_le5418975059635765733_int_o(aTP_Lamm_ag(R4),aTP_Lamm_ag(S3))
    <=> ord_less_eq_set_int(R4,S3) ) ).

% pred_subset_eq
tff(fact_6143_take__bit__eq__mask__iff__exp__dvd,axiom,
    ! [Nb: nat,K2: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Nb),K2) = aa_nat_int(bit_se2000444600071755411sk_int,Nb) )
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),one_one_int)) ) ).

% take_bit_eq_mask_iff_exp_dvd
tff(fact_6144_round__unique,axiom,
    ! [X: real,Y2: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(X),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_int_real(ring_1_of_int_real,Y2))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Y2)),aa_real_real(plus_plus_real(X),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => ( aa_real_int(archim8280529875227126926d_real,X) = Y2 ) ) ) ).

% round_unique
tff(fact_6145_round__unique,axiom,
    ! [X: rat,Y2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(X),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,Y2))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Y2)),aa_rat_rat(plus_plus_rat(X),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))))
       => ( aa_rat_int(archim7778729529865785530nd_rat,X) = Y2 ) ) ) ).

% round_unique
tff(fact_6146_in__measure,axiom,
    ! [X: num,Y2: num,F: fun_num_nat] :
      ( member7279096912039735102um_num(aa_num3156681643174609981um_num(aa_num8681986212680877308um_num(product_Pair_num_num,X),Y2),measure_num(F))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(F,X)),aa_num_nat(F,Y2)) ) ).

% in_measure
tff(fact_6147_in__measure,axiom,
    ! [X: nat,Y2: nat,F: fun_nat_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X),Y2),measure_nat(F))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,X)),aa_nat_nat(F,Y2)) ) ).

% in_measure
tff(fact_6148_in__measure,axiom,
    ! [X: int,Y2: int,F: fun_int_nat] :
      ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Y2),measure_int(F))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(F,X)),aa_int_nat(F,Y2)) ) ).

% in_measure
tff(fact_6149_of__int__round__gt,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(X),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,X))) ).

% of_int_round_gt
tff(fact_6150_of__int__round__gt,axiom,
    ! [X: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(X),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,X))) ).

% of_int_round_gt
tff(fact_6151_of__int__round__ge,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(X),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,X))) ).

% of_int_round_ge
tff(fact_6152_of__int__round__ge,axiom,
    ! [X: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(X),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,X))) ).

% of_int_round_ge
tff(fact_6153_of__int__round__le,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,X))),aa_real_real(plus_plus_real(X),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% of_int_round_le
tff(fact_6154_of__int__round__le,axiom,
    ! [X: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,X))),aa_rat_rat(plus_plus_rat(X),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))) ).

% of_int_round_le
tff(fact_6155_upto_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun_int_fun_int_o] :
      ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A0),A1))
     => ( ! [I3: int,J2: int] :
            ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,I3),J2))
           => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I3),J2)
               => aa_int_o(aa_int_fun_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I3),one_one_int)),J2) )
             => aa_int_o(aa_int_fun_int_o(P,I3),J2) ) )
       => aa_int_o(aa_int_fun_int_o(P,A0),A1) ) ) ).

% upto.pinduct
tff(fact_6156_round__numeral,axiom,
    ! [Nb: num] : aa_real_int(archim8280529875227126926d_real,aa_num_real(numeral_numeral_real,Nb)) = aa_num_int(numeral_numeral_int,Nb) ).

% round_numeral
tff(fact_6157_round__numeral,axiom,
    ! [Nb: num] : aa_rat_int(archim7778729529865785530nd_rat,aa_num_rat(numeral_numeral_rat,Nb)) = aa_num_int(numeral_numeral_int,Nb) ).

% round_numeral
tff(fact_6158_round__1,axiom,
    aa_real_int(archim8280529875227126926d_real,one_one_real) = one_one_int ).

% round_1
tff(fact_6159_round__1,axiom,
    aa_rat_int(archim7778729529865785530nd_rat,one_one_rat) = one_one_int ).

% round_1
tff(fact_6160_round__neg__numeral,axiom,
    ! [Nb: num] : aa_real_int(archim8280529875227126926d_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) ).

% round_neg_numeral
tff(fact_6161_round__neg__numeral,axiom,
    ! [Nb: num] : aa_rat_int(archim7778729529865785530nd_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) ).

% round_neg_numeral
tff(fact_6162_not__bit__Suc__0__Suc,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(suc,Nb)) ).

% not_bit_Suc_0_Suc
tff(fact_6163_bit__Suc__0__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),Nb)
    <=> ( Nb = zero_zero_nat ) ) ).

% bit_Suc_0_iff
tff(fact_6164_not__bit__Suc__0__numeral,axiom,
    ! [Nb: num] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,Nb)) ).

% not_bit_Suc_0_numeral
tff(fact_6165_round__mono,axiom,
    ! [X: rat,Y2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim7778729529865785530nd_rat,X)),aa_rat_int(archim7778729529865785530nd_rat,Y2)) ) ).

% round_mono
tff(fact_6166_bit__nat__def,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,Ma),Nb)
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ) ).

% bit_nat_def
tff(fact_6167_num_Osize__gen_I3_J,axiom,
    ! [X32: num] : aa_num_nat(size_num,aa_num_num(bit1,X32)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_num,X32)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size_gen(3)
tff(fact_6168_num_Osize__gen_I2_J,axiom,
    ! [X23: num] : aa_num_nat(size_num,aa_num_num(bit0,X23)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_num,X23)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size_gen(2)
tff(fact_6169_fold__atLeastAtMost__nat_Opinduct,axiom,
    ! [A0: fun_nat_fun_num_num,A1: nat,A22: nat,A32: num,P: fun_fu8148986511470497476_num_o] :
      ( accp_P3591886921626383704at_num(set_fo256927282339908995el_num,produc460779904311030791at_num(A0,produc1195630363706982562at_num(A1,aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,A22),A32))))
     => ( ! [F3: fun_nat_fun_num_num,A3: nat,B2: nat,Acc: num] :
            ( accp_P3591886921626383704at_num(set_fo256927282339908995el_num,produc460779904311030791at_num(F3,produc1195630363706982562at_num(A3,aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,B2),Acc))))
           => ( ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),A3)
               => aa_num_o(aa_nat_fun_num_o(aa_nat9072979527818120146_num_o(aa_fun2776124910339071917_num_o(P,F3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A3),one_one_nat)),B2),aa_num_num(aa_nat_fun_num_num(F3,A3),Acc)) )
             => aa_num_o(aa_nat_fun_num_o(aa_nat9072979527818120146_num_o(aa_fun2776124910339071917_num_o(P,F3),A3),B2),Acc) ) )
       => aa_num_o(aa_nat_fun_num_o(aa_nat9072979527818120146_num_o(aa_fun2776124910339071917_num_o(P,A0),A1),A22),A32) ) ) ).

% fold_atLeastAtMost_nat.pinduct
tff(fact_6170_fold__atLeastAtMost__nat_Opinduct,axiom,
    ! [A0: fun_nat_fun_nat_nat,A1: nat,A22: nat,A32: nat,P: fun_fu2571895429041709858_nat_o] :
      ( accp_P4694664897847628410at_nat(set_fo3699595496184130361el_nat,produc2818902965508115497at_nat(A0,produc487386426758144856at_nat(A1,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,A22),A32))))
     => ( ! [F3: fun_nat_fun_nat_nat,A3: nat,B2: nat,Acc: nat] :
            ( accp_P4694664897847628410at_nat(set_fo3699595496184130361el_nat,produc2818902965508115497at_nat(F3,produc487386426758144856at_nat(A3,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,B2),Acc))))
           => ( ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),A3)
               => aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aa_fun3751150903526668555_nat_o(P,F3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A3),one_one_nat)),B2),aa_nat_nat(aa_nat_fun_nat_nat(F3,A3),Acc)) )
             => aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aa_fun3751150903526668555_nat_o(P,F3),A3),B2),Acc) ) )
       => aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aa_fun3751150903526668555_nat_o(P,A0),A1),A22),A32) ) ) ).

% fold_atLeastAtMost_nat.pinduct
tff(fact_6171_and__int__unfold,axiom,
    ! [K2: int,L: int] :
      aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K2),L) = $ite(
        ( ( K2 = zero_zero_int )
        | ( L = zero_zero_int ) ),
        zero_zero_int,
        $ite(
          K2 = aa_int_int(uminus_uminus_int,one_one_int),
          L,
          $ite(L = aa_int_int(uminus_uminus_int,one_one_int),K2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))))) ) ) ).

% and_int_unfold
tff(fact_6172_exp__lower__Taylor__quadratic,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(one_one_real),X)),aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(exp_real,X)) ) ).

% exp_lower_Taylor_quadratic
tff(fact_6173_sqrt__sum__squares__half__less,axiom,
    ! [X: real,U: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(divide_divide_real(U),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),aa_real_real(divide_divide_real(U),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y2)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),U) ) ) ) ) ).

% sqrt_sum_squares_half_less
tff(fact_6174_and_Oidem,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),A) = A ).

% and.idem
tff(fact_6175_and_Oidem,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),A) = A ).

% and.idem
tff(fact_6176_and_Oleft__idem,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),B)) = aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),B) ).

% and.left_idem
tff(fact_6177_and_Oleft__idem,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),B) ).

% and.left_idem
tff(fact_6178_and_Oright__idem,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),B)),B) = aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),B) ).

% and.right_idem
tff(fact_6179_and_Oright__idem,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),B) ).

% and.right_idem
tff(fact_6180_real__sqrt__eq__iff,axiom,
    ! [X: real,Y2: real] :
      ( ( aa_real_real(sqrt,X) = aa_real_real(sqrt,Y2) )
    <=> ( X = Y2 ) ) ).

% real_sqrt_eq_iff
tff(fact_6181_and__zero__eq,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),zero_zero_int) = zero_zero_int ).

% and_zero_eq
tff(fact_6182_and__zero__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),zero_zero_nat) = zero_zero_nat ).

% and_zero_eq
tff(fact_6183_zero__and__eq,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,zero_zero_int),A) = zero_zero_int ).

% zero_and_eq
tff(fact_6184_zero__and__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,zero_zero_nat),A) = zero_zero_nat ).

% zero_and_eq
tff(fact_6185_bit_Oconj__zero__left,axiom,
    ! [X: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,zero_zero_int),X) = zero_zero_int ).

% bit.conj_zero_left
tff(fact_6186_bit_Oconj__zero__right,axiom,
    ! [X: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),zero_zero_int) = zero_zero_int ).

% bit.conj_zero_right
tff(fact_6187_real__sqrt__eq__zero__cancel__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(sqrt,X) = zero_zero_real )
    <=> ( X = zero_zero_real ) ) ).

% real_sqrt_eq_zero_cancel_iff
tff(fact_6188_real__sqrt__zero,axiom,
    aa_real_real(sqrt,zero_zero_real) = zero_zero_real ).

% real_sqrt_zero
tff(fact_6189_real__sqrt__less__iff,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,X)),aa_real_real(sqrt,Y2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2) ) ).

% real_sqrt_less_iff
tff(fact_6190_real__sqrt__le__iff,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,X)),aa_real_real(sqrt,Y2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y2) ) ).

% real_sqrt_le_iff
tff(fact_6191_real__sqrt__eq__1__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(sqrt,X) = one_one_real )
    <=> ( X = one_one_real ) ) ).

% real_sqrt_eq_1_iff
tff(fact_6192_real__sqrt__one,axiom,
    aa_real_real(sqrt,one_one_real) = one_one_real ).

% real_sqrt_one
tff(fact_6193_take__bit__and,axiom,
    ! [Nb: nat,A: int,B: int] : aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),B)) = aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(bit_se2923211474154528505it_int(Nb),A)),aa_int_int(bit_se2923211474154528505it_int(Nb),B)) ).

% take_bit_and
tff(fact_6194_take__bit__and,axiom,
    ! [Nb: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),B)) ).

% take_bit_and
tff(fact_6195_bit_Oconj__one__right,axiom,
    ! [X: code_integer] : bit_se3949692690581998587nteger(X,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = X ).

% bit.conj_one_right
tff(fact_6196_bit_Oconj__one__right,axiom,
    ! [X: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),aa_int_int(uminus_uminus_int,one_one_int)) = X ).

% bit.conj_one_right
tff(fact_6197_and_Oright__neutral,axiom,
    ! [A: code_integer] : bit_se3949692690581998587nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = A ).

% and.right_neutral
tff(fact_6198_and_Oright__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),aa_int_int(uminus_uminus_int,one_one_int)) = A ).

% and.right_neutral
tff(fact_6199_and_Oleft__neutral,axiom,
    ! [A: code_integer] : bit_se3949692690581998587nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),A) = A ).

% and.left_neutral
tff(fact_6200_and_Oleft__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(uminus_uminus_int,one_one_int)),A) = A ).

% and.left_neutral
tff(fact_6201_real__sqrt__lt__0__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,X)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),zero_zero_real) ) ).

% real_sqrt_lt_0_iff
tff(fact_6202_real__sqrt__gt__0__iff,axiom,
    ! [Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sqrt,Y2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y2) ) ).

% real_sqrt_gt_0_iff
tff(fact_6203_real__sqrt__ge__0__iff,axiom,
    ! [Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sqrt,Y2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y2) ) ).

% real_sqrt_ge_0_iff
tff(fact_6204_real__sqrt__le__0__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,X)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),zero_zero_real) ) ).

% real_sqrt_le_0_iff
tff(fact_6205_real__sqrt__gt__1__iff,axiom,
    ! [Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(sqrt,Y2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Y2) ) ).

% real_sqrt_gt_1_iff
tff(fact_6206_real__sqrt__lt__1__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,X)),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),one_one_real) ) ).

% real_sqrt_lt_1_iff
tff(fact_6207_real__sqrt__ge__1__iff,axiom,
    ! [Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(sqrt,Y2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Y2) ) ).

% real_sqrt_ge_1_iff
tff(fact_6208_real__sqrt__le__1__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,X)),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),one_one_real) ) ).

% real_sqrt_le_1_iff
tff(fact_6209_and__nonnegative__int__iff,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K2),L))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) ) ) ).

% and_nonnegative_int_iff
tff(fact_6210_and__negative__int__iff,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K2),L)),zero_zero_int)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ) ).

% and_negative_int_iff
tff(fact_6211_and__numerals_I8_J,axiom,
    ! [X: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,X))),one_one_int) = one_one_int ).

% and_numerals(8)
tff(fact_6212_and__numerals_I8_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),one_one_nat) = one_one_nat ).

% and_numerals(8)
tff(fact_6213_and__numerals_I2_J,axiom,
    ! [Y2: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y2))) = one_one_int ).

% and_numerals(2)
tff(fact_6214_and__numerals_I2_J,axiom,
    ! [Y2: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y2))) = one_one_nat ).

% and_numerals(2)
tff(fact_6215_real__sqrt__four,axiom,
    aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)) ).

% real_sqrt_four
tff(fact_6216_and__numerals_I1_J,axiom,
    ! [Y2: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y2))) = zero_zero_int ).

% and_numerals(1)
tff(fact_6217_and__numerals_I1_J,axiom,
    ! [Y2: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y2))) = zero_zero_nat ).

% and_numerals(1)
tff(fact_6218_and__numerals_I5_J,axiom,
    ! [X: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,X))),one_one_int) = zero_zero_int ).

% and_numerals(5)
tff(fact_6219_and__numerals_I5_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X))),one_one_nat) = zero_zero_nat ).

% and_numerals(5)
tff(fact_6220_and__numerals_I3_J,axiom,
    ! [X: num,Y2: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,X))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y2))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,X)),aa_num_int(numeral_numeral_int,Y2))) ).

% and_numerals(3)
tff(fact_6221_and__numerals_I3_J,axiom,
    ! [X: num,Y2: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y2))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,X)),aa_num_nat(numeral_numeral_nat,Y2))) ).

% and_numerals(3)
tff(fact_6222_and__minus__numerals_I6_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))),one_one_int) = one_one_int ).

% and_minus_numerals(6)
tff(fact_6223_and__minus__numerals_I2_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = one_one_int ).

% and_minus_numerals(2)
tff(fact_6224_and__numerals_I4_J,axiom,
    ! [X: num,Y2: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,X))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y2))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,X)),aa_num_int(numeral_numeral_int,Y2))) ).

% and_numerals(4)
tff(fact_6225_and__numerals_I4_J,axiom,
    ! [X: num,Y2: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y2))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,X)),aa_num_nat(numeral_numeral_nat,Y2))) ).

% and_numerals(4)
tff(fact_6226_and__numerals_I6_J,axiom,
    ! [X: num,Y2: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,X))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y2))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,X)),aa_num_int(numeral_numeral_int,Y2))) ).

% and_numerals(6)
tff(fact_6227_and__numerals_I6_J,axiom,
    ! [X: num,Y2: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y2))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,X)),aa_num_nat(numeral_numeral_nat,Y2))) ).

% and_numerals(6)
tff(fact_6228_and__minus__numerals_I5_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))),one_one_int) = zero_zero_int ).

% and_minus_numerals(5)
tff(fact_6229_and__minus__numerals_I1_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = zero_zero_int ).

% and_minus_numerals(1)
tff(fact_6230_real__sqrt__pow2__iff,axiom,
    ! [X: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = X )
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X) ) ).

% real_sqrt_pow2_iff
tff(fact_6231_real__sqrt__pow2,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = X ) ) ).

% real_sqrt_pow2
tff(fact_6232_real__sqrt__sum__squares__mult__squared__eq,axiom,
    ! [X: real,Y2: real,Xa: real,Ya: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,aa_real_real(times_times_real(aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(times_times_real(aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% real_sqrt_sum_squares_mult_squared_eq
tff(fact_6233_and__numerals_I7_J,axiom,
    ! [X: num,Y2: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,X))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,X)),aa_num_int(numeral_numeral_int,Y2)))) ).

% and_numerals(7)
tff(fact_6234_and__numerals_I7_J,axiom,
    ! [X: num,Y2: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,X)),aa_num_nat(numeral_numeral_nat,Y2)))) ).

% and_numerals(7)
tff(fact_6235_and_Oassoc,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,B),C)) ).

% and.assoc
tff(fact_6236_and_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,B),C)) ).

% and.assoc
tff(fact_6237_and_Ocommute,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),B) = aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,B),A) ).

% and.commute
tff(fact_6238_and_Ocommute,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,B),A) ).

% and.commute
tff(fact_6239_and_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,B),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),C)) = aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,B),C)) ).

% and.left_commute
tff(fact_6240_and_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),C)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,B),C)) ).

% and.left_commute
tff(fact_6241_of__int__and__eq,axiom,
    ! [K2: int,L: int] : aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K2),L)) = aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(ring_1_of_int_int,K2)),aa_int_int(ring_1_of_int_int,L)) ).

% of_int_and_eq
tff(fact_6242_real__sqrt__less__mono,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,X)),aa_real_real(sqrt,Y2)) ) ).

% real_sqrt_less_mono
tff(fact_6243_real__sqrt__le__mono,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,X)),aa_real_real(sqrt,Y2)) ) ).

% real_sqrt_le_mono
tff(fact_6244_real__sqrt__divide,axiom,
    ! [X: real,Y2: real] : aa_real_real(sqrt,aa_real_real(divide_divide_real(X),Y2)) = aa_real_real(divide_divide_real(aa_real_real(sqrt,X)),aa_real_real(sqrt,Y2)) ).

% real_sqrt_divide
tff(fact_6245_real__sqrt__mult,axiom,
    ! [X: real,Y2: real] : aa_real_real(sqrt,aa_real_real(times_times_real(X),Y2)) = aa_real_real(times_times_real(aa_real_real(sqrt,X)),aa_real_real(sqrt,Y2)) ).

% real_sqrt_mult
tff(fact_6246_real__sqrt__power,axiom,
    ! [X: real,K2: nat] : aa_real_real(sqrt,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),K2)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,X)),K2) ).

% real_sqrt_power
tff(fact_6247_real__sqrt__minus,axiom,
    ! [X: real] : aa_real_real(sqrt,aa_real_real(uminus_uminus_real,X)) = aa_real_real(uminus_uminus_real,aa_real_real(sqrt,X)) ).

% real_sqrt_minus
tff(fact_6248_bit__and__iff,axiom,
    ! [A: int,B: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),B)),Nb)
    <=> ( aa_nat_o(bit_se1146084159140164899it_int(A),Nb)
        & aa_nat_o(bit_se1146084159140164899it_int(B),Nb) ) ) ).

% bit_and_iff
tff(fact_6249_bit__and__iff,axiom,
    ! [A: nat,B: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),B)),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Nb)
        & aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,B),Nb) ) ) ).

% bit_and_iff
tff(fact_6250_bit__and__int__iff,axiom,
    ! [K2: int,L: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K2),L)),Nb)
    <=> ( aa_nat_o(bit_se1146084159140164899it_int(K2),Nb)
        & aa_nat_o(bit_se1146084159140164899it_int(L),Nb) ) ) ).

% bit_and_int_iff
tff(fact_6251_and__eq__minus__1__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( bit_se3949692690581998587nteger(A,B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
    <=> ( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
        & ( B = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ) ).

% and_eq_minus_1_iff
tff(fact_6252_and__eq__minus__1__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),B) = aa_int_int(uminus_uminus_int,one_one_int) )
    <=> ( ( A = aa_int_int(uminus_uminus_int,one_one_int) )
        & ( B = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% and_eq_minus_1_iff
tff(fact_6253_real__sqrt__gt__zero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sqrt,X)) ) ).

% real_sqrt_gt_zero
tff(fact_6254_real__sqrt__ge__zero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sqrt,X)) ) ).

% real_sqrt_ge_zero
tff(fact_6255_real__sqrt__eq__zero__cancel,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( ( aa_real_real(sqrt,X) = zero_zero_real )
       => ( X = zero_zero_real ) ) ) ).

% real_sqrt_eq_zero_cancel
tff(fact_6256_real__sqrt__ge__one,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(sqrt,X)) ) ).

% real_sqrt_ge_one
tff(fact_6257_AND__upper2_H,axiom,
    ! [Y2: int,Z2: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y2),Z2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),Y2)),Z2) ) ) ).

% AND_upper2'
tff(fact_6258_AND__upper1_H,axiom,
    ! [Y2: int,Z2: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y2),Z2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Y2),Ya)),Z2) ) ) ).

% AND_upper1'
tff(fact_6259_AND__upper2,axiom,
    ! [Y2: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),Y2)),Y2) ) ).

% AND_upper2
tff(fact_6260_AND__upper1,axiom,
    ! [X: int,Y2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),Y2)),X) ) ).

% AND_upper1
tff(fact_6261_AND__lower,axiom,
    ! [X: int,Y2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),Y2)) ) ).

% AND_lower
tff(fact_6262_take__bit__eq__mask,axiom,
    ! [Nb: nat,A: int] : aa_int_int(bit_se2923211474154528505it_int(Nb),A) = aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),aa_nat_int(bit_se2000444600071755411sk_int,Nb)) ).

% take_bit_eq_mask
tff(fact_6263_take__bit__eq__mask,axiom,
    ! [Nb: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),aa_nat_nat(bit_se2002935070580805687sk_nat,Nb)) ).

% take_bit_eq_mask
tff(fact_6264_real__div__sqrt,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_real(divide_divide_real(X),aa_real_real(sqrt,X)) = aa_real_real(sqrt,X) ) ) ).

% real_div_sqrt
tff(fact_6265_sqrt__add__le__add__sqrt,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(plus_plus_real(X),Y2))),aa_real_real(plus_plus_real(aa_real_real(sqrt,X)),aa_real_real(sqrt,Y2))) ) ) ).

% sqrt_add_le_add_sqrt
tff(fact_6266_le__real__sqrt__sumsq,axiom,
    ! [X: real,Y2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_real_real(times_times_real(X),X)),aa_real_real(times_times_real(Y2),Y2)))) ).

% le_real_sqrt_sumsq
tff(fact_6267_AND__upper2_H_H,axiom,
    ! [Y2: int,Z2: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y2),Z2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),Y2)),Z2) ) ) ).

% AND_upper2''
tff(fact_6268_AND__upper1_H_H,axiom,
    ! [Y2: int,Z2: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y2),Z2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Y2),Ya)),Z2) ) ) ).

% AND_upper1''
tff(fact_6269_and__less__eq,axiom,
    ! [L: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K2),L)),K2) ) ).

% and_less_eq
tff(fact_6270_even__and__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se3949692690581998587nteger(A,B))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),B) ) ) ).

% even_and_iff
tff(fact_6271_even__and__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),B))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B) ) ) ).

% even_and_iff
tff(fact_6272_even__and__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),B))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B) ) ) ).

% even_and_iff
tff(fact_6273_sqrt2__less__2,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% sqrt2_less_2
tff(fact_6274_even__and__iff__int,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K2),L))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K2)
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ) ) ).

% even_and_iff_int
tff(fact_6275_one__and__eq,axiom,
    ! [A: code_integer] : bit_se3949692690581998587nteger(one_one_Code_integer,A) = modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% one_and_eq
tff(fact_6276_one__and__eq,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),A) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% one_and_eq
tff(fact_6277_one__and__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,one_one_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% one_and_eq
tff(fact_6278_and__one__eq,axiom,
    ! [A: code_integer] : bit_se3949692690581998587nteger(A,one_one_Code_integer) = modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% and_one_eq
tff(fact_6279_and__one__eq,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),one_one_int) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% and_one_eq
tff(fact_6280_and__one__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),one_one_nat) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% and_one_eq
tff(fact_6281_real__less__rsqrt,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Y2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(sqrt,Y2)) ) ).

% real_less_rsqrt
tff(fact_6282_real__le__rsqrt,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Y2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(sqrt,Y2)) ) ).

% real_le_rsqrt
tff(fact_6283_sqrt__le__D,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,X)),Y2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% sqrt_le_D
tff(fact_6284_num_Osize__gen_I1_J,axiom,
    aa_num_nat(size_num,one) = zero_zero_nat ).

% num.size_gen(1)
tff(fact_6285_and__exp__eq__0__iff__not__bit,axiom,
    ! [A: int,Nb: nat] :
      ( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = zero_zero_int )
    <=> ~ aa_nat_o(bit_se1146084159140164899it_int(A),Nb) ) ).

% and_exp_eq_0_iff_not_bit
tff(fact_6286_and__exp__eq__0__iff__not__bit,axiom,
    ! [A: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = zero_zero_nat )
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Nb) ) ).

% and_exp_eq_0_iff_not_bit
tff(fact_6287_real__sqrt__unique,axiom,
    ! [Y2: real,X: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = X )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y2)
       => ( aa_real_real(sqrt,X) = Y2 ) ) ) ).

% real_sqrt_unique
tff(fact_6288_real__le__lsqrt,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,X)),Y2) ) ) ) ).

% real_le_lsqrt
tff(fact_6289_lemma__real__divide__sqrt__less,axiom,
    ! [U: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),U)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(U),aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),U) ) ).

% lemma_real_divide_sqrt_less
tff(fact_6290_real__sqrt__sum__squares__eq__cancel2,axiom,
    ! [X: real,Y2: real] :
      ( ( aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) = Y2 )
     => ( X = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel2
tff(fact_6291_real__sqrt__sum__squares__eq__cancel,axiom,
    ! [X: real,Y2: real] :
      ( ( aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) = X )
     => ( Y2 = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel
tff(fact_6292_real__sqrt__sum__squares__triangle__ineq,axiom,
    ! [A: real,C: real,B: real,D: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(plus_plus_real(A),C)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(plus_plus_real(B),D)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(plus_plus_real(aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,C),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,D),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ).

% real_sqrt_sum_squares_triangle_ineq
tff(fact_6293_real__sqrt__sum__squares__ge2,axiom,
    ! [Y2: real,X: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% real_sqrt_sum_squares_ge2
tff(fact_6294_real__sqrt__sum__squares__ge1,axiom,
    ! [X: real,Y2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% real_sqrt_sum_squares_ge1
tff(fact_6295_real__less__lsqrt,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,X)),Y2) ) ) ) ).

% real_less_lsqrt
tff(fact_6296_sqrt__sum__squares__le__sum,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(plus_plus_real(X),Y2)) ) ) ).

% sqrt_sum_squares_le_sum
tff(fact_6297_sqrt__even__pow2,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_real(sqrt,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% sqrt_even_pow2
tff(fact_6298_real__sqrt__sum__squares__mult__ge__zero,axiom,
    ! [X: real,Y2: real,Xa: real,Ya: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sqrt,aa_real_real(times_times_real(aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ).

% real_sqrt_sum_squares_mult_ge_zero
tff(fact_6299_real__sqrt__power__even,axiom,
    ! [Nb: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,X)),Nb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% real_sqrt_power_even
tff(fact_6300_arith__geo__mean__sqrt,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(times_times_real(X),Y2))),aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(X),Y2)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ).

% arith_geo_mean_sqrt
tff(fact_6301_and__int__rec,axiom,
    ! [K2: int,L: int] :
      aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K2),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,
          aa_o_int(zero_n2684676970156552555ol_int,
            ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K2)
            & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% and_int_rec
tff(fact_6302_one__le__exp__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(exp_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X) ) ).

% one_le_exp_iff
tff(fact_6303_exp__le__one__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,X)),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),zero_zero_real) ) ).

% exp_le_one_iff
tff(fact_6304_exp__less__one__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(exp_real,X)),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),zero_zero_real) ) ).

% exp_less_one_iff
tff(fact_6305_one__less__exp__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(exp_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X) ) ).

% one_less_exp_iff
tff(fact_6306_real__exp__bound__lemma,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,X)),aa_real_real(plus_plus_real(one_one_real),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X))) ) ) ).

% real_exp_bound_lemma
tff(fact_6307_exp__bound,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,X)),aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(one_one_real),X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% exp_bound
tff(fact_6308_exp__le__cancel__iff,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,X)),aa_real_real(exp_real,Y2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y2) ) ).

% exp_le_cancel_iff
tff(fact_6309_exp__zero,axiom,
    aa_complex_complex(exp_complex,zero_zero_complex) = one_one_complex ).

% exp_zero
tff(fact_6310_exp__zero,axiom,
    aa_real_real(exp_real,zero_zero_real) = one_one_real ).

% exp_zero
tff(fact_6311_exp__eq__one__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(exp_real,X) = one_one_real )
    <=> ( X = zero_zero_real ) ) ).

% exp_eq_one_iff
tff(fact_6312_and__nat__numerals_I3_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X))),aa_nat_nat(suc,zero_zero_nat)) = zero_zero_nat ).

% and_nat_numerals(3)
tff(fact_6313_and__nat__numerals_I1_J,axiom,
    ! [Y2: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y2))) = zero_zero_nat ).

% and_nat_numerals(1)
tff(fact_6314_and__nat__numerals_I4_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% and_nat_numerals(4)
tff(fact_6315_and__nat__numerals_I2_J,axiom,
    ! [Y2: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y2))) = one_one_nat ).

% and_nat_numerals(2)
tff(fact_6316_Suc__0__and__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Suc_0_and_eq
tff(fact_6317_and__Suc__0__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% and_Suc_0_eq
tff(fact_6318_exp__times__arg__commute,axiom,
    ! [A2: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(exp_complex,A2)),A2) = aa_complex_complex(times_times_complex(A2),aa_complex_complex(exp_complex,A2)) ).

% exp_times_arg_commute
tff(fact_6319_exp__times__arg__commute,axiom,
    ! [A2: real] : aa_real_real(times_times_real(aa_real_real(exp_real,A2)),A2) = aa_real_real(times_times_real(A2),aa_real_real(exp_real,A2)) ).

% exp_times_arg_commute
tff(fact_6320_and__nat__unfold,axiom,
    ! [Ma: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Ma),Nb) = $ite(
        ( ( Ma = zero_zero_nat )
        | ( Nb = zero_zero_nat ) ),
        zero_zero_nat,
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ).

% and_nat_unfold
tff(fact_6321_and__nat__rec,axiom,
    ! [Ma: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Ma),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
          aa_o_nat(zero_n2687167440665602831ol_nat,
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma)
            & ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% and_nat_rec
tff(fact_6322_mult__exp__exp,axiom,
    ! [X: complex,Y2: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(exp_complex,X)),aa_complex_complex(exp_complex,Y2)) = aa_complex_complex(exp_complex,aa_complex_complex(plus_plus_complex(X),Y2)) ).

% mult_exp_exp
tff(fact_6323_mult__exp__exp,axiom,
    ! [X: real,Y2: real] : aa_real_real(times_times_real(aa_real_real(exp_real,X)),aa_real_real(exp_real,Y2)) = aa_real_real(exp_real,aa_real_real(plus_plus_real(X),Y2)) ).

% mult_exp_exp
tff(fact_6324_exp__add__commuting,axiom,
    ! [X: complex,Y2: complex] :
      ( ( aa_complex_complex(times_times_complex(X),Y2) = aa_complex_complex(times_times_complex(Y2),X) )
     => ( aa_complex_complex(exp_complex,aa_complex_complex(plus_plus_complex(X),Y2)) = aa_complex_complex(times_times_complex(aa_complex_complex(exp_complex,X)),aa_complex_complex(exp_complex,Y2)) ) ) ).

% exp_add_commuting
tff(fact_6325_exp__add__commuting,axiom,
    ! [X: real,Y2: real] :
      ( ( aa_real_real(times_times_real(X),Y2) = aa_real_real(times_times_real(Y2),X) )
     => ( aa_real_real(exp_real,aa_real_real(plus_plus_real(X),Y2)) = aa_real_real(times_times_real(aa_real_real(exp_real,X)),aa_real_real(exp_real,Y2)) ) ) ).

% exp_add_commuting
tff(fact_6326_exp__diff,axiom,
    ! [X: complex,Y2: complex] : aa_complex_complex(exp_complex,aa_complex_complex(minus_minus_complex(X),Y2)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(exp_complex,X)),aa_complex_complex(exp_complex,Y2)) ).

% exp_diff
tff(fact_6327_exp__diff,axiom,
    ! [X: real,Y2: real] : aa_real_real(exp_real,aa_real_real(minus_minus_real(X),Y2)) = aa_real_real(divide_divide_real(aa_real_real(exp_real,X)),aa_real_real(exp_real,Y2)) ).

% exp_diff
tff(fact_6328_exp__ge__zero,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(exp_real,X)) ).

% exp_ge_zero
tff(fact_6329_not__exp__le__zero,axiom,
    ! [X: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,X)),zero_zero_real) ).

% not_exp_le_zero
tff(fact_6330_exp__minus__inverse,axiom,
    ! [X: real] : aa_real_real(times_times_real(aa_real_real(exp_real,X)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,X))) = one_one_real ).

% exp_minus_inverse
tff(fact_6331_exp__minus__inverse,axiom,
    ! [X: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(exp_complex,X)),aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,X))) = one_one_complex ).

% exp_minus_inverse
tff(fact_6332_exp__gt__one,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(exp_real,X)) ) ).

% exp_gt_one
tff(fact_6333_exp__ge__add__one__self,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(one_one_real),X)),aa_real_real(exp_real,X)) ).

% exp_ge_add_one_self
tff(fact_6334_exp__ge__add__one__self__aux,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(one_one_real),X)),aa_real_real(exp_real,X)) ) ).

% exp_ge_add_one_self_aux
tff(fact_6335_lemma__exp__total,axiom,
    ! [Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Y2)
     => ? [X4: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X4)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),aa_real_real(minus_minus_real(Y2),one_one_real))
          & ( aa_real_real(exp_real,X4) = Y2 ) ) ) ).

% lemma_exp_total
tff(fact_6336_exp__le,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,one_one_real)),aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))) ).

% exp_le
tff(fact_6337_exp__double,axiom,
    ! [Z2: complex] : aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Z2)) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(exp_complex,Z2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% exp_double
tff(fact_6338_exp__double,axiom,
    ! [Z2: real] : aa_real_real(exp_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Z2)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(exp_real,Z2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% exp_double
tff(fact_6339_exp__half__le2,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% exp_half_le2
tff(fact_6340_arsinh__real__aux,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(plus_plus_real(X),aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)))) ).

% arsinh_real_aux
tff(fact_6341_arcosh__1,axiom,
    aa_real_real(arcosh_real,one_one_real) = zero_zero_real ).

% arcosh_1
tff(fact_6342_fold__atLeastAtMost__nat_Opsimps,axiom,
    ! [F: fun_nat_fun_num_num,A: nat,B: nat,Acc2: num] :
      ( accp_P3591886921626383704at_num(set_fo256927282339908995el_num,produc460779904311030791at_num(F,produc1195630363706982562at_num(A,aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,B),Acc2))))
     => ( aa_num_num(aa_nat_fun_num_num(set_fo8365102181078989356at_num(F,A),B),Acc2) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A),Acc2,aa_num_num(aa_nat_fun_num_num(set_fo8365102181078989356at_num(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),B),aa_num_num(aa_nat_fun_num_num(F,A),Acc2))) ) ) ).

% fold_atLeastAtMost_nat.psimps
tff(fact_6343_fold__atLeastAtMost__nat_Opsimps,axiom,
    ! [F: fun_nat_fun_nat_nat,A: nat,B: nat,Acc2: nat] :
      ( accp_P4694664897847628410at_nat(set_fo3699595496184130361el_nat,produc2818902965508115497at_nat(F,produc487386426758144856at_nat(A,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,B),Acc2))))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,A),B),Acc2) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A),Acc2,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),B),aa_nat_nat(aa_nat_fun_nat_nat(F,A),Acc2))) ) ) ).

% fold_atLeastAtMost_nat.psimps
tff(fact_6344_fold__atLeastAtMost__nat_Opelims,axiom,
    ! [X: fun_nat_fun_num_num,Xa: nat,Xb: nat,Xc: num,Y2: num] :
      ( ( aa_num_num(aa_nat_fun_num_num(set_fo8365102181078989356at_num(X,Xa),Xb),Xc) = Y2 )
     => ( accp_P3591886921626383704at_num(set_fo256927282339908995el_num,produc460779904311030791at_num(X,produc1195630363706982562at_num(Xa,aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,Xb),Xc))))
       => ~ ( ( Y2 = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Xa),Xc,aa_num_num(aa_nat_fun_num_num(set_fo8365102181078989356at_num(X,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),one_one_nat)),Xb),aa_num_num(aa_nat_fun_num_num(X,Xa),Xc))) )
           => ~ accp_P3591886921626383704at_num(set_fo256927282339908995el_num,produc460779904311030791at_num(X,produc1195630363706982562at_num(Xa,aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,Xb),Xc)))) ) ) ) ).

% fold_atLeastAtMost_nat.pelims
tff(fact_6345_fold__atLeastAtMost__nat_Opelims,axiom,
    ! [X: fun_nat_fun_nat_nat,Xa: nat,Xb: nat,Xc: nat,Y2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(X,Xa),Xb),Xc) = Y2 )
     => ( accp_P4694664897847628410at_nat(set_fo3699595496184130361el_nat,produc2818902965508115497at_nat(X,produc487386426758144856at_nat(Xa,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xb),Xc))))
       => ~ ( ( Y2 = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Xa),Xc,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(X,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),one_one_nat)),Xb),aa_nat_nat(aa_nat_fun_nat_nat(X,Xa),Xc))) )
           => ~ accp_P4694664897847628410at_nat(set_fo3699595496184130361el_nat,produc2818902965508115497at_nat(X,produc487386426758144856at_nat(Xa,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xb),Xc)))) ) ) ) ).

% fold_atLeastAtMost_nat.pelims
tff(fact_6346_tanh__real__altdef,axiom,
    ! [X: real] : aa_real_real(tanh_real,X) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(one_one_real),aa_real_real(exp_real,aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),X)))),aa_real_real(plus_plus_real(one_one_real),aa_real_real(exp_real,aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),X)))) ).

% tanh_real_altdef
tff(fact_6347_ln__one__minus__pos__lower__bound,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,X)),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(ln_ln_real,aa_real_real(minus_minus_real(one_one_real),X))) ) ) ).

% ln_one_minus_pos_lower_bound
tff(fact_6348_arctan__half,axiom,
    ! [X: real] : aa_real_real(arctan,X) = aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(arctan,aa_real_real(divide_divide_real(X),aa_real_real(plus_plus_real(one_one_real),aa_real_real(sqrt,aa_real_real(plus_plus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))) ).

% arctan_half
tff(fact_6349_tanh__real__le__iff,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(tanh_real,X)),aa_real_real(tanh_real,Y2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y2) ) ).

% tanh_real_le_iff
tff(fact_6350_ln__one,axiom,
    aa_real_real(ln_ln_real,one_one_real) = zero_zero_real ).

% ln_one
tff(fact_6351_zero__le__arctan__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(arctan,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X) ) ).

% zero_le_arctan_iff
tff(fact_6352_arctan__le__zero__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arctan,X)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),zero_zero_real) ) ).

% arctan_le_zero_iff
tff(fact_6353_tanh__real__nonpos__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(tanh_real,X)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),zero_zero_real) ) ).

% tanh_real_nonpos_iff
tff(fact_6354_tanh__real__nonneg__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(tanh_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X) ) ).

% tanh_real_nonneg_iff
tff(fact_6355_ln__le__cancel__iff,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,X)),aa_real_real(ln_ln_real,Y2))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y2) ) ) ) ).

% ln_le_cancel_iff
tff(fact_6356_ln__eq__zero__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( ( aa_real_real(ln_ln_real,X) = zero_zero_real )
      <=> ( X = one_one_real ) ) ) ).

% ln_eq_zero_iff
tff(fact_6357_ln__gt__zero__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,X))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X) ) ) ).

% ln_gt_zero_iff
tff(fact_6358_ln__less__zero__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,X)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),one_one_real) ) ) ).

% ln_less_zero_iff
tff(fact_6359_ln__ge__zero__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,X))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),X) ) ) ).

% ln_ge_zero_iff
tff(fact_6360_ln__le__zero__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,X)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),one_one_real) ) ) ).

% ln_le_zero_iff
tff(fact_6361_arctan__le__iff,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arctan,X)),aa_real_real(arctan,Y2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y2) ) ).

% arctan_le_iff
tff(fact_6362_arctan__monotone_H,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arctan,X)),aa_real_real(arctan,Y2)) ) ).

% arctan_monotone'
tff(fact_6363_tanh__real__lt__1,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tanh_real,X)),one_one_real) ).

% tanh_real_lt_1
tff(fact_6364_ln__bound,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,X)),X) ) ).

% ln_bound
tff(fact_6365_ln__gt__zero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,X)) ) ).

% ln_gt_zero
tff(fact_6366_ln__less__zero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,X)),zero_zero_real) ) ) ).

% ln_less_zero
tff(fact_6367_ln__gt__zero__imp__gt__one,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,X))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X) ) ) ).

% ln_gt_zero_imp_gt_one
tff(fact_6368_ln__ge__zero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,X)) ) ).

% ln_ge_zero
tff(fact_6369_tanh__real__gt__neg1,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(tanh_real,X)) ).

% tanh_real_gt_neg1
tff(fact_6370_ln__ge__zero__imp__ge__one,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,X))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),X) ) ) ).

% ln_ge_zero_imp_ge_one
tff(fact_6371_ln__add__one__self__le__self,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,aa_real_real(plus_plus_real(one_one_real),X))),X) ) ).

% ln_add_one_self_le_self
tff(fact_6372_ln__mult,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y2)
       => ( aa_real_real(ln_ln_real,aa_real_real(times_times_real(X),Y2)) = aa_real_real(plus_plus_real(aa_real_real(ln_ln_real,X)),aa_real_real(ln_ln_real,Y2)) ) ) ) ).

% ln_mult
tff(fact_6373_ln__eq__minus__one,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( ( aa_real_real(ln_ln_real,X) = aa_real_real(minus_minus_real(X),one_one_real) )
       => ( X = one_one_real ) ) ) ).

% ln_eq_minus_one
tff(fact_6374_ln__ge__iff,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),aa_real_real(ln_ln_real,X))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Y2)),X) ) ) ).

% ln_ge_iff
tff(fact_6375_ln__x__over__x__mono,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,one_one_real)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,Y2)),Y2)),aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,X)),X)) ) ) ).

% ln_x_over_x_mono
tff(fact_6376_ln__2__less__1,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),one_one_real) ).

% ln_2_less_1
tff(fact_6377_ln__le__minus__one,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,X)),aa_real_real(minus_minus_real(X),one_one_real)) ) ).

% ln_le_minus_one
tff(fact_6378_ln__diff__le,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(aa_real_real(ln_ln_real,X)),aa_real_real(ln_ln_real,Y2))),aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(X),Y2)),Y2)) ) ) ).

% ln_diff_le
tff(fact_6379_ln__add__one__self__le__self2,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,aa_real_real(plus_plus_real(one_one_real),X))),X) ) ).

% ln_add_one_self_le_self2
tff(fact_6380_fold__atLeastAtMost__nat_Osimps,axiom,
    ! [F: fun_nat_fun_nat_nat,A: nat,B: nat,Acc2: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,A),B),Acc2) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A),Acc2,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),B),aa_nat_nat(aa_nat_fun_nat_nat(F,A),Acc2))) ).

% fold_atLeastAtMost_nat.simps
tff(fact_6381_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [X: fun_nat_fun_nat_nat,Xa: nat,Xb: nat,Xc: nat,Y2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(X,Xa),Xb),Xc) = Y2 )
     => ( Y2 = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Xa),Xc,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(X,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xa),one_one_nat)),Xb),aa_nat_nat(aa_nat_fun_nat_nat(X,Xa),Xc))) ) ) ).

% fold_atLeastAtMost_nat.elims
tff(fact_6382_tanh__ln__real,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_real(tanh_real,aa_real_real(ln_ln_real,X)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)) ) ) ).

% tanh_ln_real
tff(fact_6383_arcosh__real__def,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),X)
     => ( aa_real_real(arcosh_real,X) = aa_real_real(ln_ln_real,aa_real_real(plus_plus_real(X),aa_real_real(sqrt,aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)))) ) ) ).

% arcosh_real_def
tff(fact_6384_ln__one__minus__pos__upper__bound,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,aa_real_real(minus_minus_real(one_one_real),X))),aa_real_real(uminus_uminus_real,X)) ) ) ).

% ln_one_minus_pos_upper_bound
tff(fact_6385_ln__sqrt,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_real(ln_ln_real,aa_real_real(sqrt,X)) = aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,X)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% ln_sqrt
tff(fact_6386_tanh__altdef,axiom,
    ! [X: real] : aa_real_real(tanh_real,X) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(exp_real,X)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,X)))),aa_real_real(plus_plus_real(aa_real_real(exp_real,X)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,X)))) ).

% tanh_altdef
tff(fact_6387_tanh__altdef,axiom,
    ! [X: complex] : aa_complex_complex(tanh_complex,X) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(exp_complex,X)),aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,X)))),aa_complex_complex(plus_plus_complex(aa_complex_complex(exp_complex,X)),aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,X)))) ).

% tanh_altdef
tff(fact_6388_ln__one__plus__pos__lower__bound,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(X),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(ln_ln_real,aa_real_real(plus_plus_real(one_one_real),X))) ) ) ).

% ln_one_plus_pos_lower_bound
tff(fact_6389_artanh__def,axiom,
    ! [X: real] : aa_real_real(artanh_real,X) = aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(one_one_real),X)),aa_real_real(minus_minus_real(one_one_real),X)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% artanh_def
tff(fact_6390_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_real_real(ln_ln_real,aa_real_real(plus_plus_real(one_one_real),X))),X))),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonpos
tff(fact_6391_arsinh__real__def,axiom,
    ! [X: real] : aa_real_real(arsinh_real,X) = aa_real_real(ln_ln_real,aa_real_real(plus_plus_real(X),aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)))) ).

% arsinh_real_def
tff(fact_6392_abs__ln__one__plus__x__minus__x__bound,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X)),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_real_real(ln_ln_real,aa_real_real(plus_plus_real(one_one_real),X))),X))),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% abs_ln_one_plus_x_minus_x_bound
tff(fact_6393_arctan__double,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,X)),one_one_real)
     => ( aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(arctan,X)) = aa_real_real(arctan,aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X)),aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% arctan_double
tff(fact_6394_log__base__10__eq1,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),X) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,aa_real_real(exp_real,one_one_real))),aa_real_real(ln_ln_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))))),aa_real_real(ln_ln_real,X)) ) ) ).

% log_base_10_eq1
tff(fact_6395_abs__idempotent,axiom,
    ! [A: real] : aa_real_real(abs_abs_real,aa_real_real(abs_abs_real,A)) = aa_real_real(abs_abs_real,A) ).

% abs_idempotent
tff(fact_6396_abs__idempotent,axiom,
    ! [A: int] : aa_int_int(abs_abs_int,aa_int_int(abs_abs_int,A)) = aa_int_int(abs_abs_int,A) ).

% abs_idempotent
tff(fact_6397_abs__idempotent,axiom,
    ! [A: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(abs_abs_rat,A)) = aa_rat_rat(abs_abs_rat,A) ).

% abs_idempotent
tff(fact_6398_abs__idempotent,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(abs_abs_Code_integer(A)) = abs_abs_Code_integer(A) ).

% abs_idempotent
tff(fact_6399_abs__zero,axiom,
    abs_abs_Code_integer(zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% abs_zero
tff(fact_6400_abs__zero,axiom,
    aa_real_real(abs_abs_real,zero_zero_real) = zero_zero_real ).

% abs_zero
tff(fact_6401_abs__zero,axiom,
    aa_rat_rat(abs_abs_rat,zero_zero_rat) = zero_zero_rat ).

% abs_zero
tff(fact_6402_abs__zero,axiom,
    aa_int_int(abs_abs_int,zero_zero_int) = zero_zero_int ).

% abs_zero
tff(fact_6403_abs__eq__0,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer(A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0
tff(fact_6404_abs__eq__0,axiom,
    ! [A: real] :
      ( ( aa_real_real(abs_abs_real,A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% abs_eq_0
tff(fact_6405_abs__eq__0,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(abs_abs_rat,A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% abs_eq_0
tff(fact_6406_abs__eq__0,axiom,
    ! [A: int] :
      ( ( aa_int_int(abs_abs_int,A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% abs_eq_0
tff(fact_6407_abs__0__eq,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger = abs_abs_Code_integer(A) )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_0_eq
tff(fact_6408_abs__0__eq,axiom,
    ! [A: real] :
      ( ( zero_zero_real = aa_real_real(abs_abs_real,A) )
    <=> ( A = zero_zero_real ) ) ).

% abs_0_eq
tff(fact_6409_abs__0__eq,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = aa_rat_rat(abs_abs_rat,A) )
    <=> ( A = zero_zero_rat ) ) ).

% abs_0_eq
tff(fact_6410_abs__0__eq,axiom,
    ! [A: int] :
      ( ( zero_zero_int = aa_int_int(abs_abs_int,A) )
    <=> ( A = zero_zero_int ) ) ).

% abs_0_eq
tff(fact_6411_abs__numeral,axiom,
    ! [Nb: num] : abs_abs_Code_integer(numera6620942414471956472nteger(Nb)) = numera6620942414471956472nteger(Nb) ).

% abs_numeral
tff(fact_6412_abs__numeral,axiom,
    ! [Nb: num] : aa_real_real(abs_abs_real,aa_num_real(numeral_numeral_real,Nb)) = aa_num_real(numeral_numeral_real,Nb) ).

% abs_numeral
tff(fact_6413_abs__numeral,axiom,
    ! [Nb: num] : aa_rat_rat(abs_abs_rat,aa_num_rat(numeral_numeral_rat,Nb)) = aa_num_rat(numeral_numeral_rat,Nb) ).

% abs_numeral
tff(fact_6414_abs__numeral,axiom,
    ! [Nb: num] : aa_int_int(abs_abs_int,aa_num_int(numeral_numeral_int,Nb)) = aa_num_int(numeral_numeral_int,Nb) ).

% abs_numeral
tff(fact_6415_abs__mult__self__eq,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(A)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),A) ).

% abs_mult_self_eq
tff(fact_6416_abs__mult__self__eq,axiom,
    ! [A: real] : aa_real_real(times_times_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,A)) = aa_real_real(times_times_real(A),A) ).

% abs_mult_self_eq
tff(fact_6417_abs__mult__self__eq,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,A)) = aa_rat_rat(times_times_rat(A),A) ).

% abs_mult_self_eq
tff(fact_6418_abs__mult__self__eq,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,A)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),A) ).

% abs_mult_self_eq
tff(fact_6419_abs__1,axiom,
    abs_abs_Code_integer(one_one_Code_integer) = one_one_Code_integer ).

% abs_1
tff(fact_6420_abs__1,axiom,
    aa_complex_complex(abs_abs_complex,one_one_complex) = one_one_complex ).

% abs_1
tff(fact_6421_abs__1,axiom,
    aa_real_real(abs_abs_real,one_one_real) = one_one_real ).

% abs_1
tff(fact_6422_abs__1,axiom,
    aa_rat_rat(abs_abs_rat,one_one_rat) = one_one_rat ).

% abs_1
tff(fact_6423_abs__1,axiom,
    aa_int_int(abs_abs_int,one_one_int) = one_one_int ).

% abs_1
tff(fact_6424_abs__add__abs,axiom,
    ! [A: code_integer,B: code_integer] : abs_abs_Code_integer(plus_p5714425477246183910nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B))) = plus_p5714425477246183910nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B)) ).

% abs_add_abs
tff(fact_6425_abs__add__abs,axiom,
    ! [A: real,B: real] : aa_real_real(abs_abs_real,aa_real_real(plus_plus_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))) = aa_real_real(plus_plus_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ).

% abs_add_abs
tff(fact_6426_abs__add__abs,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))) = aa_rat_rat(plus_plus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ).

% abs_add_abs
tff(fact_6427_abs__add__abs,axiom,
    ! [A: int,B: int] : aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)) ).

% abs_add_abs
tff(fact_6428_abs__divide,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(abs_abs_complex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(abs_abs_complex,A)),aa_complex_complex(abs_abs_complex,B)) ).

% abs_divide
tff(fact_6429_abs__divide,axiom,
    ! [A: real,B: real] : aa_real_real(abs_abs_real,aa_real_real(divide_divide_real(A),B)) = aa_real_real(divide_divide_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ).

% abs_divide
tff(fact_6430_abs__divide,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(divide_divide_rat(A),B)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ).

% abs_divide
tff(fact_6431_abs__minus__cancel,axiom,
    ! [A: real] : aa_real_real(abs_abs_real,aa_real_real(uminus_uminus_real,A)) = aa_real_real(abs_abs_real,A) ).

% abs_minus_cancel
tff(fact_6432_abs__minus__cancel,axiom,
    ! [A: int] : aa_int_int(abs_abs_int,aa_int_int(uminus_uminus_int,A)) = aa_int_int(abs_abs_int,A) ).

% abs_minus_cancel
tff(fact_6433_abs__minus__cancel,axiom,
    ! [A: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(uminus_uminus_rat,A)) = aa_rat_rat(abs_abs_rat,A) ).

% abs_minus_cancel
tff(fact_6434_abs__minus__cancel,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = abs_abs_Code_integer(A) ).

% abs_minus_cancel
tff(fact_6435_abs__le__zero__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),zero_z3403309356797280102nteger)
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_le_zero_iff
tff(fact_6436_abs__le__zero__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),zero_zero_real)
    <=> ( A = zero_zero_real ) ) ).

% abs_le_zero_iff
tff(fact_6437_abs__le__zero__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),zero_zero_rat)
    <=> ( A = zero_zero_rat ) ) ).

% abs_le_zero_iff
tff(fact_6438_abs__le__zero__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),zero_zero_int)
    <=> ( A = zero_zero_int ) ) ).

% abs_le_zero_iff
tff(fact_6439_abs__le__self__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),A)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A) ) ).

% abs_le_self_iff
tff(fact_6440_abs__le__self__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% abs_le_self_iff
tff(fact_6441_abs__le__self__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% abs_le_self_iff
tff(fact_6442_abs__le__self__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),A)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).

% abs_le_self_iff
tff(fact_6443_abs__of__nonneg,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( abs_abs_Code_integer(A) = A ) ) ).

% abs_of_nonneg
tff(fact_6444_abs__of__nonneg,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_real(abs_abs_real,A) = A ) ) ).

% abs_of_nonneg
tff(fact_6445_abs__of__nonneg,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_rat(abs_abs_rat,A) = A ) ) ).

% abs_of_nonneg
tff(fact_6446_abs__of__nonneg,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_int(abs_abs_int,A) = A ) ) ).

% abs_of_nonneg
tff(fact_6447_zero__less__abs__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),abs_abs_Code_integer(A))
    <=> ( A != zero_z3403309356797280102nteger ) ) ).

% zero_less_abs_iff
tff(fact_6448_zero__less__abs__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,A))
    <=> ( A != zero_zero_real ) ) ).

% zero_less_abs_iff
tff(fact_6449_zero__less__abs__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(abs_abs_rat,A))
    <=> ( A != zero_zero_rat ) ) ).

% zero_less_abs_iff
tff(fact_6450_zero__less__abs__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(abs_abs_int,A))
    <=> ( A != zero_zero_int ) ) ).

% zero_less_abs_iff
tff(fact_6451_abs__neg__numeral,axiom,
    ! [Nb: num] : aa_real_real(abs_abs_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_num_real(numeral_numeral_real,Nb) ).

% abs_neg_numeral
tff(fact_6452_abs__neg__numeral,axiom,
    ! [Nb: num] : aa_int_int(abs_abs_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_num_int(numeral_numeral_int,Nb) ).

% abs_neg_numeral
tff(fact_6453_abs__neg__numeral,axiom,
    ! [Nb: num] : aa_rat_rat(abs_abs_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_num_rat(numeral_numeral_rat,Nb) ).

% abs_neg_numeral
tff(fact_6454_abs__neg__numeral,axiom,
    ! [Nb: num] : abs_abs_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) = numera6620942414471956472nteger(Nb) ).

% abs_neg_numeral
tff(fact_6455_abs__neg__one,axiom,
    aa_real_real(abs_abs_real,aa_real_real(uminus_uminus_real,one_one_real)) = one_one_real ).

% abs_neg_one
tff(fact_6456_abs__neg__one,axiom,
    aa_int_int(abs_abs_int,aa_int_int(uminus_uminus_int,one_one_int)) = one_one_int ).

% abs_neg_one
tff(fact_6457_abs__neg__one,axiom,
    aa_rat_rat(abs_abs_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) = one_one_rat ).

% abs_neg_one
tff(fact_6458_abs__neg__one,axiom,
    abs_abs_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = one_one_Code_integer ).

% abs_neg_one
tff(fact_6459_abs__power__minus,axiom,
    ! [A: real,Nb: nat] : aa_real_real(abs_abs_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),Nb)) = aa_real_real(abs_abs_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ).

% abs_power_minus
tff(fact_6460_abs__power__minus,axiom,
    ! [A: int,Nb: nat] : aa_int_int(abs_abs_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,A)),Nb)) = aa_int_int(abs_abs_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)) ).

% abs_power_minus
tff(fact_6461_abs__power__minus,axiom,
    ! [A: rat,Nb: nat] : aa_rat_rat(abs_abs_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),Nb)) = aa_rat_rat(abs_abs_rat,aa_nat_rat(power_power_rat(A),Nb)) ).

% abs_power_minus
tff(fact_6462_abs__power__minus,axiom,
    ! [A: code_integer,Nb: nat] : abs_abs_Code_integer(aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),Nb)) = abs_abs_Code_integer(aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)) ).

% abs_power_minus
tff(fact_6463_log__one,axiom,
    ! [A: real] : aa_real_real(log(A),one_one_real) = zero_zero_real ).

% log_one
tff(fact_6464_real__sqrt__mult__self,axiom,
    ! [A: real] : aa_real_real(times_times_real(aa_real_real(sqrt,A)),aa_real_real(sqrt,A)) = aa_real_real(abs_abs_real,A) ).

% real_sqrt_mult_self
tff(fact_6465_real__sqrt__abs2,axiom,
    ! [X: real] : aa_real_real(sqrt,aa_real_real(times_times_real(X),X)) = aa_real_real(abs_abs_real,X) ).

% real_sqrt_abs2
tff(fact_6466_divide__le__0__abs__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(A),aa_real_real(abs_abs_real,B))),zero_zero_real)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
        | ( B = zero_zero_real ) ) ) ).

% divide_le_0_abs_iff
tff(fact_6467_divide__le__0__abs__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(A),aa_rat_rat(abs_abs_rat,B))),zero_zero_rat)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
        | ( B = zero_zero_rat ) ) ) ).

% divide_le_0_abs_iff
tff(fact_6468_zero__le__divide__abs__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(A),aa_real_real(abs_abs_real,B)))
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
        | ( B = zero_zero_real ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_6469_zero__le__divide__abs__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(A),aa_rat_rat(abs_abs_rat,B)))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
        | ( B = zero_zero_rat ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_6470_abs__of__nonpos,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_real(abs_abs_real,A) = aa_real_real(uminus_uminus_real,A) ) ) ).

% abs_of_nonpos
tff(fact_6471_abs__of__nonpos,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger)
     => ( abs_abs_Code_integer(A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).

% abs_of_nonpos
tff(fact_6472_abs__of__nonpos,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_rat(abs_abs_rat,A) = aa_rat_rat(uminus_uminus_rat,A) ) ) ).

% abs_of_nonpos
tff(fact_6473_abs__of__nonpos,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_int(abs_abs_int,A) = aa_int_int(uminus_uminus_int,A) ) ) ).

% abs_of_nonpos
tff(fact_6474_zero__less__log__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(log(A),X))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X) ) ) ) ).

% zero_less_log_cancel_iff
tff(fact_6475_log__less__zero__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(log(A),X)),zero_zero_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),one_one_real) ) ) ) ).

% log_less_zero_cancel_iff
tff(fact_6476_one__less__log__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(log(A),X))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),X) ) ) ) ).

% one_less_log_cancel_iff
tff(fact_6477_log__less__one__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(log(A),X)),one_one_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),A) ) ) ) ).

% log_less_one_cancel_iff
tff(fact_6478_log__less__cancel__iff,axiom,
    ! [A: real,X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(log(A),X)),aa_real_real(log(A),Y2))
          <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2) ) ) ) ) ).

% log_less_cancel_iff
tff(fact_6479_log__eq__one,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_real(log(A),A) = one_one_real ) ) ) ).

% log_eq_one
tff(fact_6480_artanh__minus__real,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,X)),one_one_real)
     => ( aa_real_real(artanh_real,aa_real_real(uminus_uminus_real,X)) = aa_real_real(uminus_uminus_real,aa_real_real(artanh_real,X)) ) ) ).

% artanh_minus_real
tff(fact_6481_zero__less__power__abs__iff,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),Nb))
    <=> ( ( A != zero_z3403309356797280102nteger )
        | ( Nb = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_6482_zero__less__power__abs__iff,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),Nb))
    <=> ( ( A != zero_zero_real )
        | ( Nb = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_6483_zero__less__power__abs__iff,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A)),Nb))
    <=> ( ( A != zero_zero_rat )
        | ( Nb = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_6484_zero__less__power__abs__iff,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(abs_abs_int,A)),Nb))
    <=> ( ( A != zero_zero_int )
        | ( Nb = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_6485_abs__power2,axiom,
    ! [A: rat] : aa_rat_rat(abs_abs_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% abs_power2
tff(fact_6486_abs__power2,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% abs_power2
tff(fact_6487_abs__power2,axiom,
    ! [A: real] : aa_real_real(abs_abs_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% abs_power2
tff(fact_6488_abs__power2,axiom,
    ! [A: int] : aa_int_int(abs_abs_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% abs_power2
tff(fact_6489_power2__abs,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_abs
tff(fact_6490_power2__abs,axiom,
    ! [A: code_integer] : aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_abs
tff(fact_6491_power2__abs,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_abs
tff(fact_6492_power2__abs,axiom,
    ! [A: int] : aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(abs_abs_int,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_abs
tff(fact_6493_zero__le__log__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(log(A),X))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),X) ) ) ) ).

% zero_le_log_cancel_iff
tff(fact_6494_log__le__zero__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(log(A),X)),zero_zero_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),one_one_real) ) ) ) ).

% log_le_zero_cancel_iff
tff(fact_6495_one__le__log__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(log(A),X))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X) ) ) ) ).

% one_le_log_cancel_iff
tff(fact_6496_log__le__one__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(log(A),X)),one_one_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),A) ) ) ) ).

% log_le_one_cancel_iff
tff(fact_6497_log__le__cancel__iff,axiom,
    ! [A: real,X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(log(A),X)),aa_real_real(log(A),Y2))
          <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y2) ) ) ) ) ).

% log_le_cancel_iff
tff(fact_6498_power__even__abs__numeral,axiom,
    ! [W: num,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A)),aa_num_nat(numeral_numeral_nat,W)) = aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,W)) ) ) ).

% power_even_abs_numeral
tff(fact_6499_power__even__abs__numeral,axiom,
    ! [W: num,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),aa_num_nat(numeral_numeral_nat,W)) = aa_nat_Code_integer(power_8256067586552552935nteger(A),aa_num_nat(numeral_numeral_nat,W)) ) ) ).

% power_even_abs_numeral
tff(fact_6500_power__even__abs__numeral,axiom,
    ! [W: num,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),aa_num_nat(numeral_numeral_nat,W)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,W)) ) ) ).

% power_even_abs_numeral
tff(fact_6501_power__even__abs__numeral,axiom,
    ! [W: num,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
     => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(abs_abs_int,A)),aa_num_nat(numeral_numeral_nat,W)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,A),aa_num_nat(numeral_numeral_nat,W)) ) ) ).

% power_even_abs_numeral
tff(fact_6502_real__sqrt__abs,axiom,
    ! [X: real] : aa_real_real(sqrt,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_real_real(abs_abs_real,X) ).

% real_sqrt_abs
tff(fact_6503_abs__le__D1,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% abs_le_D1
tff(fact_6504_abs__le__D1,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B) ) ).

% abs_le_D1
tff(fact_6505_abs__le__D1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% abs_le_D1
tff(fact_6506_abs__le__D1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% abs_le_D1
tff(fact_6507_abs__ge__self,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(abs_abs_real,A)) ).

% abs_ge_self
tff(fact_6508_abs__ge__self,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),abs_abs_Code_integer(A)) ).

% abs_ge_self
tff(fact_6509_abs__ge__self,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(abs_abs_rat,A)) ).

% abs_ge_self
tff(fact_6510_abs__ge__self,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(abs_abs_int,A)) ).

% abs_ge_self
tff(fact_6511_abs__mult,axiom,
    ! [A: code_integer,B: code_integer] : abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(B)) ).

% abs_mult
tff(fact_6512_abs__mult,axiom,
    ! [A: real,B: real] : aa_real_real(abs_abs_real,aa_real_real(times_times_real(A),B)) = aa_real_real(times_times_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ).

% abs_mult
tff(fact_6513_abs__mult,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(times_times_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ).

% abs_mult
tff(fact_6514_abs__mult,axiom,
    ! [A: int,B: int] : aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)) ).

% abs_mult
tff(fact_6515_abs__one,axiom,
    abs_abs_Code_integer(one_one_Code_integer) = one_one_Code_integer ).

% abs_one
tff(fact_6516_abs__one,axiom,
    aa_real_real(abs_abs_real,one_one_real) = one_one_real ).

% abs_one
tff(fact_6517_abs__one,axiom,
    aa_rat_rat(abs_abs_rat,one_one_rat) = one_one_rat ).

% abs_one
tff(fact_6518_abs__one,axiom,
    aa_int_int(abs_abs_int,one_one_int) = one_one_int ).

% abs_one
tff(fact_6519_abs__minus__commute,axiom,
    ! [A: code_integer,B: code_integer] : abs_abs_Code_integer(minus_8373710615458151222nteger(A,B)) = abs_abs_Code_integer(minus_8373710615458151222nteger(B,A)) ).

% abs_minus_commute
tff(fact_6520_abs__minus__commute,axiom,
    ! [A: real,B: real] : aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(A),B)) = aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(B),A)) ).

% abs_minus_commute
tff(fact_6521_abs__minus__commute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(A),B)) = aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(B),A)) ).

% abs_minus_commute
tff(fact_6522_abs__minus__commute,axiom,
    ! [A: int,B: int] : aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) = aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)) ).

% abs_minus_commute
tff(fact_6523_power__abs,axiom,
    ! [A: rat,Nb: nat] : aa_rat_rat(abs_abs_rat,aa_nat_rat(power_power_rat(A),Nb)) = aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A)),Nb) ).

% power_abs
tff(fact_6524_power__abs,axiom,
    ! [A: code_integer,Nb: nat] : abs_abs_Code_integer(aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)) = aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),Nb) ).

% power_abs
tff(fact_6525_power__abs,axiom,
    ! [A: real,Nb: nat] : aa_real_real(abs_abs_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),Nb) ).

% power_abs
tff(fact_6526_power__abs,axiom,
    ! [A: int,Nb: nat] : aa_int_int(abs_abs_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(abs_abs_int,A)),Nb) ).

% power_abs
tff(fact_6527_abs__ge__zero,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),abs_abs_Code_integer(A)) ).

% abs_ge_zero
tff(fact_6528_abs__ge__zero,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(abs_abs_real,A)) ).

% abs_ge_zero
tff(fact_6529_abs__ge__zero,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(abs_abs_rat,A)) ).

% abs_ge_zero
tff(fact_6530_abs__ge__zero,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(abs_abs_int,A)) ).

% abs_ge_zero
tff(fact_6531_abs__of__pos,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
     => ( abs_abs_Code_integer(A) = A ) ) ).

% abs_of_pos
tff(fact_6532_abs__of__pos,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_real(abs_abs_real,A) = A ) ) ).

% abs_of_pos
tff(fact_6533_abs__of__pos,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_rat(abs_abs_rat,A) = A ) ) ).

% abs_of_pos
tff(fact_6534_abs__of__pos,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_int(abs_abs_int,A) = A ) ) ).

% abs_of_pos
tff(fact_6535_abs__not__less__zero,axiom,
    ! [A: code_integer] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(A)),zero_z3403309356797280102nteger) ).

% abs_not_less_zero
tff(fact_6536_abs__not__less__zero,axiom,
    ! [A: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,A)),zero_zero_real) ).

% abs_not_less_zero
tff(fact_6537_abs__not__less__zero,axiom,
    ! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,A)),zero_zero_rat) ).

% abs_not_less_zero
tff(fact_6538_abs__not__less__zero,axiom,
    ! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,A)),zero_zero_int) ).

% abs_not_less_zero
tff(fact_6539_abs__triangle__ineq,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(plus_p5714425477246183910nteger(A,B))),plus_p5714425477246183910nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B))) ).

% abs_triangle_ineq
tff(fact_6540_abs__triangle__ineq,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(plus_plus_real(A),B))),aa_real_real(plus_plus_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))) ).

% abs_triangle_ineq
tff(fact_6541_abs__triangle__ineq,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(plus_plus_rat(A),B))),aa_rat_rat(plus_plus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))) ).

% abs_triangle_ineq
tff(fact_6542_abs__triangle__ineq,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))) ).

% abs_triangle_ineq
tff(fact_6543_abs__mult__less,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(A)),C)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(B)),D)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(B))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),D)) ) ) ).

% abs_mult_less
tff(fact_6544_abs__mult__less,axiom,
    ! [A: real,C: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,A)),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,B)),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))),aa_real_real(times_times_real(C),D)) ) ) ).

% abs_mult_less
tff(fact_6545_abs__mult__less,axiom,
    ! [A: rat,C: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,A)),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,B)),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))),aa_rat_rat(times_times_rat(C),D)) ) ) ).

% abs_mult_less
tff(fact_6546_abs__mult__less,axiom,
    ! [A: int,C: int,B: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,A)),C)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,B)),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))),aa_int_int(aa_int_fun_int_int(times_times_int,C),D)) ) ) ).

% abs_mult_less
tff(fact_6547_abs__triangle__ineq2,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,minus_8373710615458151222nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B))),abs_abs_Code_integer(minus_8373710615458151222nteger(A,B))) ).

% abs_triangle_ineq2
tff(fact_6548_abs__triangle__ineq2,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))),aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(A),B))) ).

% abs_triangle_ineq2
tff(fact_6549_abs__triangle__ineq2,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(A),B))) ).

% abs_triangle_ineq2
tff(fact_6550_abs__triangle__ineq2,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))) ).

% abs_triangle_ineq2
tff(fact_6551_abs__triangle__ineq3,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B)))),abs_abs_Code_integer(minus_8373710615458151222nteger(A,B))) ).

% abs_triangle_ineq3
tff(fact_6552_abs__triangle__ineq3,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)))),aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(A),B))) ).

% abs_triangle_ineq3
tff(fact_6553_abs__triangle__ineq3,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(A),B))) ).

% abs_triangle_ineq3
tff(fact_6554_abs__triangle__ineq3,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))) ).

% abs_triangle_ineq3
tff(fact_6555_abs__triangle__ineq2__sym,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,minus_8373710615458151222nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B))),abs_abs_Code_integer(minus_8373710615458151222nteger(B,A))) ).

% abs_triangle_ineq2_sym
tff(fact_6556_abs__triangle__ineq2__sym,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))),aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(B),A))) ).

% abs_triangle_ineq2_sym
tff(fact_6557_abs__triangle__ineq2__sym,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(B),A))) ).

% abs_triangle_ineq2_sym
tff(fact_6558_abs__triangle__ineq2__sym,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A))) ).

% abs_triangle_ineq2_sym
tff(fact_6559_abs__leI,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B) ) ) ).

% abs_leI
tff(fact_6560_abs__leI,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B) ) ) ).

% abs_leI
tff(fact_6561_abs__leI,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B) ) ) ).

% abs_leI
tff(fact_6562_abs__leI,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B) ) ) ).

% abs_leI
tff(fact_6563_abs__le__D2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B) ) ).

% abs_le_D2
tff(fact_6564_abs__le__D2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) ) ).

% abs_le_D2
tff(fact_6565_abs__le__D2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B) ) ).

% abs_le_D2
tff(fact_6566_abs__le__D2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B) ) ).

% abs_le_D2
tff(fact_6567_abs__le__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B) ) ) ).

% abs_le_iff
tff(fact_6568_abs__le__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) ) ) ).

% abs_le_iff
tff(fact_6569_abs__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B) ) ) ).

% abs_le_iff
tff(fact_6570_abs__le__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B) ) ) ).

% abs_le_iff
tff(fact_6571_abs__ge__minus__self,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(abs_abs_real,A)) ).

% abs_ge_minus_self
tff(fact_6572_abs__ge__minus__self,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),abs_abs_Code_integer(A)) ).

% abs_ge_minus_self
tff(fact_6573_abs__ge__minus__self,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(abs_abs_rat,A)) ).

% abs_ge_minus_self
tff(fact_6574_abs__ge__minus__self,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(abs_abs_int,A)) ).

% abs_ge_minus_self
tff(fact_6575_nonzero__abs__divide,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(abs_abs_real,aa_real_real(divide_divide_real(A),B)) = aa_real_real(divide_divide_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ) ) ).

% nonzero_abs_divide
tff(fact_6576_nonzero__abs__divide,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(abs_abs_rat,aa_rat_rat(divide_divide_rat(A),B)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ) ) ).

% nonzero_abs_divide
tff(fact_6577_abs__less__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,A)),B)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,A)),B) ) ) ).

% abs_less_iff
tff(fact_6578_abs__less__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,A)),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,A)),B) ) ) ).

% abs_less_iff
tff(fact_6579_abs__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,A)),B)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A)),B) ) ) ).

% abs_less_iff
tff(fact_6580_abs__less__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(A)),B)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) ) ) ).

% abs_less_iff
tff(fact_6581_dense__eq0__I,axiom,
    ! [X: real] :
      ( ! [E2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X)),E2) )
     => ( X = zero_zero_real ) ) ).

% dense_eq0_I
tff(fact_6582_dense__eq0__I,axiom,
    ! [X: rat] :
      ( ! [E2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,X)),E2) )
     => ( X = zero_zero_rat ) ) ).

% dense_eq0_I
tff(fact_6583_abs__eq__mult,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
          | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),zero_z3403309356797280102nteger) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
          | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B),zero_z3403309356797280102nteger) ) )
     => ( abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(B)) ) ) ).

% abs_eq_mult
tff(fact_6584_abs__eq__mult,axiom,
    ! [A: real,B: real] :
      ( ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          | aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
          | aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) ) )
     => ( aa_real_real(abs_abs_real,aa_real_real(times_times_real(A),B)) = aa_real_real(times_times_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ) ) ).

% abs_eq_mult
tff(fact_6585_abs__eq__mult,axiom,
    ! [A: rat,B: rat] :
      ( ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
          | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
          | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),zero_zero_rat) ) )
     => ( aa_rat_rat(abs_abs_rat,aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(times_times_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ) ) ).

% abs_eq_mult
tff(fact_6586_abs__eq__mult,axiom,
    ! [A: int,B: int] :
      ( ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
          | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) ) )
     => ( aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)) ) ) ).

% abs_eq_mult
tff(fact_6587_abs__mult__pos,axiom,
    ! [X: code_integer,Y2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(Y2)),X) = abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y2),X)) ) ) ).

% abs_mult_pos
tff(fact_6588_abs__mult__pos,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_real(times_times_real(aa_real_real(abs_abs_real,Y2)),X) = aa_real_real(abs_abs_real,aa_real_real(times_times_real(Y2),X)) ) ) ).

% abs_mult_pos
tff(fact_6589_abs__mult__pos,axiom,
    ! [X: rat,Y2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X)
     => ( aa_rat_rat(times_times_rat(aa_rat_rat(abs_abs_rat,Y2)),X) = aa_rat_rat(abs_abs_rat,aa_rat_rat(times_times_rat(Y2),X)) ) ) ).

% abs_mult_pos
tff(fact_6590_abs__mult__pos,axiom,
    ! [X: int,Y2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,Y2)),X) = aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,Y2),X)) ) ) ).

% abs_mult_pos
tff(fact_6591_abs__eq__iff_H,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(abs_abs_real,A) = B )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
        & ( ( A = B )
          | ( A = aa_real_real(uminus_uminus_real,B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_6592_abs__eq__iff_H,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( abs_abs_Code_integer(A) = B )
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B)
        & ( ( A = B )
          | ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_6593_abs__eq__iff_H,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(abs_abs_rat,A) = B )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
        & ( ( A = B )
          | ( A = aa_rat_rat(uminus_uminus_rat,B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_6594_abs__eq__iff_H,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(abs_abs_int,A) = B )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
        & ( ( A = B )
          | ( A = aa_int_int(uminus_uminus_int,B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_6595_eq__abs__iff_H,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(abs_abs_real,B) )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
        & ( ( B = A )
          | ( B = aa_real_real(uminus_uminus_real,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_6596_eq__abs__iff_H,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = abs_abs_Code_integer(B) )
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
        & ( ( B = A )
          | ( B = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_6597_eq__abs__iff_H,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(abs_abs_rat,B) )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
        & ( ( B = A )
          | ( B = aa_rat_rat(uminus_uminus_rat,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_6598_eq__abs__iff_H,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(abs_abs_int,B) )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
        & ( ( B = A )
          | ( B = aa_int_int(uminus_uminus_int,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_6599_abs__minus__le__zero,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(abs_abs_real,A))),zero_zero_real) ).

% abs_minus_le_zero
tff(fact_6600_abs__minus__le__zero,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,abs_abs_Code_integer(A))),zero_z3403309356797280102nteger) ).

% abs_minus_le_zero
tff(fact_6601_abs__minus__le__zero,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(abs_abs_rat,A))),zero_zero_rat) ).

% abs_minus_le_zero
tff(fact_6602_abs__minus__le__zero,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_int_int(abs_abs_int,A))),zero_zero_int) ).

% abs_minus_le_zero
tff(fact_6603_zero__le__power__abs,axiom,
    ! [A: code_integer,Nb: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),Nb)) ).

% zero_le_power_abs
tff(fact_6604_zero__le__power__abs,axiom,
    ! [A: real,Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),Nb)) ).

% zero_le_power_abs
tff(fact_6605_zero__le__power__abs,axiom,
    ! [A: rat,Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A)),Nb)) ).

% zero_le_power_abs
tff(fact_6606_zero__le__power__abs,axiom,
    ! [A: int,Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(abs_abs_int,A)),Nb)) ).

% zero_le_power_abs
tff(fact_6607_abs__div__pos,axiom,
    ! [Y2: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y2)
     => ( aa_real_real(divide_divide_real(aa_real_real(abs_abs_real,X)),Y2) = aa_real_real(abs_abs_real,aa_real_real(divide_divide_real(X),Y2)) ) ) ).

% abs_div_pos
tff(fact_6608_abs__div__pos,axiom,
    ! [Y2: rat,X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y2)
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(abs_abs_rat,X)),Y2) = aa_rat_rat(abs_abs_rat,aa_rat_rat(divide_divide_rat(X),Y2)) ) ) ).

% abs_div_pos
tff(fact_6609_abs__if,axiom,
    ! [A: real] :
      aa_real_real(abs_abs_real,A) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real),aa_real_real(uminus_uminus_real,A),A) ).

% abs_if
tff(fact_6610_abs__if,axiom,
    ! [A: int] :
      aa_int_int(abs_abs_int,A) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int),aa_int_int(uminus_uminus_int,A),A) ).

% abs_if
tff(fact_6611_abs__if,axiom,
    ! [A: rat] :
      aa_rat_rat(abs_abs_rat,A) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat),aa_rat_rat(uminus_uminus_rat,A),A) ).

% abs_if
tff(fact_6612_abs__if,axiom,
    ! [A: code_integer] :
      abs_abs_Code_integer(A) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),A) ).

% abs_if
tff(fact_6613_abs__of__neg,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_real(abs_abs_real,A) = aa_real_real(uminus_uminus_real,A) ) ) ).

% abs_of_neg
tff(fact_6614_abs__of__neg,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_int(abs_abs_int,A) = aa_int_int(uminus_uminus_int,A) ) ) ).

% abs_of_neg
tff(fact_6615_abs__of__neg,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_rat(abs_abs_rat,A) = aa_rat_rat(uminus_uminus_rat,A) ) ) ).

% abs_of_neg
tff(fact_6616_abs__of__neg,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)
     => ( abs_abs_Code_integer(A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).

% abs_of_neg
tff(fact_6617_abs__if__raw,axiom,
    ! [X2: real] :
      aa_real_real(abs_abs_real,X2) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,X2),zero_zero_real),aa_real_real(uminus_uminus_real,X2),X2) ).

% abs_if_raw
tff(fact_6618_abs__if__raw,axiom,
    ! [X2: int] :
      aa_int_int(abs_abs_int,X2) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,X2),zero_zero_int),aa_int_int(uminus_uminus_int,X2),X2) ).

% abs_if_raw
tff(fact_6619_abs__if__raw,axiom,
    ! [X2: rat] :
      aa_rat_rat(abs_abs_rat,X2) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X2),zero_zero_rat),aa_rat_rat(uminus_uminus_rat,X2),X2) ).

% abs_if_raw
tff(fact_6620_abs__if__raw,axiom,
    ! [X2: code_integer] :
      abs_abs_Code_integer(X2) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X2),zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X2),X2) ).

% abs_if_raw
tff(fact_6621_abs__triangle__ineq4,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(A,B))),plus_p5714425477246183910nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B))) ).

% abs_triangle_ineq4
tff(fact_6622_abs__triangle__ineq4,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(A),B))),aa_real_real(plus_plus_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))) ).

% abs_triangle_ineq4
tff(fact_6623_abs__triangle__ineq4,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(A),B))),aa_rat_rat(plus_plus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))) ).

% abs_triangle_ineq4
tff(fact_6624_abs__triangle__ineq4,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))) ).

% abs_triangle_ineq4
tff(fact_6625_abs__diff__triangle__ineq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(plus_p5714425477246183910nteger(A,B),plus_p5714425477246183910nteger(C,D)))),plus_p5714425477246183910nteger(abs_abs_Code_integer(minus_8373710615458151222nteger(A,C)),abs_abs_Code_integer(minus_8373710615458151222nteger(B,D)))) ).

% abs_diff_triangle_ineq
tff(fact_6626_abs__diff__triangle__ineq,axiom,
    ! [A: real,B: real,C: real,D: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(A),B)),aa_real_real(plus_plus_real(C),D)))),aa_real_real(plus_plus_real(aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(A),C))),aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(B),D)))) ).

% abs_diff_triangle_ineq
tff(fact_6627_abs__diff__triangle__ineq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),B)),aa_rat_rat(plus_plus_rat(C),D)))),aa_rat_rat(plus_plus_rat(aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(A),C))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(B),D)))) ).

% abs_diff_triangle_ineq
tff(fact_6628_abs__diff__triangle__ineq,axiom,
    ! [A: int,B: int,C: int,D: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),D)))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D)))) ).

% abs_diff_triangle_ineq
tff(fact_6629_abs__diff__le__iff,axiom,
    ! [X: code_integer,A: code_integer,R: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(X,A))),R)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,minus_8373710615458151222nteger(A,R)),X)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),plus_p5714425477246183910nteger(A,R)) ) ) ).

% abs_diff_le_iff
tff(fact_6630_abs__diff__le__iff,axiom,
    ! [X: real,A: real,R: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(X),A))),R)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(A),R)),X)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(plus_plus_real(A),R)) ) ) ).

% abs_diff_le_iff
tff(fact_6631_abs__diff__le__iff,axiom,
    ! [X: rat,A: rat,R: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(X),A))),R)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),R)),X)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_rat_rat(plus_plus_rat(A),R)) ) ) ).

% abs_diff_le_iff
tff(fact_6632_abs__diff__le__iff,axiom,
    ! [X: int,A: int,R: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),A))),R)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),R)),X)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),R)) ) ) ).

% abs_diff_le_iff
tff(fact_6633_abs__diff__less__iff,axiom,
    ! [X: code_integer,A: code_integer,R: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(X,A))),R)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,minus_8373710615458151222nteger(A,R)),X)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),plus_p5714425477246183910nteger(A,R)) ) ) ).

% abs_diff_less_iff
tff(fact_6634_abs__diff__less__iff,axiom,
    ! [X: real,A: real,R: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(X),A))),R)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(A),R)),X)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(plus_plus_real(A),R)) ) ) ).

% abs_diff_less_iff
tff(fact_6635_abs__diff__less__iff,axiom,
    ! [X: rat,A: rat,R: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(X),A))),R)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(A),R)),X)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_rat_rat(plus_plus_rat(A),R)) ) ) ).

% abs_diff_less_iff
tff(fact_6636_abs__diff__less__iff,axiom,
    ! [X: int,A: int,R: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),A))),R)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),R)),X)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,X),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),R)) ) ) ).

% abs_diff_less_iff
tff(fact_6637_abs__real__def,axiom,
    ! [A: real] :
      aa_real_real(abs_abs_real,A) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real),aa_real_real(uminus_uminus_real,A),A) ).

% abs_real_def
tff(fact_6638_log__ln,axiom,
    ! [X: real] : aa_real_real(ln_ln_real,X) = aa_real_real(log(aa_real_real(exp_real,one_one_real)),X) ).

% log_ln
tff(fact_6639_sin__bound__lemma,axiom,
    ! [X: real,Y2: real,U: real,V: real] :
      ( ( X = Y2 )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,U)),V)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(X),U)),Y2))),V) ) ) ).

% sin_bound_lemma
tff(fact_6640_abs__add__one__gt__zero,axiom,
    ! [X: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),plus_p5714425477246183910nteger(one_one_Code_integer,abs_abs_Code_integer(X))) ).

% abs_add_one_gt_zero
tff(fact_6641_abs__add__one__gt__zero,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(plus_plus_real(one_one_real),aa_real_real(abs_abs_real,X))) ).

% abs_add_one_gt_zero
tff(fact_6642_abs__add__one__gt__zero,axiom,
    ! [X: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(one_one_rat),aa_rat_rat(abs_abs_rat,X))) ).

% abs_add_one_gt_zero
tff(fact_6643_abs__add__one__gt__zero,axiom,
    ! [X: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(abs_abs_int,X))) ).

% abs_add_one_gt_zero
tff(fact_6644_log__base__change,axiom,
    ! [A: real,B: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_real(log(B),X) = aa_real_real(divide_divide_real(aa_real_real(log(A),X)),aa_real_real(log(A),B)) ) ) ) ).

% log_base_change
tff(fact_6645_of__int__leD,axiom,
    ! [Nb: int,X: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(ring_18347121197199848620nteger(Nb))),X)
     => ( ( Nb = zero_zero_int )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),X) ) ) ).

% of_int_leD
tff(fact_6646_of__int__leD,axiom,
    ! [Nb: int,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_int_real(ring_1_of_int_real,Nb))),X)
     => ( ( Nb = zero_zero_int )
        | aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),X) ) ) ).

% of_int_leD
tff(fact_6647_of__int__leD,axiom,
    ! [Nb: int,X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_int_rat(ring_1_of_int_rat,Nb))),X)
     => ( ( Nb = zero_zero_int )
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),X) ) ) ).

% of_int_leD
tff(fact_6648_of__int__leD,axiom,
    ! [Nb: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(ring_1_of_int_int,Nb))),X)
     => ( ( Nb = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),X) ) ) ).

% of_int_leD
tff(fact_6649_of__int__lessD,axiom,
    ! [Nb: int,X: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(ring_18347121197199848620nteger(Nb))),X)
     => ( ( Nb = zero_zero_int )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),X) ) ) ).

% of_int_lessD
tff(fact_6650_of__int__lessD,axiom,
    ! [Nb: int,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_int_real(ring_1_of_int_real,Nb))),X)
     => ( ( Nb = zero_zero_int )
        | aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X) ) ) ).

% of_int_lessD
tff(fact_6651_of__int__lessD,axiom,
    ! [Nb: int,X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_int_rat(ring_1_of_int_rat,Nb))),X)
     => ( ( Nb = zero_zero_int )
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),X) ) ) ).

% of_int_lessD
tff(fact_6652_of__int__lessD,axiom,
    ! [Nb: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(ring_1_of_int_int,Nb))),X)
     => ( ( Nb = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),X) ) ) ).

% of_int_lessD
tff(fact_6653_lemma__interval,axiom,
    ! [A: real,X: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),B)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [Y5: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(X),Y5))),D2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Y5)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y5),B) ) ) ) ) ) ).

% lemma_interval
tff(fact_6654_round__diff__minimal,axiom,
    ! [Z2: real,Ma: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(Z2),aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,Z2))))),aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(Z2),aa_int_real(ring_1_of_int_real,Ma)))) ).

% round_diff_minimal
tff(fact_6655_round__diff__minimal,axiom,
    ! [Z2: rat,Ma: int] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(Z2),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,Z2))))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(Z2),aa_int_rat(ring_1_of_int_rat,Ma)))) ).

% round_diff_minimal
tff(fact_6656_abs__le__square__iff,axiom,
    ! [X: code_integer,Y2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(X)),abs_abs_Code_integer(Y2))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_le_square_iff
tff(fact_6657_abs__le__square__iff,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X)),aa_real_real(abs_abs_real,Y2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_le_square_iff
tff(fact_6658_abs__le__square__iff,axiom,
    ! [X: rat,Y2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,X)),aa_rat_rat(abs_abs_rat,Y2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_le_square_iff
tff(fact_6659_abs__le__square__iff,axiom,
    ! [X: int,Y2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,X)),aa_int_int(abs_abs_int,Y2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_le_square_iff
tff(fact_6660_abs__square__eq__1,axiom,
    ! [X: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_Code_integer )
    <=> ( abs_abs_Code_integer(X) = one_one_Code_integer ) ) ).

% abs_square_eq_1
tff(fact_6661_abs__square__eq__1,axiom,
    ! [X: rat] :
      ( ( aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_rat )
    <=> ( aa_rat_rat(abs_abs_rat,X) = one_one_rat ) ) ).

% abs_square_eq_1
tff(fact_6662_abs__square__eq__1,axiom,
    ! [X: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_real )
    <=> ( aa_real_real(abs_abs_real,X) = one_one_real ) ) ).

% abs_square_eq_1
tff(fact_6663_abs__square__eq__1,axiom,
    ! [X: int] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_int )
    <=> ( aa_int_int(abs_abs_int,X) = one_one_int ) ) ).

% abs_square_eq_1
tff(fact_6664_power__even__abs,axiom,
    ! [Nb: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A)),Nb) = aa_nat_rat(power_power_rat(A),Nb) ) ) ).

% power_even_abs
tff(fact_6665_power__even__abs,axiom,
    ! [Nb: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),Nb) = aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb) ) ) ).

% power_even_abs
tff(fact_6666_power__even__abs,axiom,
    ! [Nb: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),Nb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb) ) ) ).

% power_even_abs
tff(fact_6667_power__even__abs,axiom,
    ! [Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(abs_abs_int,A)),Nb) = aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb) ) ) ).

% power_even_abs
tff(fact_6668_log__mult,axiom,
    ! [A: real,X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y2)
           => ( aa_real_real(log(A),aa_real_real(times_times_real(X),Y2)) = aa_real_real(plus_plus_real(aa_real_real(log(A),X)),aa_real_real(log(A),Y2)) ) ) ) ) ) ).

% log_mult
tff(fact_6669_log__divide,axiom,
    ! [A: real,X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y2)
           => ( aa_real_real(log(A),aa_real_real(divide_divide_real(X),Y2)) = aa_real_real(minus_minus_real(aa_real_real(log(A),X)),aa_real_real(log(A),Y2)) ) ) ) ) ) ).

% log_divide
tff(fact_6670_power2__le__iff__abs__le,axiom,
    ! [Y2: code_integer,X: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(X)),Y2) ) ) ).

% power2_le_iff_abs_le
tff(fact_6671_power2__le__iff__abs__le,axiom,
    ! [Y2: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X)),Y2) ) ) ).

% power2_le_iff_abs_le
tff(fact_6672_power2__le__iff__abs__le,axiom,
    ! [Y2: rat,X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,X)),Y2) ) ) ).

% power2_le_iff_abs_le
tff(fact_6673_power2__le__iff__abs__le,axiom,
    ! [Y2: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,X)),Y2) ) ) ).

% power2_le_iff_abs_le
tff(fact_6674_abs__square__le__1,axiom,
    ! [X: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_Code_integer)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(X)),one_one_Code_integer) ) ).

% abs_square_le_1
tff(fact_6675_abs__square__le__1,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real) ) ).

% abs_square_le_1
tff(fact_6676_abs__square__le__1,axiom,
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,X)),one_one_rat) ) ).

% abs_square_le_1
tff(fact_6677_abs__square__le__1,axiom,
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,X)),one_one_int) ) ).

% abs_square_le_1
tff(fact_6678_abs__square__less__1,axiom,
    ! [X: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_Code_integer)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(X)),one_one_Code_integer) ) ).

% abs_square_less_1
tff(fact_6679_abs__square__less__1,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,X)),one_one_real) ) ).

% abs_square_less_1
tff(fact_6680_abs__square__less__1,axiom,
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,X)),one_one_rat) ) ).

% abs_square_less_1
tff(fact_6681_abs__square__less__1,axiom,
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,X)),one_one_int) ) ).

% abs_square_less_1
tff(fact_6682_power__mono__even,axiom,
    ! [Nb: nat,A: code_integer,B: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(B))
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(B),Nb)) ) ) ).

% power_mono_even
tff(fact_6683_power__mono__even,axiom,
    ! [Nb: nat,A: real,B: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)) ) ) ).

% power_mono_even
tff(fact_6684_power__mono__even,axiom,
    ! [Nb: nat,A: rat,B: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb)) ) ) ).

% power_mono_even
tff(fact_6685_power__mono__even,axiom,
    ! [Nb: nat,A: int,B: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,B),Nb)) ) ) ).

% power_mono_even
tff(fact_6686_log__eq__div__ln__mult__log,axiom,
    ! [A: real,B: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
         => ( ( B != one_one_real )
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
             => ( aa_real_real(log(A),X) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,B)),aa_real_real(ln_ln_real,A))),aa_real_real(log(B),X)) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
tff(fact_6687_sqrt__ge__absD,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X)),aa_real_real(sqrt,Y2))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Y2) ) ).

% sqrt_ge_absD
tff(fact_6688_sqrt__sum__squares__le__sum__abs,axiom,
    ! [X: real,Y2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(plus_plus_real(aa_real_real(abs_abs_real,X)),aa_real_real(abs_abs_real,Y2))) ).

% sqrt_sum_squares_le_sum_abs
tff(fact_6689_real__sqrt__ge__abs2,axiom,
    ! [Y2: real,X: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y2)),aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% real_sqrt_ge_abs2
tff(fact_6690_real__sqrt__ge__abs1,axiom,
    ! [X: real,Y2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X)),aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% real_sqrt_ge_abs1
tff(fact_6691_arctan__add,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Y2)),one_one_real)
       => ( aa_real_real(plus_plus_real(aa_real_real(arctan,X)),aa_real_real(arctan,Y2)) = aa_real_real(arctan,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(X),Y2)),aa_real_real(minus_minus_real(one_one_real),aa_real_real(times_times_real(X),Y2)))) ) ) ) ).

% arctan_add
tff(fact_6692_of__int__round__abs__le,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,X))),X))),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% of_int_round_abs_le
tff(fact_6693_of__int__round__abs__le,axiom,
    ! [X: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,X))),X))),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ).

% of_int_round_abs_le
tff(fact_6694_round__unique_H,axiom,
    ! [X: real,Nb: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(X),aa_int_real(ring_1_of_int_real,Nb)))),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => ( aa_real_int(archim8280529875227126926d_real,X) = Nb ) ) ).

% round_unique'
tff(fact_6695_round__unique_H,axiom,
    ! [X: rat,Nb: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(X),aa_int_rat(ring_1_of_int_rat,Nb)))),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))
     => ( aa_rat_int(archim7778729529865785530nd_rat,X) = Nb ) ) ).

% round_unique'
tff(fact_6696_cos__x__y__le__one,axiom,
    ! [X: real,Y2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(divide_divide_real(X),aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),one_one_real) ).

% cos_x_y_le_one
tff(fact_6697_real__sqrt__sum__squares__less,axiom,
    ! [X: real,U: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,X)),aa_real_real(divide_divide_real(U),aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Y2)),aa_real_real(divide_divide_real(U),aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),U) ) ) ).

% real_sqrt_sum_squares_less
tff(fact_6698_log__base__10__eq2,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),X) = aa_real_real(times_times_real(aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_real_real(exp_real,one_one_real))),aa_real_real(ln_ln_real,X)) ) ) ).

% log_base_10_eq2
tff(fact_6699_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_real_real(ln_ln_real,aa_real_real(plus_plus_real(one_one_real),X))),X))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonneg
tff(fact_6700_abs__sqrt__wlog,axiom,
    ! [P: fun_Co2081214437782444048eger_o,X: code_integer] :
      ( ! [X4: code_integer] :
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X4)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(P,X4),aa_nat_Code_integer(power_8256067586552552935nteger(X4),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(P,abs_abs_Code_integer(X)),aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_sqrt_wlog
tff(fact_6701_abs__sqrt__wlog,axiom,
    ! [P: fun_real_fun_real_o,X: real] :
      ( ! [X4: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X4)
         => aa_real_o(aa_real_fun_real_o(P,X4),aa_nat_real(aa_real_fun_nat_real(power_power_real,X4),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
     => aa_real_o(aa_real_fun_real_o(P,aa_real_real(abs_abs_real,X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_sqrt_wlog
tff(fact_6702_abs__sqrt__wlog,axiom,
    ! [P: fun_rat_fun_rat_o,X: rat] :
      ( ! [X4: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X4)
         => aa_rat_o(aa_rat_fun_rat_o(P,X4),aa_nat_rat(power_power_rat(X4),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
     => aa_rat_o(aa_rat_fun_rat_o(P,aa_rat_rat(abs_abs_rat,X)),aa_nat_rat(power_power_rat(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_sqrt_wlog
tff(fact_6703_abs__sqrt__wlog,axiom,
    ! [P: fun_int_fun_int_o,X: int] :
      ( ! [X4: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X4)
         => aa_int_o(aa_int_fun_int_o(P,X4),aa_nat_int(aa_int_fun_nat_int(power_power_int,X4),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
     => aa_int_o(aa_int_fun_int_o(P,aa_int_int(abs_abs_int,X)),aa_nat_int(aa_int_fun_nat_int(power_power_int,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_sqrt_wlog
tff(fact_6704_log2__of__power__le,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ma)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Ma))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ).

% log2_of_power_le
tff(fact_6705_machin,axiom,
    aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_real_real(arctan,aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit0,one))))))),aa_real_real(arctan,aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,one))))))))))) ).

% machin
tff(fact_6706_machin__Euler,axiom,
    aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_real_real(arctan,aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit1,one))))))),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(arctan,aa_real_real(divide_divide_real(aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))),aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))))))))) = aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))) ).

% machin_Euler
tff(fact_6707_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [X: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_nat_real(semiri5074537144036343181t_real,Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(one_one_real),aa_real_real(divide_divide_real(X),aa_nat_real(semiri5074537144036343181t_real,Nb)))),Nb)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,X))) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
tff(fact_6708_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [Nb: nat,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Nb))),X)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(plus_plus_real(one_one_real),aa_real_real(divide_divide_real(X),aa_nat_real(semiri5074537144036343181t_real,Nb)))),Nb)),aa_real_real(exp_real,X)) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
tff(fact_6709_of__nat__eq__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Ma) = aa_nat_real(semiri5074537144036343181t_real,Nb) )
    <=> ( Ma = Nb ) ) ).

% of_nat_eq_iff
tff(fact_6710_of__nat__eq__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Ma) = aa_nat_int(semiri1314217659103216013at_int,Nb) )
    <=> ( Ma = Nb ) ) ).

% of_nat_eq_iff
tff(fact_6711_of__nat__eq__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( aa_nat_Extended_enat(semiri4216267220026989637d_enat,Ma) = aa_nat_Extended_enat(semiri4216267220026989637d_enat,Nb) )
    <=> ( Ma = Nb ) ) ).

% of_nat_eq_iff
tff(fact_6712_of__nat__eq__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,Ma) = aa_nat_nat(semiri1316708129612266289at_nat,Nb) )
    <=> ( Ma = Nb ) ) ).

% of_nat_eq_iff
tff(fact_6713_abs__of__nat,axiom,
    ! [Nb: nat] : aa_rat_rat(abs_abs_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb)) = aa_nat_rat(semiri681578069525770553at_rat,Nb) ).

% abs_of_nat
tff(fact_6714_abs__of__nat,axiom,
    ! [Nb: nat] : abs_abs_Code_integer(aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)) = aa_nat_Code_integer(semiri4939895301339042750nteger,Nb) ).

% abs_of_nat
tff(fact_6715_abs__of__nat,axiom,
    ! [Nb: nat] : aa_real_real(abs_abs_real,aa_nat_real(semiri5074537144036343181t_real,Nb)) = aa_nat_real(semiri5074537144036343181t_real,Nb) ).

% abs_of_nat
tff(fact_6716_abs__of__nat,axiom,
    ! [Nb: nat] : aa_int_int(abs_abs_int,aa_nat_int(semiri1314217659103216013at_int,Nb)) = aa_nat_int(semiri1314217659103216013at_int,Nb) ).

% abs_of_nat
tff(fact_6717_zdvd1__eq,axiom,
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X),one_one_int)
    <=> ( aa_int_int(abs_abs_int,X) = one_one_int ) ) ).

% zdvd1_eq
tff(fact_6718_of__nat__0,axiom,
    aa_nat_complex(semiri8010041392384452111omplex,zero_zero_nat) = zero_zero_complex ).

% of_nat_0
tff(fact_6719_of__nat__0,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,zero_zero_nat) = zero_zero_rat ).

% of_nat_0
tff(fact_6720_of__nat__0,axiom,
    aa_nat_real(semiri5074537144036343181t_real,zero_zero_nat) = zero_zero_real ).

% of_nat_0
tff(fact_6721_of__nat__0,axiom,
    aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% of_nat_0
tff(fact_6722_of__nat__0,axiom,
    aa_nat_Extended_enat(semiri4216267220026989637d_enat,zero_zero_nat) = zero_z5237406670263579293d_enat ).

% of_nat_0
tff(fact_6723_of__nat__0,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,zero_zero_nat) = zero_zero_nat ).

% of_nat_0
tff(fact_6724_of__nat__0__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( zero_zero_complex = aa_nat_complex(semiri8010041392384452111omplex,Nb) )
    <=> ( zero_zero_nat = Nb ) ) ).

% of_nat_0_eq_iff
tff(fact_6725_of__nat__0__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( zero_zero_rat = aa_nat_rat(semiri681578069525770553at_rat,Nb) )
    <=> ( zero_zero_nat = Nb ) ) ).

% of_nat_0_eq_iff
tff(fact_6726_of__nat__0__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( zero_zero_real = aa_nat_real(semiri5074537144036343181t_real,Nb) )
    <=> ( zero_zero_nat = Nb ) ) ).

% of_nat_0_eq_iff
tff(fact_6727_of__nat__0__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( zero_zero_int = aa_nat_int(semiri1314217659103216013at_int,Nb) )
    <=> ( zero_zero_nat = Nb ) ) ).

% of_nat_0_eq_iff
tff(fact_6728_of__nat__0__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( zero_z5237406670263579293d_enat = aa_nat_Extended_enat(semiri4216267220026989637d_enat,Nb) )
    <=> ( zero_zero_nat = Nb ) ) ).

% of_nat_0_eq_iff
tff(fact_6729_of__nat__0__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( zero_zero_nat = aa_nat_nat(semiri1316708129612266289at_nat,Nb) )
    <=> ( zero_zero_nat = Nb ) ) ).

% of_nat_0_eq_iff
tff(fact_6730_of__nat__eq__0__iff,axiom,
    ! [Ma: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,Ma) = zero_zero_complex )
    <=> ( Ma = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_6731_of__nat__eq__0__iff,axiom,
    ! [Ma: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,Ma) = zero_zero_rat )
    <=> ( Ma = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_6732_of__nat__eq__0__iff,axiom,
    ! [Ma: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Ma) = zero_zero_real )
    <=> ( Ma = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_6733_of__nat__eq__0__iff,axiom,
    ! [Ma: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Ma) = zero_zero_int )
    <=> ( Ma = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_6734_of__nat__eq__0__iff,axiom,
    ! [Ma: nat] :
      ( ( aa_nat_Extended_enat(semiri4216267220026989637d_enat,Ma) = zero_z5237406670263579293d_enat )
    <=> ( Ma = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_6735_of__nat__eq__0__iff,axiom,
    ! [Ma: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,Ma) = zero_zero_nat )
    <=> ( Ma = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_6736_of__nat__less__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Ma)),aa_nat_rat(semiri681578069525770553at_rat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb) ) ).

% of_nat_less_iff
tff(fact_6737_of__nat__less__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Ma)),aa_nat_real(semiri5074537144036343181t_real,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb) ) ).

% of_nat_less_iff
tff(fact_6738_of__nat__less__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Ma)),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb) ) ).

% of_nat_less_iff
tff(fact_6739_of__nat__less__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,Ma)),aa_nat_Extended_enat(semiri4216267220026989637d_enat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb) ) ).

% of_nat_less_iff
tff(fact_6740_of__nat__less__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Ma)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb) ) ).

% of_nat_less_iff
tff(fact_6741_of__nat__le__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Ma)),aa_nat_real(semiri5074537144036343181t_real,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ).

% of_nat_le_iff
tff(fact_6742_of__nat__le__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,Ma)),aa_nat_Extended_enat(semiri4216267220026989637d_enat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ).

% of_nat_le_iff
tff(fact_6743_of__nat__le__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,Ma)),aa_nat_rat(semiri681578069525770553at_rat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ).

% of_nat_le_iff
tff(fact_6744_of__nat__le__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,Ma)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ).

% of_nat_le_iff
tff(fact_6745_of__nat__le__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Ma)),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ).

% of_nat_le_iff
tff(fact_6746_of__nat__numeral,axiom,
    ! [Nb: num] : aa_nat_complex(semiri8010041392384452111omplex,aa_num_nat(numeral_numeral_nat,Nb)) = numera6690914467698888265omplex(Nb) ).

% of_nat_numeral
tff(fact_6747_of__nat__numeral,axiom,
    ! [Nb: num] : aa_nat_rat(semiri681578069525770553at_rat,aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_rat(numeral_numeral_rat,Nb) ).

% of_nat_numeral
tff(fact_6748_of__nat__numeral,axiom,
    ! [Nb: num] : aa_nat_real(semiri5074537144036343181t_real,aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_real(numeral_numeral_real,Nb) ).

% of_nat_numeral
tff(fact_6749_of__nat__numeral,axiom,
    ! [Nb: num] : aa_nat_int(semiri1314217659103216013at_int,aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_int(numeral_numeral_int,Nb) ).

% of_nat_numeral
tff(fact_6750_of__nat__numeral,axiom,
    ! [Nb: num] : aa_nat_Extended_enat(semiri4216267220026989637d_enat,aa_num_nat(numeral_numeral_nat,Nb)) = numera1916890842035813515d_enat(Nb) ).

% of_nat_numeral
tff(fact_6751_of__nat__numeral,axiom,
    ! [Nb: num] : aa_nat_nat(semiri1316708129612266289at_nat,aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_nat(numeral_numeral_nat,Nb) ).

% of_nat_numeral
tff(fact_6752_of__nat__add,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,Ma)),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ).

% of_nat_add
tff(fact_6753_of__nat__add,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb)) = aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,Ma)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ).

% of_nat_add
tff(fact_6754_of__nat__add,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Ma)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% of_nat_add
tff(fact_6755_of__nat__add,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_Extended_enat(semiri4216267220026989637d_enat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb)) = plus_p3455044024723400733d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,Ma),aa_nat_Extended_enat(semiri4216267220026989637d_enat,Nb)) ).

% of_nat_add
tff(fact_6756_of__nat__add,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Ma)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% of_nat_add
tff(fact_6757_of__nat__mult,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Nb)) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Ma)),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ).

% of_nat_mult
tff(fact_6758_of__nat__mult,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Nb)) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Ma)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ).

% of_nat_mult
tff(fact_6759_of__nat__mult,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Ma)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% of_nat_mult
tff(fact_6760_of__nat__mult,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_Extended_enat(semiri4216267220026989637d_enat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Nb)) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,Ma)),aa_nat_Extended_enat(semiri4216267220026989637d_enat,Nb)) ).

% of_nat_mult
tff(fact_6761_of__nat__mult,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Ma)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% of_nat_mult
tff(fact_6762_of__nat__1,axiom,
    aa_nat_complex(semiri8010041392384452111omplex,one_one_nat) = one_one_complex ).

% of_nat_1
tff(fact_6763_of__nat__1,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,one_one_nat) = one_one_rat ).

% of_nat_1
tff(fact_6764_of__nat__1,axiom,
    aa_nat_real(semiri5074537144036343181t_real,one_one_nat) = one_one_real ).

% of_nat_1
tff(fact_6765_of__nat__1,axiom,
    aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).

% of_nat_1
tff(fact_6766_of__nat__1,axiom,
    aa_nat_Extended_enat(semiri4216267220026989637d_enat,one_one_nat) = one_on7984719198319812577d_enat ).

% of_nat_1
tff(fact_6767_of__nat__1,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,one_one_nat) = one_one_nat ).

% of_nat_1
tff(fact_6768_of__nat__1__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( one_one_complex = aa_nat_complex(semiri8010041392384452111omplex,Nb) )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_6769_of__nat__1__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( one_one_rat = aa_nat_rat(semiri681578069525770553at_rat,Nb) )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_6770_of__nat__1__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( one_one_real = aa_nat_real(semiri5074537144036343181t_real,Nb) )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_6771_of__nat__1__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( one_one_int = aa_nat_int(semiri1314217659103216013at_int,Nb) )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_6772_of__nat__1__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( one_on7984719198319812577d_enat = aa_nat_Extended_enat(semiri4216267220026989637d_enat,Nb) )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_6773_of__nat__1__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( one_one_nat = aa_nat_nat(semiri1316708129612266289at_nat,Nb) )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_6774_of__nat__eq__1__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,Nb) = one_one_complex )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_6775_of__nat__eq__1__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,Nb) = one_one_rat )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_6776_of__nat__eq__1__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Nb) = one_one_real )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_6777_of__nat__eq__1__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Nb) = one_one_int )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_6778_of__nat__eq__1__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_Extended_enat(semiri4216267220026989637d_enat,Nb) = one_on7984719198319812577d_enat )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_6779_of__nat__eq__1__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,Nb) = one_one_nat )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_6780_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,X) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_nat_complex(semiri8010041392384452111omplex,B)),W) )
    <=> ( X = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_6781_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,X) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W) )
    <=> ( X = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_6782_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,X) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(semiri1314217659103216013at_int,B)),W) )
    <=> ( X = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_6783_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( aa_nat_Extended_enat(semiri4216267220026989637d_enat,X) = aa_nat_Extended_enat(power_8040749407984259932d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,B)),W) )
    <=> ( X = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_6784_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,X) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W) )
    <=> ( X = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_6785_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_nat_complex(semiri8010041392384452111omplex,B)),W) = aa_nat_complex(semiri8010041392384452111omplex,X) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = X ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_6786_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W) = aa_nat_real(semiri5074537144036343181t_real,X) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = X ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_6787_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(semiri1314217659103216013at_int,B)),W) = aa_nat_int(semiri1314217659103216013at_int,X) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = X ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_6788_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( aa_nat_Extended_enat(power_8040749407984259932d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,B)),W) = aa_nat_Extended_enat(semiri4216267220026989637d_enat,X) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = X ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_6789_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W) = aa_nat_nat(semiri1316708129612266289at_nat,X) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = X ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_6790_of__nat__power,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ma),Nb)) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_nat_complex(semiri8010041392384452111omplex,Ma)),Nb) ).

% of_nat_power
tff(fact_6791_of__nat__power,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ma),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,Ma)),Nb) ).

% of_nat_power
tff(fact_6792_of__nat__power,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ma),Nb)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(semiri1314217659103216013at_int,Ma)),Nb) ).

% of_nat_power
tff(fact_6793_of__nat__power,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_Extended_enat(semiri4216267220026989637d_enat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ma),Nb)) = aa_nat_Extended_enat(power_8040749407984259932d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,Ma)),Nb) ).

% of_nat_power
tff(fact_6794_of__nat__power,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ma),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,Ma)),Nb) ).

% of_nat_power
tff(fact_6795_zabs__less__one__iff,axiom,
    ! [Z2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,Z2)),one_one_int)
    <=> ( Z2 = zero_zero_int ) ) ).

% zabs_less_one_iff
tff(fact_6796_of__nat__of__bool,axiom,
    ! [P: $o] : aa_nat_real(semiri5074537144036343181t_real,aa_o_nat(zero_n2687167440665602831ol_nat,(P))) = zero_n3304061248610475627l_real((P)) ).

% of_nat_of_bool
tff(fact_6797_of__nat__of__bool,axiom,
    ! [P: $o] : aa_nat_Extended_enat(semiri4216267220026989637d_enat,aa_o_nat(zero_n2687167440665602831ol_nat,(P))) = zero_n1046097342994218471d_enat((P)) ).

% of_nat_of_bool
tff(fact_6798_of__nat__of__bool,axiom,
    ! [P: $o] : aa_nat_nat(semiri1316708129612266289at_nat,aa_o_nat(zero_n2687167440665602831ol_nat,(P))) = aa_o_nat(zero_n2687167440665602831ol_nat,(P)) ).

% of_nat_of_bool
tff(fact_6799_of__nat__of__bool,axiom,
    ! [P: $o] : aa_nat_int(semiri1314217659103216013at_int,aa_o_nat(zero_n2687167440665602831ol_nat,(P))) = aa_o_int(zero_n2684676970156552555ol_int,(P)) ).

% of_nat_of_bool
tff(fact_6800_of__nat__of__bool,axiom,
    ! [P: $o] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_o_nat(zero_n2687167440665602831ol_nat,(P))) = zero_n356916108424825756nteger((P)) ).

% of_nat_of_bool
tff(fact_6801_of__nat__le__0__iff,axiom,
    ! [Ma: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Ma)),zero_zero_real)
    <=> ( Ma = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_6802_of__nat__le__0__iff,axiom,
    ! [Ma: nat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,Ma)),zero_z5237406670263579293d_enat)
    <=> ( Ma = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_6803_of__nat__le__0__iff,axiom,
    ! [Ma: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,Ma)),zero_zero_rat)
    <=> ( Ma = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_6804_of__nat__le__0__iff,axiom,
    ! [Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,Ma)),zero_zero_nat)
    <=> ( Ma = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_6805_of__nat__le__0__iff,axiom,
    ! [Ma: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Ma)),zero_zero_int)
    <=> ( Ma = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_6806_of__nat__Suc,axiom,
    ! [Ma: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,Ma)) = aa_complex_complex(plus_plus_complex(one_one_complex),aa_nat_complex(semiri8010041392384452111omplex,Ma)) ).

% of_nat_Suc
tff(fact_6807_of__nat__Suc,axiom,
    ! [Ma: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,Ma)) = aa_rat_rat(plus_plus_rat(one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,Ma)) ).

% of_nat_Suc
tff(fact_6808_of__nat__Suc,axiom,
    ! [Ma: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Ma)) = aa_real_real(plus_plus_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,Ma)) ).

% of_nat_Suc
tff(fact_6809_of__nat__Suc,axiom,
    ! [Ma: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Ma)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_nat_int(semiri1314217659103216013at_int,Ma)) ).

% of_nat_Suc
tff(fact_6810_of__nat__Suc,axiom,
    ! [Ma: nat] : aa_nat_Extended_enat(semiri4216267220026989637d_enat,aa_nat_nat(suc,Ma)) = plus_p3455044024723400733d_enat(one_on7984719198319812577d_enat,aa_nat_Extended_enat(semiri4216267220026989637d_enat,Ma)) ).

% of_nat_Suc
tff(fact_6811_of__nat__Suc,axiom,
    ! [Ma: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,Ma)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(semiri1316708129612266289at_nat,Ma)) ).

% of_nat_Suc
tff(fact_6812_of__nat__0__less__iff,axiom,
    ! [Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(semiri681578069525770553at_rat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% of_nat_0_less_iff
tff(fact_6813_of__nat__0__less__iff,axiom,
    ! [Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(semiri5074537144036343181t_real,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% of_nat_0_less_iff
tff(fact_6814_of__nat__0__less__iff,axiom,
    ! [Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% of_nat_0_less_iff
tff(fact_6815_of__nat__0__less__iff,axiom,
    ! [Nb: nat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),aa_nat_Extended_enat(semiri4216267220026989637d_enat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% of_nat_0_less_iff
tff(fact_6816_of__nat__0__less__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% of_nat_0_less_iff
tff(fact_6817_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,X)),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_6818_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_6819_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(semiri1314217659103216013at_int,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_6820_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_6821_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W)),aa_nat_rat(semiri681578069525770553at_rat,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),X) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_6822_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W)),aa_nat_real(semiri5074537144036343181t_real,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),X) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_6823_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(semiri1314217659103216013at_int,B)),W)),aa_nat_int(semiri1314217659103216013at_int,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),X) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_6824_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W)),aa_nat_nat(semiri1316708129612266289at_nat,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),X) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_6825_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_6826_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,X)),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_6827_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_6828_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(semiri1314217659103216013at_int,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_6829_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W)),aa_nat_real(semiri5074537144036343181t_real,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),X) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_6830_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W)),aa_nat_rat(semiri681578069525770553at_rat,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),X) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_6831_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W)),aa_nat_nat(semiri1316708129612266289at_nat,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),X) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_6832_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(semiri1314217659103216013at_int,B)),W)),aa_nat_int(semiri1314217659103216013at_int,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),X) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_6833_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y2: nat,X: num,Nb: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,Y2) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,numera6690914467698888265omplex(X)),Nb) )
    <=> ( Y2 = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),Nb) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_6834_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y2: nat,X: num,Nb: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,Y2) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X)),Nb) )
    <=> ( Y2 = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),Nb) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_6835_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y2: nat,X: num,Nb: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Y2) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,X)),Nb) )
    <=> ( Y2 = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),Nb) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_6836_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y2: nat,X: num,Nb: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Y2) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X)),Nb) )
    <=> ( Y2 = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),Nb) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_6837_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y2: nat,X: num,Nb: nat] :
      ( ( aa_nat_Extended_enat(semiri4216267220026989637d_enat,Y2) = aa_nat_Extended_enat(power_8040749407984259932d_enat(numera1916890842035813515d_enat(X)),Nb) )
    <=> ( Y2 = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),Nb) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_6838_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y2: nat,X: num,Nb: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,Y2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),Nb) )
    <=> ( Y2 = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),Nb) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_6839_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,Nb: nat,Y2: nat] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,numera6690914467698888265omplex(X)),Nb) = aa_nat_complex(semiri8010041392384452111omplex,Y2) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),Nb) = Y2 ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_6840_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,Nb: nat,Y2: nat] :
      ( ( aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X)),Nb) = aa_nat_rat(semiri681578069525770553at_rat,Y2) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),Nb) = Y2 ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_6841_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,Nb: nat,Y2: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,X)),Nb) = aa_nat_real(semiri5074537144036343181t_real,Y2) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),Nb) = Y2 ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_6842_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,Nb: nat,Y2: nat] :
      ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X)),Nb) = aa_nat_int(semiri1314217659103216013at_int,Y2) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),Nb) = Y2 ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_6843_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,Nb: nat,Y2: nat] :
      ( ( aa_nat_Extended_enat(power_8040749407984259932d_enat(numera1916890842035813515d_enat(X)),Nb) = aa_nat_Extended_enat(semiri4216267220026989637d_enat,Y2) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),Nb) = Y2 ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_6844_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,Nb: nat,Y2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),Nb) = aa_nat_nat(semiri1316708129612266289at_nat,Y2) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),Nb) = Y2 ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_6845_real__of__nat__less__numeral__iff,axiom,
    ! [Nb: nat,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_num_real(numeral_numeral_real,W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_num_nat(numeral_numeral_nat,W)) ) ).

% real_of_nat_less_numeral_iff
tff(fact_6846_numeral__less__real__of__nat__iff,axiom,
    ! [W: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,W)),aa_nat_real(semiri5074537144036343181t_real,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,W)),Nb) ) ).

% numeral_less_real_of_nat_iff
tff(fact_6847_numeral__le__real__of__nat__iff,axiom,
    ! [Nb: num,Ma: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,Ma))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Nb)),Ma) ) ).

% numeral_le_real_of_nat_iff
tff(fact_6848_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,X)),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),X)
        | ( Nb = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_6849_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,X)),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),X)
        | ( Nb = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_6850_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_nat_int(semiri1314217659103216013at_int,X)),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),X)
        | ( Nb = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_6851_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),X)
        | ( Nb = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_6852_log__pow__cancel,axiom,
    ! [A: real,B: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_real(log(A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),B)) = aa_nat_real(semiri5074537144036343181t_real,B) ) ) ) ).

% log_pow_cancel
tff(fact_6853_even__of__nat,axiom,
    ! [Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ) ).

% even_of_nat
tff(fact_6854_even__of__nat,axiom,
    ! [Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ) ).

% even_of_nat
tff(fact_6855_even__of__nat,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ) ).

% even_of_nat
tff(fact_6856_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I2: num,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,X)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I2)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),Nb)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_6857_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I2: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,I2)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),Nb)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_6858_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I2: num,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,I2)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),Nb)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_6859_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I2: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),Nb)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_6860_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I2: num,Nb: nat,X: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I2)),Nb)),aa_nat_rat(semiri681578069525770553at_rat,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),Nb)),X) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_6861_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I2: num,Nb: nat,X: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,I2)),Nb)),aa_nat_real(semiri5074537144036343181t_real,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),Nb)),X) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_6862_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I2: num,Nb: nat,X: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,I2)),Nb)),aa_nat_int(semiri1314217659103216013at_int,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),Nb)),X) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_6863_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I2: num,Nb: nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),Nb)),aa_nat_nat(semiri1316708129612266289at_nat,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),Nb)),X) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_6864_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I2: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,I2)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),Nb)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_6865_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I2: num,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,X)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I2)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),Nb)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_6866_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I2: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),Nb)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_6867_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I2: num,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,I2)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),Nb)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_6868_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I2: num,Nb: nat,X: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,I2)),Nb)),aa_nat_real(semiri5074537144036343181t_real,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),Nb)),X) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_6869_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I2: num,Nb: nat,X: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I2)),Nb)),aa_nat_rat(semiri681578069525770553at_rat,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),Nb)),X) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_6870_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I2: num,Nb: nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),Nb)),aa_nat_nat(semiri1316708129612266289at_nat,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),Nb)),X) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_6871_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I2: num,Nb: nat,X: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,I2)),Nb)),aa_nat_int(semiri1314217659103216013at_int,X))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I2)),Nb)),X) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_6872_real__arch__simple,axiom,
    ! [X: real] :
    ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_nat_real(semiri5074537144036343181t_real,N)) ).

% real_arch_simple
tff(fact_6873_real__arch__simple,axiom,
    ! [X: rat] :
    ? [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_nat_rat(semiri681578069525770553at_rat,N)) ).

% real_arch_simple
tff(fact_6874_reals__Archimedean2,axiom,
    ! [X: rat] :
    ? [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_nat_rat(semiri681578069525770553at_rat,N)) ).

% reals_Archimedean2
tff(fact_6875_reals__Archimedean2,axiom,
    ! [X: real] :
    ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_nat_real(semiri5074537144036343181t_real,N)) ).

% reals_Archimedean2
tff(fact_6876_mult__of__nat__commute,axiom,
    ! [X: nat,Y2: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,X)),Y2) = aa_rat_rat(times_times_rat(Y2),aa_nat_rat(semiri681578069525770553at_rat,X)) ).

% mult_of_nat_commute
tff(fact_6877_mult__of__nat__commute,axiom,
    ! [X: nat,Y2: real] : aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,X)),Y2) = aa_real_real(times_times_real(Y2),aa_nat_real(semiri5074537144036343181t_real,X)) ).

% mult_of_nat_commute
tff(fact_6878_mult__of__nat__commute,axiom,
    ! [X: nat,Y2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,X)),Y2) = aa_int_int(aa_int_fun_int_int(times_times_int,Y2),aa_nat_int(semiri1314217659103216013at_int,X)) ).

% mult_of_nat_commute
tff(fact_6879_mult__of__nat__commute,axiom,
    ! [X: nat,Y2: extended_enat] : aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,X)),Y2) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(Y2),aa_nat_Extended_enat(semiri4216267220026989637d_enat,X)) ).

% mult_of_nat_commute
tff(fact_6880_mult__of__nat__commute,axiom,
    ! [X: nat,Y2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,X)),Y2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y2),aa_nat_nat(semiri1316708129612266289at_nat,X)) ).

% mult_of_nat_commute
tff(fact_6881_of__nat__less__of__int__iff,axiom,
    ! [Nb: nat,X: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb)),aa_int_rat(ring_1_of_int_rat,X))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),X) ) ).

% of_nat_less_of_int_iff
tff(fact_6882_of__nat__less__of__int__iff,axiom,
    ! [Nb: nat,X: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_int_real(ring_1_of_int_real,X))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),X) ) ).

% of_nat_less_of_int_iff
tff(fact_6883_of__nat__less__of__int__iff,axiom,
    ! [Nb: nat,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_int_int(ring_1_of_int_int,X))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),X) ) ).

% of_nat_less_of_int_iff
tff(fact_6884_of__nat__0__le__iff,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(semiri5074537144036343181t_real,Nb)) ).

% of_nat_0_le_iff
tff(fact_6885_of__nat__0__le__iff,axiom,
    ! [Nb: nat] : aa_Extended_enat_o(ord_le2932123472753598470d_enat(zero_z5237406670263579293d_enat),aa_nat_Extended_enat(semiri4216267220026989637d_enat,Nb)) ).

% of_nat_0_le_iff
tff(fact_6886_of__nat__0__le__iff,axiom,
    ! [Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ).

% of_nat_0_le_iff
tff(fact_6887_of__nat__0__le__iff,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% of_nat_0_le_iff
tff(fact_6888_of__nat__0__le__iff,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% of_nat_0_le_iff
tff(fact_6889_of__nat__less__0__iff,axiom,
    ! [Ma: nat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Ma)),zero_zero_rat) ).

% of_nat_less_0_iff
tff(fact_6890_of__nat__less__0__iff,axiom,
    ! [Ma: nat] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Ma)),zero_zero_real) ).

% of_nat_less_0_iff
tff(fact_6891_of__nat__less__0__iff,axiom,
    ! [Ma: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Ma)),zero_zero_int) ).

% of_nat_less_0_iff
tff(fact_6892_of__nat__less__0__iff,axiom,
    ! [Ma: nat] : ~ aa_Extended_enat_o(ord_le72135733267957522d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,Ma)),zero_z5237406670263579293d_enat) ).

% of_nat_less_0_iff
tff(fact_6893_of__nat__less__0__iff,axiom,
    ! [Ma: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Ma)),zero_zero_nat) ).

% of_nat_less_0_iff
tff(fact_6894_of__nat__neq__0,axiom,
    ! [Nb: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,Nb)) != zero_zero_complex ).

% of_nat_neq_0
tff(fact_6895_of__nat__neq__0,axiom,
    ! [Nb: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,Nb)) != zero_zero_rat ).

% of_nat_neq_0
tff(fact_6896_of__nat__neq__0,axiom,
    ! [Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb)) != zero_zero_real ).

% of_nat_neq_0
tff(fact_6897_of__nat__neq__0,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb)) != zero_zero_int ).

% of_nat_neq_0
tff(fact_6898_of__nat__neq__0,axiom,
    ! [Nb: nat] : aa_nat_Extended_enat(semiri4216267220026989637d_enat,aa_nat_nat(suc,Nb)) != zero_z5237406670263579293d_enat ).

% of_nat_neq_0
tff(fact_6899_of__nat__neq__0,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,Nb)) != zero_zero_nat ).

% of_nat_neq_0
tff(fact_6900_div__mult2__eq_H,axiom,
    ! [A: int,Ma: nat,Nb: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Ma)),aa_nat_int(semiri1314217659103216013at_int,Nb))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(semiri1314217659103216013at_int,Ma))),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% div_mult2_eq'
tff(fact_6901_div__mult2__eq_H,axiom,
    ! [A: nat,Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Ma)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,Ma))),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% div_mult2_eq'
tff(fact_6902_of__nat__less__imp__less,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Ma)),aa_nat_rat(semiri681578069525770553at_rat,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb) ) ).

% of_nat_less_imp_less
tff(fact_6903_of__nat__less__imp__less,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Ma)),aa_nat_real(semiri5074537144036343181t_real,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb) ) ).

% of_nat_less_imp_less
tff(fact_6904_of__nat__less__imp__less,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Ma)),aa_nat_int(semiri1314217659103216013at_int,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb) ) ).

% of_nat_less_imp_less
tff(fact_6905_of__nat__less__imp__less,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,Ma)),aa_nat_Extended_enat(semiri4216267220026989637d_enat,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb) ) ).

% of_nat_less_imp_less
tff(fact_6906_of__nat__less__imp__less,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Ma)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb) ) ).

% of_nat_less_imp_less
tff(fact_6907_less__imp__of__nat__less,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Ma)),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ) ).

% less_imp_of_nat_less
tff(fact_6908_less__imp__of__nat__less,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Ma)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ).

% less_imp_of_nat_less
tff(fact_6909_less__imp__of__nat__less,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Ma)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ) ).

% less_imp_of_nat_less
tff(fact_6910_less__imp__of__nat__less,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
     => aa_Extended_enat_o(ord_le72135733267957522d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,Ma)),aa_nat_Extended_enat(semiri4216267220026989637d_enat,Nb)) ) ).

% less_imp_of_nat_less
tff(fact_6911_less__imp__of__nat__less,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Ma)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ) ).

% less_imp_of_nat_less
tff(fact_6912_of__nat__mono,axiom,
    ! [I2: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,I2)),aa_nat_real(semiri5074537144036343181t_real,J)) ) ).

% of_nat_mono
tff(fact_6913_of__nat__mono,axiom,
    ! [I2: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => aa_Extended_enat_o(ord_le2932123472753598470d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,I2)),aa_nat_Extended_enat(semiri4216267220026989637d_enat,J)) ) ).

% of_nat_mono
tff(fact_6914_of__nat__mono,axiom,
    ! [I2: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,I2)),aa_nat_rat(semiri681578069525770553at_rat,J)) ) ).

% of_nat_mono
tff(fact_6915_of__nat__mono,axiom,
    ! [I2: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,I2)),aa_nat_nat(semiri1316708129612266289at_nat,J)) ) ).

% of_nat_mono
tff(fact_6916_of__nat__mono,axiom,
    ! [I2: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,I2)),aa_nat_int(semiri1314217659103216013at_int,J)) ) ).

% of_nat_mono
tff(fact_6917_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),Nb)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(semiri1314217659103216013at_int,Ma)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
tff(fact_6918_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1316708129612266289at_nat,Ma)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
tff(fact_6919_of__nat__dvd__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(semiri4939895301339042750nteger,Ma)),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ma),Nb) ) ).

% of_nat_dvd_iff
tff(fact_6920_of__nat__dvd__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(semiri1314217659103216013at_int,Ma)),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ma),Nb) ) ).

% of_nat_dvd_iff
tff(fact_6921_of__nat__dvd__iff,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(semiri1316708129612266289at_nat,Ma)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ma),Nb) ) ).

% of_nat_dvd_iff
tff(fact_6922_abs__zmult__eq__1,axiom,
    ! [Ma: int,Nb: int] :
      ( ( aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,Ma),Nb)) = one_one_int )
     => ( aa_int_int(abs_abs_int,Ma) = one_one_int ) ) ).

% abs_zmult_eq_1
tff(fact_6923_pi__ge__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),pi) ).

% pi_ge_zero
tff(fact_6924_of__nat__mod,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),Nb)) = modulo364778990260209775nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Ma),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)) ).

% of_nat_mod
tff(fact_6925_of__nat__mod,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),Nb)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(semiri1314217659103216013at_int,Ma)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% of_nat_mod
tff(fact_6926_of__nat__mod,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(semiri1316708129612266289at_nat,Ma)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% of_nat_mod
tff(fact_6927_take__bit__of__nat,axiom,
    ! [Nb: nat,Ma: nat] : aa_int_int(bit_se2923211474154528505it_int(Nb),aa_nat_int(semiri1314217659103216013at_int,Ma)) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Ma)) ).

% take_bit_of_nat
tff(fact_6928_take__bit__of__nat,axiom,
    ! [Nb: nat,Ma: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_nat_nat(semiri1316708129612266289at_nat,Ma)) = aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Ma)) ).

% take_bit_of_nat
tff(fact_6929_of__nat__and__eq,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Ma),Nb)) = aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_nat_int(semiri1314217659103216013at_int,Ma)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% of_nat_and_eq
tff(fact_6930_of__nat__and__eq,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Ma),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(semiri1316708129612266289at_nat,Ma)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% of_nat_and_eq
tff(fact_6931_bit__of__nat__iff__bit,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_nat_int(semiri1314217659103216013at_int,Ma)),Nb)
    <=> aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,Ma),Nb) ) ).

% bit_of_nat_iff_bit
tff(fact_6932_bit__of__nat__iff__bit,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(semiri1316708129612266289at_nat,Ma)),Nb)
    <=> aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,Ma),Nb) ) ).

% bit_of_nat_iff_bit
tff(fact_6933_of__nat__mask__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(bit_se2002935070580805687sk_nat,Nb)) = aa_nat_nat(bit_se2002935070580805687sk_nat,Nb) ).

% of_nat_mask_eq
tff(fact_6934_of__nat__mask__eq,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(bit_se2002935070580805687sk_nat,Nb)) = aa_nat_int(bit_se2000444600071755411sk_int,Nb) ).

% of_nat_mask_eq
tff(fact_6935_ex__less__of__nat__mult,axiom,
    ! [X: rat,Y2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),X)
     => ? [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y2),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),X)) ) ).

% ex_less_of_nat_mult
tff(fact_6936_ex__less__of__nat__mult,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),X)) ) ).

% ex_less_of_nat_mult
tff(fact_6937_of__nat__diff,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma)
     => ( aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb)) = aa_complex_complex(minus_minus_complex(aa_nat_complex(semiri8010041392384452111omplex,Ma)),aa_nat_complex(semiri8010041392384452111omplex,Nb)) ) ) ).

% of_nat_diff
tff(fact_6938_of__nat__diff,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma)
     => ( aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(semiri681578069525770553at_rat,Ma)),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ) ) ).

% of_nat_diff
tff(fact_6939_of__nat__diff,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma)
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb)) = aa_real_real(minus_minus_real(aa_nat_real(semiri5074537144036343181t_real,Ma)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ).

% of_nat_diff
tff(fact_6940_of__nat__diff,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma)
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,Ma)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ) ) ).

% of_nat_diff
tff(fact_6941_of__nat__diff,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma)
     => ( aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Ma)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ) ) ).

% of_nat_diff
tff(fact_6942_exp__of__nat__mult,axiom,
    ! [Nb: nat,X: complex] : aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),X)) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(exp_complex,X)),Nb) ).

% exp_of_nat_mult
tff(fact_6943_exp__of__nat__mult,axiom,
    ! [Nb: nat,X: real] : aa_real_real(exp_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),X)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(exp_real,X)),Nb) ).

% exp_of_nat_mult
tff(fact_6944_exp__of__nat2__mult,axiom,
    ! [X: complex,Nb: nat] : aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(X),aa_nat_complex(semiri8010041392384452111omplex,Nb))) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(exp_complex,X)),Nb) ).

% exp_of_nat2_mult
tff(fact_6945_exp__of__nat2__mult,axiom,
    ! [X: real,Nb: nat] : aa_real_real(exp_real,aa_real_real(times_times_real(X),aa_nat_real(semiri5074537144036343181t_real,Nb))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(exp_real,X)),Nb) ).

% exp_of_nat2_mult
tff(fact_6946_reals__Archimedean3,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ! [Y5: real] :
        ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y5),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),X)) ) ).

% reals_Archimedean3
tff(fact_6947_dvd__imp__le__int,axiom,
    ! [I2: int,D: int] :
      ( ( I2 != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),I2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,D)),aa_int_int(abs_abs_int,I2)) ) ) ).

% dvd_imp_le_int
tff(fact_6948_real__of__nat__div4,axiom,
    ! [Nb: nat,X: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),X))),aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,X))) ).

% real_of_nat_div4
tff(fact_6949_abs__mod__less,axiom,
    ! [L: int,K2: int] :
      ( ( L != zero_zero_int )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),L))),aa_int_int(abs_abs_int,L)) ) ).

% abs_mod_less
tff(fact_6950_real__of__nat__div,axiom,
    ! [D: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),Nb)
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),D)) = aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,D)) ) ) ).

% real_of_nat_div
tff(fact_6951_mod__mult2__eq_H,axiom,
    ! [A: code_integer,Ma: nat,Nb: nat] : modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Ma)),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Ma)),modulo364778990260209775nteger(divide6298287555418463151nteger(A,aa_nat_Code_integer(semiri4939895301339042750nteger,Ma)),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))),modulo364778990260209775nteger(A,aa_nat_Code_integer(semiri4939895301339042750nteger,Ma))) ).

% mod_mult2_eq'
tff(fact_6952_mod__mult2__eq_H,axiom,
    ! [A: int,Ma: nat,Nb: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Ma)),aa_nat_int(semiri1314217659103216013at_int,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Ma)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(semiri1314217659103216013at_int,Ma))),aa_nat_int(semiri1314217659103216013at_int,Nb)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_nat_int(semiri1314217659103216013at_int,Ma))) ).

% mod_mult2_eq'
tff(fact_6953_mod__mult2__eq_H,axiom,
    ! [A: nat,Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Ma)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Ma)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,Ma))),aa_nat_nat(semiri1316708129612266289at_nat,Nb)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,Ma))) ).

% mod_mult2_eq'
tff(fact_6954_field__char__0__class_Oof__nat__div,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),Nb)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(semiri8010041392384452111omplex,Ma)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),Nb)))),aa_nat_complex(semiri8010041392384452111omplex,Nb)) ).

% field_char_0_class.of_nat_div
tff(fact_6955_field__char__0__class_Oof__nat__div,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),Nb)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_nat_rat(semiri681578069525770553at_rat,Ma)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),Nb)))),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ).

% field_char_0_class.of_nat_div
tff(fact_6956_field__char__0__class_Oof__nat__div,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),Nb)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_nat_real(semiri5074537144036343181t_real,Ma)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),Nb)))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ).

% field_char_0_class.of_nat_div
tff(fact_6957_pi__less__4,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))) ).

% pi_less_4
tff(fact_6958_nat__less__real__le,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Ma)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,Ma)) ) ).

% nat_less_real_le
tff(fact_6959_nat__le__real__less,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,Ma)),one_one_real)) ) ).

% nat_le_real_less
tff(fact_6960_pi__ge__two,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi) ).

% pi_ge_two
tff(fact_6961_zdvd__mult__cancel1,axiom,
    ! [Ma: int,Nb: int] :
      ( ( Ma != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,Ma),Nb)),Ma)
      <=> ( aa_int_int(abs_abs_int,Nb) = one_one_int ) ) ) ).

% zdvd_mult_cancel1
tff(fact_6962_pi__half__neq__two,axiom,
    aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) != aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)) ).

% pi_half_neq_two
tff(fact_6963_log__of__power__eq,axiom,
    ! [Ma: nat,B: real,Nb: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Ma) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => ( aa_nat_real(semiri5074537144036343181t_real,Nb) = aa_real_real(log(B),aa_nat_real(semiri5074537144036343181t_real,Ma)) ) ) ) ).

% log_of_power_eq
tff(fact_6964_less__log__of__power,axiom,
    ! [B: real,Nb: nat,Ma: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)),Ma)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(log(B),Ma)) ) ) ).

% less_log_of_power
tff(fact_6965_real__of__nat__div__aux,axiom,
    ! [X: nat,D: nat] : aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,X)),aa_nat_real(semiri5074537144036343181t_real,D)) = aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),D))),aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),D))),aa_nat_real(semiri5074537144036343181t_real,D))) ).

% real_of_nat_div_aux
tff(fact_6966_nat__approx__posE,axiom,
    ! [E: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E)
     => ~ ! [N: nat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N)))),E) ) ).

% nat_approx_posE
tff(fact_6967_nat__approx__posE,axiom,
    ! [E: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
     => ~ ! [N: nat] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N)))),E) ) ).

% nat_approx_posE
tff(fact_6968_of__nat__less__two__power,axiom,
    ! [Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Nb)) ).

% of_nat_less_two_power
tff(fact_6969_of__nat__less__two__power,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Nb)) ).

% of_nat_less_two_power
tff(fact_6970_of__nat__less__two__power,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% of_nat_less_two_power
tff(fact_6971_inverse__of__nat__le,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma)
     => ( ( Nb != zero_zero_nat )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,Ma))),aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,Nb))) ) ) ).

% inverse_of_nat_le
tff(fact_6972_inverse__of__nat__le,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma)
     => ( ( Nb != zero_zero_nat )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,Ma))),aa_rat_rat(divide_divide_rat(one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,Nb))) ) ) ).

% inverse_of_nat_le
tff(fact_6973_exp__divide__power__eq,axiom,
    ! [Nb: nat,X: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(exp_complex,aa_complex_complex(divide1717551699836669952omplex(X),aa_nat_complex(semiri8010041392384452111omplex,Nb)))),Nb) = aa_complex_complex(exp_complex,X) ) ) ).

% exp_divide_power_eq
tff(fact_6974_exp__divide__power__eq,axiom,
    ! [Nb: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(exp_real,aa_real_real(divide_divide_real(X),aa_nat_real(semiri5074537144036343181t_real,Nb)))),Nb) = aa_real_real(exp_real,X) ) ) ).

% exp_divide_power_eq
tff(fact_6975_even__abs__add__iff,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,K2)),L))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),L)) ) ).

% even_abs_add_iff
tff(fact_6976_even__add__abs__iff,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),aa_int_int(abs_abs_int,L)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),L)) ) ).

% even_add_abs_iff
tff(fact_6977_real__archimedian__rdiv__eq__0,axiom,
    ! [X: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => ( ! [M2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M2)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,M2)),X)),C) )
         => ( X = zero_zero_real ) ) ) ) ).

% real_archimedian_rdiv_eq_0
tff(fact_6978_pi__half__neq__zero,axiom,
    aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) != zero_zero_real ).

% pi_half_neq_zero
tff(fact_6979_pi__half__less__two,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% pi_half_less_two
tff(fact_6980_pi__half__le__two,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% pi_half_le_two
tff(fact_6981_real__of__nat__div2,axiom,
    ! [Nb: nat,X: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,X))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),X)))) ).

% real_of_nat_div2
tff(fact_6982_le__log__of__power,axiom,
    ! [B: real,Nb: nat,Ma: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)),Ma)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(log(B),Ma)) ) ) ).

% le_log_of_power
tff(fact_6983_ln__realpow,axiom,
    ! [X: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_real(ln_ln_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Nb)) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(ln_ln_real,X)) ) ) ).

% ln_realpow
tff(fact_6984_log__nat__power,axiom,
    ! [X: real,B: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_real(log(B),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Nb)) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(log(B),X)) ) ) ).

% log_nat_power
tff(fact_6985_real__of__nat__div3,axiom,
    ! [Nb: nat,X: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,X))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),X)))),one_one_real) ).

% real_of_nat_div3
tff(fact_6986_nat__intermed__int__val,axiom,
    ! [Ma: nat,Nb: nat,F: fun_nat_int,K2: int] :
      ( ! [I3: nat] :
          ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),I3)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Nb) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,I3))),aa_nat_int(F,I3)))),one_one_int) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,Ma)),K2)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),aa_nat_int(F,Nb))
           => ? [I3: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),I3)
                & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),Nb)
                & ( aa_nat_int(F,I3) = K2 ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_6987_incr__lemma,axiom,
    ! [D: int,Z2: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Z2))),one_one_int)),D))) ) ).

% incr_lemma
tff(fact_6988_decr__lemma,axiom,
    ! [D: int,X: int,Z2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Z2))),one_one_int)),D))),Z2) ) ).

% decr_lemma
tff(fact_6989_pi__half__gt__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% pi_half_gt_zero
tff(fact_6990_linear__plus__1__le__power,axiom,
    ! [X: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),X)),one_one_real)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(plus_plus_real(X),one_one_real)),Nb)) ) ).

% linear_plus_1_le_power
tff(fact_6991_pi__half__ge__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% pi_half_ge_zero
tff(fact_6992_log2__of__power__eq,axiom,
    ! [Ma: nat,Nb: nat] :
      ( ( Ma = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) )
     => ( aa_nat_real(semiri5074537144036343181t_real,Nb) = aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Ma)) ) ) ).

% log2_of_power_eq
tff(fact_6993_log__of__power__less,axiom,
    ! [Ma: nat,B: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Ma)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ma)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(log(B),aa_nat_real(semiri5074537144036343181t_real,Ma))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ) ).

% log_of_power_less
tff(fact_6994_Bernoulli__inequality,axiom,
    ! [X: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(one_one_real),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),X))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(plus_plus_real(one_one_real),X)),Nb)) ) ).

% Bernoulli_inequality
tff(fact_6995_m2pi__less__pi,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))),pi) ).

% m2pi_less_pi
tff(fact_6996_arctan__ubound,axiom,
    ! [Y2: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Y2)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% arctan_ubound
tff(fact_6997_arctan__one,axiom,
    aa_real_real(arctan,one_one_real) = aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))) ).

% arctan_one
tff(fact_6998_nat__ivt__aux,axiom,
    ! [Nb: nat,F: fun_nat_int,K2: int] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Nb)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,I3))),aa_nat_int(F,I3)))),one_one_int) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,zero_zero_nat)),K2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),aa_nat_int(F,Nb))
         => ? [I3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),Nb)
              & ( aa_nat_int(F,I3) = K2 ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_6999_log__of__power__le,axiom,
    ! [Ma: nat,B: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Ma)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ma)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(log(B),aa_nat_real(semiri5074537144036343181t_real,Ma))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ) ).

% log_of_power_le
tff(fact_7000_minus__pi__half__less__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),zero_zero_real) ).

% minus_pi_half_less_zero
tff(fact_7001_arctan__bounded,axiom,
    ! [Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arctan,Y2))
      & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Y2)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ).

% arctan_bounded
tff(fact_7002_arctan__lbound,axiom,
    ! [Y2: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arctan,Y2)) ).

% arctan_lbound
tff(fact_7003_nat0__intermed__int__val,axiom,
    ! [Nb: nat,F: fun_nat_int,K2: int] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Nb)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I3),one_one_nat))),aa_nat_int(F,I3)))),one_one_int) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,zero_zero_nat)),K2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),aa_nat_int(F,Nb))
         => ? [I3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),Nb)
              & ( aa_nat_int(F,I3) = K2 ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_7004_less__log2__of__power,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),Ma)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Ma))) ) ).

% less_log2_of_power
tff(fact_7005_le__log2__of__power,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),Ma)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Ma))) ) ).

% le_log2_of_power
tff(fact_7006_Bernoulli__inequality__even,axiom,
    ! [Nb: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(one_one_real),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),X))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(plus_plus_real(one_one_real),X)),Nb)) ) ).

% Bernoulli_inequality_even
tff(fact_7007_log2__of__power__less,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ma)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Ma))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ).

% log2_of_power_less
tff(fact_7008_of__nat__code__if,axiom,
    ! [Nb: nat] :
      aa_nat_complex(semiri8010041392384452111omplex,Nb) = $ite(Nb = zero_zero_nat,zero_zero_complex,produc1917071388513777916omplex(aTP_Lamm_dv,divmod_nat(Nb,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% of_nat_code_if
tff(fact_7009_of__nat__code__if,axiom,
    ! [Nb: nat] :
      aa_nat_rat(semiri681578069525770553at_rat,Nb) = $ite(Nb = zero_zero_nat,zero_zero_rat,produc6207742614233964070at_rat(aTP_Lamm_dw,divmod_nat(Nb,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% of_nat_code_if
tff(fact_7010_of__nat__code__if,axiom,
    ! [Nb: nat] :
      aa_nat_real(semiri5074537144036343181t_real,Nb) = $ite(Nb = zero_zero_nat,zero_zero_real,produc1703576794950452218t_real(aTP_Lamm_dx,divmod_nat(Nb,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% of_nat_code_if
tff(fact_7011_of__nat__code__if,axiom,
    ! [Nb: nat] :
      aa_nat_int(semiri1314217659103216013at_int,Nb) = $ite(Nb = zero_zero_nat,zero_zero_int,produc6840382203811409530at_int(aTP_Lamm_dy,divmod_nat(Nb,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% of_nat_code_if
tff(fact_7012_of__nat__code__if,axiom,
    ! [Nb: nat] :
      aa_nat_Extended_enat(semiri4216267220026989637d_enat,Nb) = $ite(Nb = zero_zero_nat,zero_z5237406670263579293d_enat,produc2676513652042109336d_enat(aTP_Lamm_dz,divmod_nat(Nb,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% of_nat_code_if
tff(fact_7013_of__nat__code__if,axiom,
    ! [Nb: nat] :
      aa_nat_nat(semiri1316708129612266289at_nat,Nb) = $ite(Nb = zero_zero_nat,zero_zero_nat,aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(aTP_Lamm_ea),divmod_nat(Nb,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% of_nat_code_if
tff(fact_7014_monoseq__arctan__series,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real)
     => topolo6980174941875973593q_real(aa_real_fun_nat_real(aTP_Lamm_eb,X)) ) ).

% monoseq_arctan_series
tff(fact_7015_lemma__termdiff3,axiom,
    ! [H: real,Z2: real,K5: real,Nb: nat] :
      ( ( H != zero_zero_real )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z2)),K5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(plus_plus_real(Z2),H))),K5)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(plus_plus_real(Z2),H)),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z2),Nb))),H)),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))))),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,K5),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(real_V7735802525324610683m_real,H))) ) ) ) ).

% lemma_termdiff3
tff(fact_7016_lemma__termdiff3,axiom,
    ! [H: complex,Z2: complex,K5: real,Nb: nat] :
      ( ( H != zero_zero_complex )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z2)),K5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(Z2),H))),K5)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(plus_plus_complex(Z2),H)),Nb)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Z2),Nb))),H)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Z2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))))),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,K5),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_complex_real(real_V1022390504157884413omplex,H))) ) ) ) ).

% lemma_termdiff3
tff(fact_7017_sin__cos__npi,axiom,
    ! [Nb: nat] : aa_real_real(sin_real,aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb) ).

% sin_cos_npi
tff(fact_7018_ln__series,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
       => ( aa_real_real(ln_ln_real,X) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_ec,X)) ) ) ) ).

% ln_series
tff(fact_7019_arctan__series,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real)
     => ( aa_real_real(arctan,X) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_ed,X)) ) ) ).

% arctan_series
tff(fact_7020_int__eq__iff__numeral,axiom,
    ! [Ma: nat,V: num] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Ma) = aa_num_int(numeral_numeral_int,V) )
    <=> ( Ma = aa_num_nat(numeral_numeral_nat,V) ) ) ).

% int_eq_iff_numeral
tff(fact_7021_negative__zle,axiom,
    ! [Nb: nat,Ma: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Nb))),aa_nat_int(semiri1314217659103216013at_int,Ma)) ).

% negative_zle
tff(fact_7022_sin__npi,axiom,
    ! [Nb: nat] : aa_real_real(sin_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),pi)) = zero_zero_real ).

% sin_npi
tff(fact_7023_sin__npi2,axiom,
    ! [Nb: nat] : aa_real_real(sin_real,aa_real_real(times_times_real(pi),aa_nat_real(semiri5074537144036343181t_real,Nb))) = zero_zero_real ).

% sin_npi2
tff(fact_7024_sin__npi__int,axiom,
    ! [Nb: int] : aa_real_real(sin_real,aa_real_real(times_times_real(pi),aa_int_real(ring_1_of_int_real,Nb))) = zero_zero_real ).

% sin_npi_int
tff(fact_7025_powser__zero,axiom,
    ! [F: fun_nat_complex] : suminf_complex(aTP_Lamm_ee(F)) = aa_nat_complex(F,zero_zero_nat) ).

% powser_zero
tff(fact_7026_powser__zero,axiom,
    ! [F: fun_nat_real] : suminf_real(aTP_Lamm_ef(F)) = aa_nat_real(F,zero_zero_nat) ).

% powser_zero
tff(fact_7027_sin__two__pi,axiom,
    aa_real_real(sin_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)) = zero_zero_real ).

% sin_two_pi
tff(fact_7028_sin__pi__half,axiom,
    aa_real_real(sin_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = one_one_real ).

% sin_pi_half
tff(fact_7029_sin__periodic,axiom,
    ! [X: real] : aa_real_real(sin_real,aa_real_real(plus_plus_real(X),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) = aa_real_real(sin_real,X) ).

% sin_periodic
tff(fact_7030_sin__2npi,axiom,
    ! [Nb: nat] : aa_real_real(sin_real,aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)) = zero_zero_real ).

% sin_2npi
tff(fact_7031_sin__2pi__minus,axiom,
    ! [X: real] : aa_real_real(sin_real,aa_real_real(minus_minus_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),X)) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,X)) ).

% sin_2pi_minus
tff(fact_7032_sin__int__2pin,axiom,
    ! [Nb: int] : aa_real_real(sin_real,aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),aa_int_real(ring_1_of_int_real,Nb))) = zero_zero_real ).

% sin_int_2pin
tff(fact_7033_sin__3over2__pi,axiom,
    aa_real_real(sin_real,aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),pi)) = aa_real_real(uminus_uminus_real,one_one_real) ).

% sin_3over2_pi
tff(fact_7034_nat__int__comparison_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
    <=> ( aa_nat_int(semiri1314217659103216013at_int,A) = aa_nat_int(semiri1314217659103216013at_int,B) ) ) ).

% nat_int_comparison(1)
tff(fact_7035_int__if,axiom,
    ! [P: $o,A: nat,B: nat] :
      aa_nat_int(semiri1314217659103216013at_int,
        $ite((P),A,B)) = $ite((P),aa_nat_int(semiri1314217659103216013at_int,A),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% int_if
tff(fact_7036_of__nat__eq__enat,axiom,
    ! [Nb: nat] : aa_nat_Extended_enat(semiri4216267220026989637d_enat,Nb) = aa_nat_Extended_enat(extended_enat2,Nb) ).

% of_nat_eq_enat
tff(fact_7037_sin__x__le__x,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,X)),X) ) ).

% sin_x_le_x
tff(fact_7038_sin__le__one,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,X)),one_one_real) ).

% sin_le_one
tff(fact_7039_abs__sin__x__le__abs__x,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(sin_real,X))),aa_real_real(abs_abs_real,X)) ).

% abs_sin_x_le_abs_x
tff(fact_7040_int__ops_I1_J,axiom,
    aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% int_ops(1)
tff(fact_7041_complex__mod__minus__le__complex__mod,axiom,
    ! [X: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_complex_real(real_V1022390504157884413omplex,X))),aa_complex_real(real_V1022390504157884413omplex,X)) ).

% complex_mod_minus_le_complex_mod
tff(fact_7042_int__ops_I3_J,axiom,
    ! [Nb: num] : aa_nat_int(semiri1314217659103216013at_int,aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_int(numeral_numeral_int,Nb) ).

% int_ops(3)
tff(fact_7043_nat__int__comparison_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ) ).

% nat_int_comparison(2)
tff(fact_7044_zle__int,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Ma)),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb) ) ).

% zle_int
tff(fact_7045_nat__int__comparison_I3_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ) ).

% nat_int_comparison(3)
tff(fact_7046_complex__mod__triangle__ineq2,axiom,
    ! [B: complex,A: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(B),A))),aa_complex_real(real_V1022390504157884413omplex,B))),aa_complex_real(real_V1022390504157884413omplex,A)) ).

% complex_mod_triangle_ineq2
tff(fact_7047_zero__le__imp__eq__int,axiom,
    ! [K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
     => ? [N: nat] : K2 = aa_nat_int(semiri1314217659103216013at_int,N) ) ).

% zero_le_imp_eq_int
tff(fact_7048_nonneg__int__cases,axiom,
    ! [K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
     => ~ ! [N: nat] : K2 != aa_nat_int(semiri1314217659103216013at_int,N) ) ).

% nonneg_int_cases
tff(fact_7049_zadd__int__left,axiom,
    ! [Ma: nat,Nb: nat,Z2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Ma)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),Z2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb))),Z2) ).

% zadd_int_left
tff(fact_7050_int__plus,axiom,
    ! [Nb: nat,Ma: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Ma)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_nat_int(semiri1314217659103216013at_int,Ma)) ).

% int_plus
tff(fact_7051_int__ops_I5_J,axiom,
    ! [A: nat,B: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% int_ops(5)
tff(fact_7052_int__ops_I7_J,axiom,
    ! [A: nat,B: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% int_ops(7)
tff(fact_7053_int__ops_I2_J,axiom,
    aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).

% int_ops(2)
tff(fact_7054_zle__iff__zadd,axiom,
    ! [W: int,Z2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Z2)
    <=> ? [N4: nat] : Z2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,W),aa_nat_int(semiri1314217659103216013at_int,N4)) ) ).

% zle_iff_zadd
tff(fact_7055_zdiv__int,axiom,
    ! [A: nat,B: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% zdiv_int
tff(fact_7056_zmod__int,axiom,
    ! [A: nat,B: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% zmod_int
tff(fact_7057_nat__less__as__int,axiom,
    ! [X2: nat,Xa3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Xa3)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X2)),aa_nat_int(semiri1314217659103216013at_int,Xa3)) ) ).

% nat_less_as_int
tff(fact_7058_nat__leq__as__int,axiom,
    ! [X2: nat,Xa3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Xa3)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X2)),aa_nat_int(semiri1314217659103216013at_int,Xa3)) ) ).

% nat_leq_as_int
tff(fact_7059_sin__x__ge__neg__x,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,X)),aa_real_real(sin_real,X)) ) ).

% sin_x_ge_neg_x
tff(fact_7060_sin__ge__zero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),pi)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,X)) ) ) ).

% sin_ge_zero
tff(fact_7061_sin__ge__minus__one,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(sin_real,X)) ).

% sin_ge_minus_one
tff(fact_7062_abs__sin__le__one,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(sin_real,X))),one_one_real) ).

% abs_sin_le_one
tff(fact_7063_int__cases4,axiom,
    ! [Ma: int] :
      ( ! [N: nat] : Ma != aa_nat_int(semiri1314217659103216013at_int,N)
     => ~ ! [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
           => ( Ma != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) ) ) ) ).

% int_cases4
tff(fact_7064_int__zle__neg,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Ma)))
    <=> ( ( Nb = zero_zero_nat )
        & ( Ma = zero_zero_nat ) ) ) ).

% int_zle_neg
tff(fact_7065_int__Suc,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int) ).

% int_Suc
tff(fact_7066_int__ops_I4_J,axiom,
    ! [A: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,A)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,A)),one_one_int) ).

% int_ops(4)
tff(fact_7067_nonpos__int__cases,axiom,
    ! [K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),zero_zero_int)
     => ~ ! [N: nat] : K2 != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) ) ).

% nonpos_int_cases
tff(fact_7068_negative__zle__0,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Nb))),zero_zero_int) ).

% negative_zle_0
tff(fact_7069_norm__exp,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(exp_real,X))),aa_real_real(exp_real,aa_real_real(real_V7735802525324610683m_real,X))) ).

% norm_exp
tff(fact_7070_norm__exp,axiom,
    ! [X: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(exp_complex,X))),aa_real_real(exp_real,aa_complex_real(real_V1022390504157884413omplex,X))) ).

% norm_exp
tff(fact_7071_sin__zero__iff__int2,axiom,
    ! [X: real] :
      ( ( aa_real_real(sin_real,X) = zero_zero_real )
    <=> ? [I: int] : X = aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,I)),pi) ) ).

% sin_zero_iff_int2
tff(fact_7072_zero__less__imp__eq__int,axiom,
    ! [K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K2)
     => ? [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
          & ( K2 = aa_nat_int(semiri1314217659103216013at_int,N) ) ) ) ).

% zero_less_imp_eq_int
tff(fact_7073_pos__int__cases,axiom,
    ! [K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K2)
     => ~ ! [N: nat] :
            ( ( K2 = aa_nat_int(semiri1314217659103216013at_int,N) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ) ).

% pos_int_cases
tff(fact_7074_int__cases3,axiom,
    ! [K2: int] :
      ( ( K2 != zero_zero_int )
     => ( ! [N: nat] :
            ( ( K2 = aa_nat_int(semiri1314217659103216013at_int,N) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) )
       => ~ ! [N: nat] :
              ( ( K2 = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) )
             => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ) ) ).

% int_cases3
tff(fact_7075_zmult__zless__mono2__lemma,axiom,
    ! [I2: int,J: int,K2: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,I2),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,K2)),I2)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,K2)),J)) ) ) ).

% zmult_zless_mono2_lemma
tff(fact_7076_not__zle__0__negative,axiom,
    ! [Nb: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb)))) ).

% not_zle_0_negative
tff(fact_7077_int__ops_I6_J,axiom,
    ! [A: nat,B: nat] :
      aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)),zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B))) ).

% int_ops(6)
tff(fact_7078_sin__gt__zero__02,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sin_real,X)) ) ) ).

% sin_gt_zero_02
tff(fact_7079_neg__int__cases,axiom,
    ! [K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int)
     => ~ ! [N: nat] :
            ( ( K2 = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ) ).

% neg_int_cases
tff(fact_7080_zdiff__int__split,axiom,
    ! [P: fun_int_o,X: nat,Y2: nat] :
      ( aa_int_o(P,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),Y2)))
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y2),X)
         => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Y2))) )
        & ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y2)
         => aa_int_o(P,zero_zero_int) ) ) ) ).

% zdiff_int_split
tff(fact_7081_monoseq__realpow,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),one_one_real)
       => topolo6980174941875973593q_real(aa_real_fun_nat_real(power_power_real,X)) ) ) ).

% monoseq_realpow
tff(fact_7082_sin__pi__divide__n__ge__0,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero_nat )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,aa_real_real(divide_divide_real(pi),aa_nat_real(semiri5074537144036343181t_real,Nb)))) ) ).

% sin_pi_divide_n_ge_0
tff(fact_7083_sin__45,axiom,
    aa_real_real(sin_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one))))) = aa_real_real(divide_divide_real(aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% sin_45
tff(fact_7084_sin__gt__zero2,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sin_real,X)) ) ) ).

% sin_gt_zero2
tff(fact_7085_sin__lt__zero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,pi),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,X)),zero_zero_real) ) ) ).

% sin_lt_zero
tff(fact_7086_sin__30,axiom,
    aa_real_real(sin_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,one))))) = aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% sin_30
tff(fact_7087_sin__inj__pi,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( ( aa_real_real(sin_real,X) = aa_real_real(sin_real,Y2) )
             => ( X = Y2 ) ) ) ) ) ) ).

% sin_inj_pi
tff(fact_7088_sin__mono__le__eq,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,X)),aa_real_real(sin_real,Y2))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y2) ) ) ) ) ) ).

% sin_mono_le_eq
tff(fact_7089_sin__monotone__2pi__le,axiom,
    ! [Y2: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Y2)),aa_real_real(sin_real,X)) ) ) ) ).

% sin_monotone_2pi_le
tff(fact_7090_sin__60,axiom,
    aa_real_real(sin_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)))) = aa_real_real(divide_divide_real(aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% sin_60
tff(fact_7091_exp__bound__half,axiom,
    ! [Z2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z2)),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(exp_real,Z2))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ).

% exp_bound_half
tff(fact_7092_exp__bound__half,axiom,
    ! [Z2: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z2)),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(exp_complex,Z2))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ).

% exp_bound_half
tff(fact_7093_sin__le__zero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,pi),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,X)),zero_zero_real) ) ) ).

% sin_le_zero
tff(fact_7094_sin__less__zero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,X)),zero_zero_real) ) ) ).

% sin_less_zero
tff(fact_7095_sin__mono__less__eq,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,X)),aa_real_real(sin_real,Y2))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2) ) ) ) ) ) ).

% sin_mono_less_eq
tff(fact_7096_sin__monotone__2pi,axiom,
    ! [Y2: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,Y2)),aa_real_real(sin_real,X)) ) ) ) ).

% sin_monotone_2pi
tff(fact_7097_sin__total,axiom,
    ! [Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),one_one_real)
       => ? [X4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X4)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
            & ( aa_real_real(sin_real,X4) = Y2 )
            & ! [Y5: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y5)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y5),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
                  & ( aa_real_real(sin_real,Y5) = Y2 ) )
               => ( Y5 = X4 ) ) ) ) ) ).

% sin_total
tff(fact_7098_sin__pi__divide__n__gt__0,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sin_real,aa_real_real(divide_divide_real(pi),aa_nat_real(semiri5074537144036343181t_real,Nb)))) ) ).

% sin_pi_divide_n_gt_0
tff(fact_7099_sin__arctan,axiom,
    ! [X: real] : aa_real_real(sin_real,aa_real_real(arctan,X)) = aa_real_real(divide_divide_real(X),aa_real_real(sqrt,aa_real_real(plus_plus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% sin_arctan
tff(fact_7100_exp__bound__lemma,axiom,
    ! [Z2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z2)),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(exp_real,Z2))),aa_real_real(plus_plus_real(one_one_real),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(real_V7735802525324610683m_real,Z2)))) ) ).

% exp_bound_lemma
tff(fact_7101_exp__bound__lemma,axiom,
    ! [Z2: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z2)),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(exp_complex,Z2))),aa_real_real(plus_plus_real(one_one_real),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_complex_real(real_V1022390504157884413omplex,Z2)))) ) ).

% exp_bound_lemma
tff(fact_7102_sin__zero__iff__int,axiom,
    ! [X: real] :
      ( ( aa_real_real(sin_real,X) = zero_zero_real )
    <=> ? [I: int] :
          ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),I)
          & ( X = aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,I)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).

% sin_zero_iff_int
tff(fact_7103_sin__zero__lemma,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( ( aa_real_real(sin_real,X) = zero_zero_real )
       => ? [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
            & ( X = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ) ).

% sin_zero_lemma
tff(fact_7104_sin__zero__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(sin_real,X) = zero_zero_real )
    <=> ( ? [N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4)
            & ( X = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N4)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) )
        | ? [N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4)
            & ( X = aa_real_real(uminus_uminus_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N4)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ) ) ).

% sin_zero_iff
tff(fact_7105_pi__series,axiom,
    aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))) = suminf_real(aTP_Lamm_eg) ).

% pi_series
tff(fact_7106_norm__divide__numeral,axiom,
    ! [A: real,W: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(divide_divide_real(A),aa_num_real(numeral_numeral_real,W))) = aa_real_real(divide_divide_real(aa_real_real(real_V7735802525324610683m_real,A)),aa_num_real(numeral_numeral_real,W)) ).

% norm_divide_numeral
tff(fact_7107_norm__divide__numeral,axiom,
    ! [A: complex,W: num] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(divide1717551699836669952omplex(A),numera6690914467698888265omplex(W))) = aa_real_real(divide_divide_real(aa_complex_real(real_V1022390504157884413omplex,A)),aa_num_real(numeral_numeral_real,W)) ).

% norm_divide_numeral
tff(fact_7108_norm__mult__numeral1,axiom,
    ! [W: num,A: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,W)),A)) = aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,W)),aa_real_real(real_V7735802525324610683m_real,A)) ).

% norm_mult_numeral1
tff(fact_7109_norm__mult__numeral1,axiom,
    ! [W: num,A: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),A)) = aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,W)),aa_complex_real(real_V1022390504157884413omplex,A)) ).

% norm_mult_numeral1
tff(fact_7110_norm__mult__numeral2,axiom,
    ! [A: real,W: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(times_times_real(A),aa_num_real(numeral_numeral_real,W))) = aa_real_real(times_times_real(aa_real_real(real_V7735802525324610683m_real,A)),aa_num_real(numeral_numeral_real,W)) ).

% norm_mult_numeral2
tff(fact_7111_norm__mult__numeral2,axiom,
    ! [A: complex,W: num] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(W))) = aa_real_real(times_times_real(aa_complex_real(real_V1022390504157884413omplex,A)),aa_num_real(numeral_numeral_real,W)) ).

% norm_mult_numeral2
tff(fact_7112_norm__neg__numeral,axiom,
    ! [W: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) = aa_num_real(numeral_numeral_real,W) ).

% norm_neg_numeral
tff(fact_7113_norm__neg__numeral,axiom,
    ! [W: num] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) = aa_num_real(numeral_numeral_real,W) ).

% norm_neg_numeral
tff(fact_7114_norm__le__zero__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,X)),zero_zero_real)
    <=> ( X = zero_zero_real ) ) ).

% norm_le_zero_iff
tff(fact_7115_norm__le__zero__iff,axiom,
    ! [X: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,X)),zero_zero_real)
    <=> ( X = zero_zero_complex ) ) ).

% norm_le_zero_iff
tff(fact_7116_suminf__geometric,axiom,
    ! [C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,C)),one_one_real)
     => ( suminf_real(aa_real_fun_nat_real(power_power_real,C)) = aa_real_real(divide_divide_real(one_one_real),aa_real_real(minus_minus_real(one_one_real),C)) ) ) ).

% suminf_geometric
tff(fact_7117_suminf__geometric,axiom,
    ! [C: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,C)),one_one_real)
     => ( suminf_complex(aa_com8181001469867462503omplex(power_power_complex,C)) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),aa_complex_complex(minus_minus_complex(one_one_complex),C)) ) ) ).

% suminf_geometric
tff(fact_7118_suminf__zero,axiom,
    suminf_complex(aTP_Lamm_eh) = zero_zero_complex ).

% suminf_zero
tff(fact_7119_suminf__zero,axiom,
    suminf_real(aTP_Lamm_ei) = zero_zero_real ).

% suminf_zero
tff(fact_7120_suminf__zero,axiom,
    suminf_nat(aTP_Lamm_av) = zero_zero_nat ).

% suminf_zero
tff(fact_7121_suminf__zero,axiom,
    suminf_int(aTP_Lamm_ej) = zero_zero_int ).

% suminf_zero
tff(fact_7122_norm__one,axiom,
    aa_real_real(real_V7735802525324610683m_real,one_one_real) = one_one_real ).

% norm_one
tff(fact_7123_norm__one,axiom,
    aa_complex_real(real_V1022390504157884413omplex,one_one_complex) = one_one_real ).

% norm_one
tff(fact_7124_norm__numeral,axiom,
    ! [W: num] : aa_real_real(real_V7735802525324610683m_real,aa_num_real(numeral_numeral_real,W)) = aa_num_real(numeral_numeral_real,W) ).

% norm_numeral
tff(fact_7125_norm__numeral,axiom,
    ! [W: num] : aa_complex_real(real_V1022390504157884413omplex,numera6690914467698888265omplex(W)) = aa_num_real(numeral_numeral_real,W) ).

% norm_numeral
tff(fact_7126_norm__ge__zero,axiom,
    ! [X: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(real_V1022390504157884413omplex,X)) ).

% norm_ge_zero
tff(fact_7127_norm__mult,axiom,
    ! [X: real,Y2: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(times_times_real(X),Y2)) = aa_real_real(times_times_real(aa_real_real(real_V7735802525324610683m_real,X)),aa_real_real(real_V7735802525324610683m_real,Y2)) ).

% norm_mult
tff(fact_7128_norm__mult,axiom,
    ! [X: complex,Y2: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(X),Y2)) = aa_real_real(times_times_real(aa_complex_real(real_V1022390504157884413omplex,X)),aa_complex_real(real_V1022390504157884413omplex,Y2)) ).

% norm_mult
tff(fact_7129_norm__divide,axiom,
    ! [A: real,B: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(divide_divide_real(A),B)) = aa_real_real(divide_divide_real(aa_real_real(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B)) ).

% norm_divide
tff(fact_7130_norm__divide,axiom,
    ! [A: complex,B: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = aa_real_real(divide_divide_real(aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B)) ).

% norm_divide
tff(fact_7131_norm__power,axiom,
    ! [X: real,Nb: nat] : aa_real_real(real_V7735802525324610683m_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(real_V7735802525324610683m_real,X)),Nb) ).

% norm_power
tff(fact_7132_norm__power,axiom,
    ! [X: complex,Nb: nat] : aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,X),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,X)),Nb) ).

% norm_power
tff(fact_7133_norm__uminus__minus,axiom,
    ! [X: real,Y2: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,X)),Y2)) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(plus_plus_real(X),Y2)) ).

% norm_uminus_minus
tff(fact_7134_norm__uminus__minus,axiom,
    ! [X: complex,Y2: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,X)),Y2)) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(X),Y2)) ).

% norm_uminus_minus
tff(fact_7135_nonzero__norm__divide,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(real_V7735802525324610683m_real,aa_real_real(divide_divide_real(A),B)) = aa_real_real(divide_divide_real(aa_real_real(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B)) ) ) ).

% nonzero_norm_divide
tff(fact_7136_nonzero__norm__divide,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = aa_real_real(divide_divide_real(aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B)) ) ) ).

% nonzero_norm_divide
tff(fact_7137_power__eq__imp__eq__norm,axiom,
    ! [W: real,Nb: nat,Z2: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,W),Nb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Z2),Nb) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_real_real(real_V7735802525324610683m_real,W) = aa_real_real(real_V7735802525324610683m_real,Z2) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_7138_power__eq__imp__eq__norm,axiom,
    ! [W: complex,Nb: nat,Z2: complex] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,W),Nb) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Z2),Nb) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_complex_real(real_V1022390504157884413omplex,W) = aa_complex_real(real_V1022390504157884413omplex,Z2) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_7139_norm__mult__less,axiom,
    ! [X: real,R: real,Y2: real,S: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X)),R)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Y2)),S)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(times_times_real(X),Y2))),aa_real_real(times_times_real(R),S)) ) ) ).

% norm_mult_less
tff(fact_7140_norm__mult__less,axiom,
    ! [X: complex,R: real,Y2: complex,S: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X)),R)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Y2)),S)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(X),Y2))),aa_real_real(times_times_real(R),S)) ) ) ).

% norm_mult_less
tff(fact_7141_norm__mult__ineq,axiom,
    ! [X: real,Y2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(times_times_real(X),Y2))),aa_real_real(times_times_real(aa_real_real(real_V7735802525324610683m_real,X)),aa_real_real(real_V7735802525324610683m_real,Y2))) ).

% norm_mult_ineq
tff(fact_7142_norm__mult__ineq,axiom,
    ! [X: complex,Y2: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(X),Y2))),aa_real_real(times_times_real(aa_complex_real(real_V1022390504157884413omplex,X)),aa_complex_real(real_V1022390504157884413omplex,Y2))) ).

% norm_mult_ineq
tff(fact_7143_norm__triangle__lt,axiom,
    ! [X: real,Y2: real,E: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(aa_real_real(real_V7735802525324610683m_real,X)),aa_real_real(real_V7735802525324610683m_real,Y2))),E)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(plus_plus_real(X),Y2))),E) ) ).

% norm_triangle_lt
tff(fact_7144_norm__triangle__lt,axiom,
    ! [X: complex,Y2: complex,E: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(aa_complex_real(real_V1022390504157884413omplex,X)),aa_complex_real(real_V1022390504157884413omplex,Y2))),E)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(X),Y2))),E) ) ).

% norm_triangle_lt
tff(fact_7145_norm__add__less,axiom,
    ! [X: real,R: real,Y2: real,S: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X)),R)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Y2)),S)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(plus_plus_real(X),Y2))),aa_real_real(plus_plus_real(R),S)) ) ) ).

% norm_add_less
tff(fact_7146_norm__add__less,axiom,
    ! [X: complex,R: real,Y2: complex,S: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X)),R)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Y2)),S)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(X),Y2))),aa_real_real(plus_plus_real(R),S)) ) ) ).

% norm_add_less
tff(fact_7147_norm__triangle__mono,axiom,
    ! [A: real,R: real,B: real,S: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,A)),R)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,B)),S)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(plus_plus_real(A),B))),aa_real_real(plus_plus_real(R),S)) ) ) ).

% norm_triangle_mono
tff(fact_7148_norm__triangle__mono,axiom,
    ! [A: complex,R: real,B: complex,S: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,A)),R)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,B)),S)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(A),B))),aa_real_real(plus_plus_real(R),S)) ) ) ).

% norm_triangle_mono
tff(fact_7149_norm__triangle__ineq,axiom,
    ! [X: real,Y2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(plus_plus_real(X),Y2))),aa_real_real(plus_plus_real(aa_real_real(real_V7735802525324610683m_real,X)),aa_real_real(real_V7735802525324610683m_real,Y2))) ).

% norm_triangle_ineq
tff(fact_7150_norm__triangle__ineq,axiom,
    ! [X: complex,Y2: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(X),Y2))),aa_real_real(plus_plus_real(aa_complex_real(real_V1022390504157884413omplex,X)),aa_complex_real(real_V1022390504157884413omplex,Y2))) ).

% norm_triangle_ineq
tff(fact_7151_norm__triangle__le,axiom,
    ! [X: real,Y2: real,E: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_real_real(real_V7735802525324610683m_real,X)),aa_real_real(real_V7735802525324610683m_real,Y2))),E)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(plus_plus_real(X),Y2))),E) ) ).

% norm_triangle_le
tff(fact_7152_norm__triangle__le,axiom,
    ! [X: complex,Y2: complex,E: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_complex_real(real_V1022390504157884413omplex,X)),aa_complex_real(real_V1022390504157884413omplex,Y2))),E)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(X),Y2))),E) ) ).

% norm_triangle_le
tff(fact_7153_norm__add__leD,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(plus_plus_real(A),B))),C)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,B)),aa_real_real(plus_plus_real(aa_real_real(real_V7735802525324610683m_real,A)),C)) ) ).

% norm_add_leD
tff(fact_7154_norm__add__leD,axiom,
    ! [A: complex,B: complex,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(A),B))),C)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,B)),aa_real_real(plus_plus_real(aa_complex_real(real_V1022390504157884413omplex,A)),C)) ) ).

% norm_add_leD
tff(fact_7155_norm__power__ineq,axiom,
    ! [X: real,Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(real_V7735802525324610683m_real,X)),Nb)) ).

% norm_power_ineq
tff(fact_7156_norm__power__ineq,axiom,
    ! [X: complex,Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,X),Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,X)),Nb)) ).

% norm_power_ineq
tff(fact_7157_norm__triangle__le__diff,axiom,
    ! [X: real,Y2: real,E: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_real_real(real_V7735802525324610683m_real,X)),aa_real_real(real_V7735802525324610683m_real,Y2))),E)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(X),Y2))),E) ) ).

% norm_triangle_le_diff
tff(fact_7158_norm__triangle__le__diff,axiom,
    ! [X: complex,Y2: complex,E: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_complex_real(real_V1022390504157884413omplex,X)),aa_complex_real(real_V1022390504157884413omplex,Y2))),E)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(X),Y2))),E) ) ).

% norm_triangle_le_diff
tff(fact_7159_norm__diff__triangle__le,axiom,
    ! [X: real,Y2: real,E1: real,Z2: real,E22: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(X),Y2))),E1)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(Y2),Z2))),E22)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(X),Z2))),aa_real_real(plus_plus_real(E1),E22)) ) ) ).

% norm_diff_triangle_le
tff(fact_7160_norm__diff__triangle__le,axiom,
    ! [X: complex,Y2: complex,E1: real,Z2: complex,E22: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(X),Y2))),E1)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Y2),Z2))),E22)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(X),Z2))),aa_real_real(plus_plus_real(E1),E22)) ) ) ).

% norm_diff_triangle_le
tff(fact_7161_norm__triangle__ineq4,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(A),B))),aa_real_real(plus_plus_real(aa_real_real(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B))) ).

% norm_triangle_ineq4
tff(fact_7162_norm__triangle__ineq4,axiom,
    ! [A: complex,B: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(A),B))),aa_real_real(plus_plus_real(aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B))) ).

% norm_triangle_ineq4
tff(fact_7163_norm__triangle__sub,axiom,
    ! [X: real,Y2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,X)),aa_real_real(plus_plus_real(aa_real_real(real_V7735802525324610683m_real,Y2)),aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(X),Y2)))) ).

% norm_triangle_sub
tff(fact_7164_norm__triangle__sub,axiom,
    ! [X: complex,Y2: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,X)),aa_real_real(plus_plus_real(aa_complex_real(real_V1022390504157884413omplex,Y2)),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(X),Y2)))) ).

% norm_triangle_sub
tff(fact_7165_norm__diff__ineq,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(aa_real_real(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(plus_plus_real(A),B))) ).

% norm_diff_ineq
tff(fact_7166_norm__diff__ineq,axiom,
    ! [A: complex,B: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(A),B))) ).

% norm_diff_ineq
tff(fact_7167_norm__triangle__ineq2,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(aa_real_real(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(A),B))) ).

% norm_triangle_ineq2
tff(fact_7168_norm__triangle__ineq2,axiom,
    ! [A: complex,B: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(A),B))) ).

% norm_triangle_ineq2
tff(fact_7169_power__eq__1__iff,axiom,
    ! [W: real,Nb: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,W),Nb) = one_one_real )
     => ( ( aa_real_real(real_V7735802525324610683m_real,W) = one_one_real )
        | ( Nb = zero_zero_nat ) ) ) ).

% power_eq_1_iff
tff(fact_7170_power__eq__1__iff,axiom,
    ! [W: complex,Nb: nat] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,W),Nb) = one_one_complex )
     => ( ( aa_complex_real(real_V1022390504157884413omplex,W) = one_one_real )
        | ( Nb = zero_zero_nat ) ) ) ).

% power_eq_1_iff
tff(fact_7171_norm__diff__triangle__ineq,axiom,
    ! [A: real,B: real,C: real,D: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(A),B)),aa_real_real(plus_plus_real(C),D)))),aa_real_real(plus_plus_real(aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(A),C))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(B),D)))) ).

% norm_diff_triangle_ineq
tff(fact_7172_norm__diff__triangle__ineq,axiom,
    ! [A: complex,B: complex,C: complex,D: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(A),B)),aa_complex_complex(plus_plus_complex(C),D)))),aa_real_real(plus_plus_real(aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(A),C))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(B),D)))) ).

% norm_diff_triangle_ineq
tff(fact_7173_norm__triangle__ineq3,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_real_real(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B)))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(A),B))) ).

% norm_triangle_ineq3
tff(fact_7174_norm__triangle__ineq3,axiom,
    ! [A: complex,B: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B)))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(A),B))) ).

% norm_triangle_ineq3
tff(fact_7175_square__norm__one,axiom,
    ! [X: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_real )
     => ( aa_real_real(real_V7735802525324610683m_real,X) = one_one_real ) ) ).

% square_norm_one
tff(fact_7176_square__norm__one,axiom,
    ! [X: complex] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_complex )
     => ( aa_complex_real(real_V1022390504157884413omplex,X) = one_one_real ) ) ).

% square_norm_one
tff(fact_7177_norm__power__diff,axiom,
    ! [Z2: real,W: real,Ma: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z2)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,W)),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Z2),Ma)),aa_nat_real(aa_real_fun_nat_real(power_power_real,W),Ma)))),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Ma)),aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(Z2),W)))) ) ) ).

% norm_power_diff
tff(fact_7178_norm__power__diff,axiom,
    ! [Z2: complex,W: complex,Ma: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z2)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,W)),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Z2),Ma)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,W),Ma)))),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Ma)),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Z2),W)))) ) ) ).

% norm_power_diff
tff(fact_7179_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
       => ( ( aa_real_int(archim7802044766580827645g_real,aa_real_real(log(aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int) )
        <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)),K2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
tff(fact_7180_summable__arctan__series,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real)
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_ed,X)) ) ).

% summable_arctan_series
tff(fact_7181_cos__pi__eq__zero,axiom,
    ! [Ma: nat] : aa_real_real(cos_real,aa_real_real(divide_divide_real(aa_real_real(times_times_real(pi),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma))))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = zero_zero_real ).

% cos_pi_eq_zero
tff(fact_7182_sincos__total__2pi,axiom,
    ! [X: real,Y2: real] :
      ( ( aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
     => ~ ! [T3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T3)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,T3),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))
             => ( ( X = aa_real_real(cos_real,T3) )
               => ( Y2 != aa_real_real(sin_real,T3) ) ) ) ) ) ).

% sincos_total_2pi
tff(fact_7183_sin__tan,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,X)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => ( aa_real_real(sin_real,X) = aa_real_real(divide_divide_real(aa_real_real(tan_real,X)),aa_real_real(sqrt,aa_real_real(plus_plus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(tan_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% sin_tan
tff(fact_7184_of__int__ceiling__cancel,axiom,
    ! [X: rat] :
      ( ( aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,X)) = X )
    <=> ? [N4: int] : X = aa_int_rat(ring_1_of_int_rat,N4) ) ).

% of_int_ceiling_cancel
tff(fact_7185_of__int__ceiling__cancel,axiom,
    ! [X: real] :
      ( ( aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,X)) = X )
    <=> ? [N4: int] : X = aa_int_real(ring_1_of_int_real,N4) ) ).

% of_int_ceiling_cancel
tff(fact_7186_summable__zero,axiom,
    summable_complex(aTP_Lamm_eh) ).

% summable_zero
tff(fact_7187_summable__zero,axiom,
    summable_real(aTP_Lamm_ei) ).

% summable_zero
tff(fact_7188_summable__zero,axiom,
    summable_nat(aTP_Lamm_av) ).

% summable_zero
tff(fact_7189_summable__zero,axiom,
    summable_int(aTP_Lamm_ej) ).

% summable_zero
tff(fact_7190_summable__single,axiom,
    ! [I2: nat,F: fun_nat_complex] : summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ek(I2),F)) ).

% summable_single
tff(fact_7191_summable__single,axiom,
    ! [I2: nat,F: fun_nat_real] : summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_el(I2),F)) ).

% summable_single
tff(fact_7192_summable__single,axiom,
    ! [I2: nat,F: fun_nat_nat] : summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_em(I2),F)) ).

% summable_single
tff(fact_7193_summable__single,axiom,
    ! [I2: nat,F: fun_nat_int] : summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_en(I2),F)) ).

% summable_single
tff(fact_7194_summable__iff__shift,axiom,
    ! [F: fun_nat_real,K2: nat] :
      ( summable_real(aa_nat_fun_nat_real(aTP_Lamm_eo(F),K2))
    <=> summable_real(F) ) ).

% summable_iff_shift
tff(fact_7195_summable__iff__shift,axiom,
    ! [F: fun_nat_complex,K2: nat] :
      ( summable_complex(aa_nat6454746962644518025omplex(aTP_Lamm_ep(F),K2))
    <=> summable_complex(F) ) ).

% summable_iff_shift
tff(fact_7196_cos__zero,axiom,
    aa_complex_complex(cos_complex,zero_zero_complex) = one_one_complex ).

% cos_zero
tff(fact_7197_cos__zero,axiom,
    aa_real_real(cos_real,zero_zero_real) = one_one_real ).

% cos_zero
tff(fact_7198_ceiling__numeral,axiom,
    ! [V: num] : aa_real_int(archim7802044766580827645g_real,aa_num_real(numeral_numeral_real,V)) = aa_num_int(numeral_numeral_int,V) ).

% ceiling_numeral
tff(fact_7199_ceiling__numeral,axiom,
    ! [V: num] : aa_rat_int(archim2889992004027027881ng_rat,aa_num_rat(numeral_numeral_rat,V)) = aa_num_int(numeral_numeral_int,V) ).

% ceiling_numeral
tff(fact_7200_ceiling__one,axiom,
    aa_rat_int(archim2889992004027027881ng_rat,one_one_rat) = one_one_int ).

% ceiling_one
tff(fact_7201_ceiling__one,axiom,
    aa_real_int(archim7802044766580827645g_real,one_one_real) = one_one_int ).

% ceiling_one
tff(fact_7202_summable__cmult__iff,axiom,
    ! [C: complex,F: fun_nat_complex] :
      ( summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_eq(C),F))
    <=> ( ( C = zero_zero_complex )
        | summable_complex(F) ) ) ).

% summable_cmult_iff
tff(fact_7203_summable__cmult__iff,axiom,
    ! [C: real,F: fun_nat_real] :
      ( summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_er(C),F))
    <=> ( ( C = zero_zero_real )
        | summable_real(F) ) ) ).

% summable_cmult_iff
tff(fact_7204_summable__divide__iff,axiom,
    ! [F: fun_nat_complex,C: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_es(F),C))
    <=> ( ( C = zero_zero_complex )
        | summable_complex(F) ) ) ).

% summable_divide_iff
tff(fact_7205_summable__divide__iff,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_et(F),C))
    <=> ( ( C = zero_zero_real )
        | summable_real(F) ) ) ).

% summable_divide_iff
tff(fact_7206_ceiling__add__of__int,axiom,
    ! [X: rat,Z2: int] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(plus_plus_rat(X),aa_int_rat(ring_1_of_int_rat,Z2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim2889992004027027881ng_rat,X)),Z2) ).

% ceiling_add_of_int
tff(fact_7207_ceiling__add__of__int,axiom,
    ! [X: real,Z2: int] : aa_real_int(archim7802044766580827645g_real,aa_real_real(plus_plus_real(X),aa_int_real(ring_1_of_int_real,Z2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,X)),Z2) ).

% ceiling_add_of_int
tff(fact_7208_cos__pi,axiom,
    aa_real_real(cos_real,pi) = aa_real_real(uminus_uminus_real,one_one_real) ).

% cos_pi
tff(fact_7209_sin__cos__squared__add3,axiom,
    ! [X: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,X)),aa_complex_complex(cos_complex,X))),aa_complex_complex(times_times_complex(aa_complex_complex(sin_complex,X)),aa_complex_complex(sin_complex,X))) = one_one_complex ).

% sin_cos_squared_add3
tff(fact_7210_sin__cos__squared__add3,axiom,
    ! [X: real] : aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(cos_real,X)),aa_real_real(cos_real,X))),aa_real_real(times_times_real(aa_real_real(sin_real,X)),aa_real_real(sin_real,X))) = one_one_real ).

% sin_cos_squared_add3
tff(fact_7211_ceiling__le__zero,axiom,
    ! [X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X)),zero_zero_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),zero_zero_real) ) ).

% ceiling_le_zero
tff(fact_7212_ceiling__le__zero,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,X)),zero_zero_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),zero_zero_rat) ) ).

% ceiling_le_zero
tff(fact_7213_zero__less__ceiling,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_rat_int(archim2889992004027027881ng_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),X) ) ).

% zero_less_ceiling
tff(fact_7214_zero__less__ceiling,axiom,
    ! [X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_real_int(archim7802044766580827645g_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X) ) ).

% zero_less_ceiling
tff(fact_7215_ceiling__le__numeral,axiom,
    ! [X: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X)),aa_num_int(numeral_numeral_int,V))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_num_real(numeral_numeral_real,V)) ) ).

% ceiling_le_numeral
tff(fact_7216_ceiling__le__numeral,axiom,
    ! [X: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_num_int(numeral_numeral_int,V))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_num_rat(numeral_numeral_rat,V)) ) ).

% ceiling_le_numeral
tff(fact_7217_ceiling__less__one,axiom,
    ! [X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,X)),one_one_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),zero_zero_real) ) ).

% ceiling_less_one
tff(fact_7218_ceiling__less__one,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X)),one_one_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),zero_zero_rat) ) ).

% ceiling_less_one
tff(fact_7219_one__le__ceiling,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_rat_int(archim2889992004027027881ng_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),X) ) ).

% one_le_ceiling
tff(fact_7220_one__le__ceiling,axiom,
    ! [X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_real_int(archim7802044766580827645g_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X) ) ).

% one_le_ceiling
tff(fact_7221_numeral__less__ceiling,axiom,
    ! [V: num,X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,V)),aa_real_int(archim7802044766580827645g_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,V)),X) ) ).

% numeral_less_ceiling
tff(fact_7222_numeral__less__ceiling,axiom,
    ! [V: num,X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,V)),aa_rat_int(archim2889992004027027881ng_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,V)),X) ) ).

% numeral_less_ceiling
tff(fact_7223_ceiling__le__one,axiom,
    ! [X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X)),one_one_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),one_one_real) ) ).

% ceiling_le_one
tff(fact_7224_ceiling__le__one,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,X)),one_one_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),one_one_rat) ) ).

% ceiling_le_one
tff(fact_7225_one__less__ceiling,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_rat_int(archim2889992004027027881ng_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),X) ) ).

% one_less_ceiling
tff(fact_7226_one__less__ceiling,axiom,
    ! [X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_real_int(archim7802044766580827645g_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X) ) ).

% one_less_ceiling
tff(fact_7227_ceiling__add__numeral,axiom,
    ! [X: real,V: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(plus_plus_real(X),aa_num_real(numeral_numeral_real,V))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,X)),aa_num_int(numeral_numeral_int,V)) ).

% ceiling_add_numeral
tff(fact_7228_ceiling__add__numeral,axiom,
    ! [X: rat,V: num] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(plus_plus_rat(X),aa_num_rat(numeral_numeral_rat,V))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_num_int(numeral_numeral_int,V)) ).

% ceiling_add_numeral
tff(fact_7229_ceiling__neg__numeral,axiom,
    ! [V: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)) ).

% ceiling_neg_numeral
tff(fact_7230_ceiling__neg__numeral,axiom,
    ! [V: num] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)) ).

% ceiling_neg_numeral
tff(fact_7231_ceiling__add__one,axiom,
    ! [X: rat] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(plus_plus_rat(X),one_one_rat)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim2889992004027027881ng_rat,X)),one_one_int) ).

% ceiling_add_one
tff(fact_7232_ceiling__add__one,axiom,
    ! [X: real] : aa_real_int(archim7802044766580827645g_real,aa_real_real(plus_plus_real(X),one_one_real)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,X)),one_one_int) ).

% ceiling_add_one
tff(fact_7233_ceiling__diff__numeral,axiom,
    ! [X: real,V: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(minus_minus_real(X),aa_num_real(numeral_numeral_real,V))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(archim7802044766580827645g_real,X)),aa_num_int(numeral_numeral_int,V)) ).

% ceiling_diff_numeral
tff(fact_7234_ceiling__diff__numeral,axiom,
    ! [X: rat,V: num] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(minus_minus_rat(X),aa_num_rat(numeral_numeral_rat,V))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_num_int(numeral_numeral_int,V)) ).

% ceiling_diff_numeral
tff(fact_7235_ceiling__diff__one,axiom,
    ! [X: rat] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(minus_minus_rat(X),one_one_rat)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_rat_int(archim2889992004027027881ng_rat,X)),one_one_int) ).

% ceiling_diff_one
tff(fact_7236_ceiling__diff__one,axiom,
    ! [X: real] : aa_real_int(archim7802044766580827645g_real,aa_real_real(minus_minus_real(X),one_one_real)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(archim7802044766580827645g_real,X)),one_one_int) ).

% ceiling_diff_one
tff(fact_7237_ceiling__numeral__power,axiom,
    ! [X: num,Nb: nat] : aa_real_int(archim7802044766580827645g_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,X)),Nb)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X)),Nb) ).

% ceiling_numeral_power
tff(fact_7238_ceiling__numeral__power,axiom,
    ! [X: num,Nb: nat] : aa_rat_int(archim2889992004027027881ng_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X)),Nb)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X)),Nb) ).

% ceiling_numeral_power
tff(fact_7239_tan__npi,axiom,
    ! [Nb: nat] : aa_real_real(tan_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),pi)) = zero_zero_real ).

% tan_npi
tff(fact_7240_tan__periodic__n,axiom,
    ! [X: real,Nb: num] : aa_real_real(tan_real,aa_real_real(plus_plus_real(X),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,Nb)),pi))) = aa_real_real(tan_real,X) ).

% tan_periodic_n
tff(fact_7241_tan__periodic__nat,axiom,
    ! [X: real,Nb: nat] : aa_real_real(tan_real,aa_real_real(plus_plus_real(X),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),pi))) = aa_real_real(tan_real,X) ).

% tan_periodic_nat
tff(fact_7242_tan__periodic__int,axiom,
    ! [X: real,I2: int] : aa_real_real(tan_real,aa_real_real(plus_plus_real(X),aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,I2)),pi))) = aa_real_real(tan_real,X) ).

% tan_periodic_int
tff(fact_7243_summable__geometric__iff,axiom,
    ! [C: real] :
      ( summable_real(aa_real_fun_nat_real(power_power_real,C))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,C)),one_one_real) ) ).

% summable_geometric_iff
tff(fact_7244_summable__geometric__iff,axiom,
    ! [C: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(power_power_complex,C))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,C)),one_one_real) ) ).

% summable_geometric_iff
tff(fact_7245_ceiling__less__zero,axiom,
    ! [X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,X)),zero_zero_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(uminus_uminus_real,one_one_real)) ) ).

% ceiling_less_zero
tff(fact_7246_ceiling__less__zero,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X)),zero_zero_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ) ).

% ceiling_less_zero
tff(fact_7247_zero__le__ceiling,axiom,
    ! [X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(archim7802044766580827645g_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X) ) ).

% zero_le_ceiling
tff(fact_7248_zero__le__ceiling,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_rat_int(archim2889992004027027881ng_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),X) ) ).

% zero_le_ceiling
tff(fact_7249_ceiling__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(divide_divide_real(aa_num_real(numeral_numeral_real,A)),aa_num_real(numeral_numeral_real,B))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,A))),aa_num_int(numeral_numeral_int,B))) ).

% ceiling_divide_eq_div_numeral
tff(fact_7250_ceiling__less__numeral,axiom,
    ! [X: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,X)),aa_num_int(numeral_numeral_int,V))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(minus_minus_real(aa_num_real(numeral_numeral_real,V)),one_one_real)) ) ).

% ceiling_less_numeral
tff(fact_7251_ceiling__less__numeral,axiom,
    ! [X: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_num_int(numeral_numeral_int,V))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_rat_rat(minus_minus_rat(aa_num_rat(numeral_numeral_rat,V)),one_one_rat)) ) ).

% ceiling_less_numeral
tff(fact_7252_numeral__le__ceiling,axiom,
    ! [V: num,X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,V)),aa_real_int(archim7802044766580827645g_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(aa_num_real(numeral_numeral_real,V)),one_one_real)),X) ) ).

% numeral_le_ceiling
tff(fact_7253_numeral__le__ceiling,axiom,
    ! [V: num,X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,V)),aa_rat_int(archim2889992004027027881ng_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_num_rat(numeral_numeral_rat,V)),one_one_rat)),X) ) ).

% numeral_le_ceiling
tff(fact_7254_ceiling__le__neg__numeral,axiom,
    ! [X: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))) ) ).

% ceiling_le_neg_numeral
tff(fact_7255_ceiling__le__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))) ) ).

% ceiling_le_neg_numeral
tff(fact_7256_neg__numeral__less__ceiling,axiom,
    ! [V: num,X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_real_int(archim7802044766580827645g_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),X) ) ).

% neg_numeral_less_ceiling
tff(fact_7257_neg__numeral__less__ceiling,axiom,
    ! [V: num,X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_rat_int(archim2889992004027027881ng_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),X) ) ).

% neg_numeral_less_ceiling
tff(fact_7258_cos__pi__half,axiom,
    aa_real_real(cos_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = zero_zero_real ).

% cos_pi_half
tff(fact_7259_cos__two__pi,axiom,
    aa_real_real(cos_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)) = one_one_real ).

% cos_two_pi
tff(fact_7260_cos__periodic,axiom,
    ! [X: real] : aa_real_real(cos_real,aa_real_real(plus_plus_real(X),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) = aa_real_real(cos_real,X) ).

% cos_periodic
tff(fact_7261_cos__2pi__minus,axiom,
    ! [X: real] : aa_real_real(cos_real,aa_real_real(minus_minus_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),X)) = aa_real_real(cos_real,X) ).

% cos_2pi_minus
tff(fact_7262_tan__periodic,axiom,
    ! [X: real] : aa_real_real(tan_real,aa_real_real(plus_plus_real(X),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) = aa_real_real(tan_real,X) ).

% tan_periodic
tff(fact_7263_cos__npi2,axiom,
    ! [Nb: nat] : aa_real_real(cos_real,aa_real_real(times_times_real(pi),aa_nat_real(semiri5074537144036343181t_real,Nb))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb) ).

% cos_npi2
tff(fact_7264_cos__npi,axiom,
    ! [Nb: nat] : aa_real_real(cos_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),pi)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb) ).

% cos_npi
tff(fact_7265_ceiling__minus__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(aa_num_real(numeral_numeral_real,A)),aa_num_real(numeral_numeral_real,B)))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,A)),aa_num_int(numeral_numeral_int,B))) ).

% ceiling_minus_divide_eq_div_numeral
tff(fact_7266_sin__cos__squared__add,axiom,
    ! [X: real] : aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real ).

% sin_cos_squared_add
tff(fact_7267_sin__cos__squared__add,axiom,
    ! [X: complex] : aa_complex_complex(plus_plus_complex(aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(sin_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(cos_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_complex ).

% sin_cos_squared_add
tff(fact_7268_sin__cos__squared__add2,axiom,
    ! [X: real] : aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real ).

% sin_cos_squared_add2
tff(fact_7269_sin__cos__squared__add2,axiom,
    ! [X: complex] : aa_complex_complex(plus_plus_complex(aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(cos_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(sin_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_complex ).

% sin_cos_squared_add2
tff(fact_7270_cos__2npi,axiom,
    ! [Nb: nat] : aa_real_real(cos_real,aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)) = one_one_real ).

% cos_2npi
tff(fact_7271_cos__int__2pin,axiom,
    ! [Nb: int] : aa_real_real(cos_real,aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),aa_int_real(ring_1_of_int_real,Nb))) = one_one_real ).

% cos_int_2pin
tff(fact_7272_ceiling__less__neg__numeral,axiom,
    ! [X: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),one_one_real)) ) ).

% ceiling_less_neg_numeral
tff(fact_7273_ceiling__less__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),one_one_rat)) ) ).

% ceiling_less_neg_numeral
tff(fact_7274_neg__numeral__le__ceiling,axiom,
    ! [V: num,X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_real_int(archim7802044766580827645g_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),one_one_real)),X) ) ).

% neg_numeral_le_ceiling
tff(fact_7275_neg__numeral__le__ceiling,axiom,
    ! [V: num,X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_rat_int(archim2889992004027027881ng_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),one_one_rat)),X) ) ).

% neg_numeral_le_ceiling
tff(fact_7276_cos__3over2__pi,axiom,
    aa_real_real(cos_real,aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),pi)) = zero_zero_real ).

% cos_3over2_pi
tff(fact_7277_cos__npi__int,axiom,
    ! [Nb: int] :
      aa_real_real(cos_real,aa_real_real(times_times_real(pi),aa_int_real(ring_1_of_int_real,Nb))) = $ite(aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb),one_one_real,aa_real_real(uminus_uminus_real,one_one_real)) ).

% cos_npi_int
tff(fact_7278_summable__norm__cancel,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(aTP_Lamm_eu(F))
     => summable_real(F) ) ).

% summable_norm_cancel
tff(fact_7279_summable__norm__cancel,axiom,
    ! [F: fun_nat_complex] :
      ( summable_real(aTP_Lamm_ev(F))
     => summable_complex(F) ) ).

% summable_norm_cancel
tff(fact_7280_tan__def,axiom,
    ! [X2: complex] : aa_complex_complex(tan_complex,X2) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(sin_complex,X2)),aa_complex_complex(cos_complex,X2)) ).

% tan_def
tff(fact_7281_tan__def,axiom,
    ! [X2: real] : aa_real_real(tan_real,X2) = aa_real_real(divide_divide_real(aa_real_real(sin_real,X2)),aa_real_real(cos_real,X2)) ).

% tan_def
tff(fact_7282_summable__const__iff,axiom,
    ! [C: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_ew,C))
    <=> ( C = zero_zero_complex ) ) ).

% summable_const_iff
tff(fact_7283_summable__const__iff,axiom,
    ! [C: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_ex,C))
    <=> ( C = zero_zero_real ) ) ).

% summable_const_iff
tff(fact_7284_summable__comparison__test_H,axiom,
    ! [G: fun_nat_real,N2: nat,F: fun_nat_real] :
      ( summable_real(G)
     => ( ! [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,N))),aa_nat_real(G,N)) )
       => summable_real(F) ) ) ).

% summable_comparison_test'
tff(fact_7285_summable__comparison__test_H,axiom,
    ! [G: fun_nat_real,N2: nat,F: fun_nat_complex] :
      ( summable_real(G)
     => ( ! [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,N))),aa_nat_real(G,N)) )
       => summable_complex(F) ) ) ).

% summable_comparison_test'
tff(fact_7286_summable__comparison__test,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ? [N5: nat] :
        ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N5),N)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,N))),aa_nat_real(G,N)) )
     => ( summable_real(G)
       => summable_real(F) ) ) ).

% summable_comparison_test
tff(fact_7287_summable__comparison__test,axiom,
    ! [F: fun_nat_complex,G: fun_nat_real] :
      ( ? [N5: nat] :
        ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N5),N)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,N))),aa_nat_real(G,N)) )
     => ( summable_real(G)
       => summable_complex(F) ) ) ).

% summable_comparison_test
tff(fact_7288_summable__mult,axiom,
    ! [F: fun_nat_complex,C: complex] :
      ( summable_complex(F)
     => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_ey(F),C)) ) ).

% summable_mult
tff(fact_7289_summable__mult,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(F)
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_ez(F),C)) ) ).

% summable_mult
tff(fact_7290_summable__mult2,axiom,
    ! [F: fun_nat_complex,C: complex] :
      ( summable_complex(F)
     => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_fa(F),C)) ) ).

% summable_mult2
tff(fact_7291_summable__mult2,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(F)
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_fb(F),C)) ) ).

% summable_mult2
tff(fact_7292_summable__add,axiom,
    ! [F: fun_nat_complex,G: fun_nat_complex] :
      ( summable_complex(F)
     => ( summable_complex(G)
       => summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_fc(F),G)) ) ) ).

% summable_add
tff(fact_7293_summable__add,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( summable_real(F)
     => ( summable_real(G)
       => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_fd(F),G)) ) ) ).

% summable_add
tff(fact_7294_summable__add,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat] :
      ( summable_nat(F)
     => ( summable_nat(G)
       => summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_fe(F),G)) ) ) ).

% summable_add
tff(fact_7295_summable__add,axiom,
    ! [F: fun_nat_int,G: fun_nat_int] :
      ( summable_int(F)
     => ( summable_int(G)
       => summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_ff(F),G)) ) ) ).

% summable_add
tff(fact_7296_summable__diff,axiom,
    ! [F: fun_nat_complex,G: fun_nat_complex] :
      ( summable_complex(F)
     => ( summable_complex(G)
       => summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_fg(F),G)) ) ) ).

% summable_diff
tff(fact_7297_summable__diff,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( summable_real(F)
     => ( summable_real(G)
       => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_fh(F),G)) ) ) ).

% summable_diff
tff(fact_7298_summable__divide,axiom,
    ! [F: fun_nat_complex,C: complex] :
      ( summable_complex(F)
     => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_es(F),C)) ) ).

% summable_divide
tff(fact_7299_summable__divide,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(F)
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_et(F),C)) ) ).

% summable_divide
tff(fact_7300_summable__minus,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => summable_real(aTP_Lamm_fi(F)) ) ).

% summable_minus
tff(fact_7301_summable__minus,axiom,
    ! [F: fun_nat_complex] :
      ( summable_complex(F)
     => summable_complex(aTP_Lamm_fj(F)) ) ).

% summable_minus
tff(fact_7302_summable__minus__iff,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(aTP_Lamm_fi(F))
    <=> summable_real(F) ) ).

% summable_minus_iff
tff(fact_7303_summable__minus__iff,axiom,
    ! [F: fun_nat_complex] :
      ( summable_complex(aTP_Lamm_fj(F))
    <=> summable_complex(F) ) ).

% summable_minus_iff
tff(fact_7304_summable__Suc__iff,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(aTP_Lamm_fk(F))
    <=> summable_real(F) ) ).

% summable_Suc_iff
tff(fact_7305_summable__Suc__iff,axiom,
    ! [F: fun_nat_complex] :
      ( summable_complex(aTP_Lamm_fl(F))
    <=> summable_complex(F) ) ).

% summable_Suc_iff
tff(fact_7306_summable__ignore__initial__segment,axiom,
    ! [F: fun_nat_real,K2: nat] :
      ( summable_real(F)
     => summable_real(aa_nat_fun_nat_real(aTP_Lamm_eo(F),K2)) ) ).

% summable_ignore_initial_segment
tff(fact_7307_summable__ignore__initial__segment,axiom,
    ! [F: fun_nat_complex,K2: nat] :
      ( summable_complex(F)
     => summable_complex(aa_nat6454746962644518025omplex(aTP_Lamm_ep(F),K2)) ) ).

% summable_ignore_initial_segment
tff(fact_7308_summable__rabs__cancel,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(aTP_Lamm_fm(F))
     => summable_real(F) ) ).

% summable_rabs_cancel
tff(fact_7309_powser__insidea,axiom,
    ! [F: fun_nat_real,X: real,Z2: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_fn(F),X))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Z2)),aa_real_real(real_V7735802525324610683m_real,X))
       => summable_real(aa_real_fun_nat_real(aTP_Lamm_fo(F),Z2)) ) ) ).

% powser_insidea
tff(fact_7310_powser__insidea,axiom,
    ! [F: fun_nat_complex,X: complex,Z2: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_fp(F),X))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Z2)),aa_complex_real(real_V1022390504157884413omplex,X))
       => summable_real(aa_com20334259897932645t_real(aTP_Lamm_fq(F),Z2)) ) ) ).

% powser_insidea
tff(fact_7311_suminf__le,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N)),aa_nat_real(G,N))
     => ( summable_real(F)
       => ( summable_real(G)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,suminf_real(F)),suminf_real(G)) ) ) ) ).

% suminf_le
tff(fact_7312_suminf__le,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,N)),aa_nat_nat(G,N))
     => ( summable_nat(F)
       => ( summable_nat(G)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,suminf_nat(F)),suminf_nat(G)) ) ) ) ).

% suminf_le
tff(fact_7313_suminf__le,axiom,
    ! [F: fun_nat_int,G: fun_nat_int] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,N)),aa_nat_int(G,N))
     => ( summable_int(F)
       => ( summable_int(G)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,suminf_int(F)),suminf_int(G)) ) ) ) ).

% suminf_le
tff(fact_7314_ceiling__mono,axiom,
    ! [Y2: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),X)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Y2)),aa_real_int(archim7802044766580827645g_real,X)) ) ).

% ceiling_mono
tff(fact_7315_ceiling__mono,axiom,
    ! [Y2: rat,X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y2),X)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,Y2)),aa_rat_int(archim2889992004027027881ng_rat,X)) ) ).

% ceiling_mono
tff(fact_7316_le__of__int__ceiling,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,X))) ).

% le_of_int_ceiling
tff(fact_7317_le__of__int__ceiling,axiom,
    ! [X: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,X))) ).

% le_of_int_ceiling
tff(fact_7318_ceiling__less__cancel,axiom,
    ! [X: rat,Y2: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_rat_int(archim2889992004027027881ng_rat,Y2))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y2) ) ).

% ceiling_less_cancel
tff(fact_7319_ceiling__less__cancel,axiom,
    ! [X: real,Y2: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,X)),aa_real_int(archim7802044766580827645g_real,Y2))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2) ) ).

% ceiling_less_cancel
tff(fact_7320_cos__le__one,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,X)),one_one_real) ).

% cos_le_one
tff(fact_7321_summable__mult__D,axiom,
    ! [C: complex,F: fun_nat_complex] :
      ( summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_eq(C),F))
     => ( ( C != zero_zero_complex )
       => summable_complex(F) ) ) ).

% summable_mult_D
tff(fact_7322_summable__mult__D,axiom,
    ! [C: real,F: fun_nat_real] :
      ( summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_er(C),F))
     => ( ( C != zero_zero_real )
       => summable_real(F) ) ) ).

% summable_mult_D
tff(fact_7323_polar__Ex,axiom,
    ! [X: real,Y2: real] :
    ? [R3: real,A3: real] :
      ( ( X = aa_real_real(times_times_real(R3),aa_real_real(cos_real,A3)) )
      & ( Y2 = aa_real_real(times_times_real(R3),aa_real_real(sin_real,A3)) ) ) ).

% polar_Ex
tff(fact_7324_summable__zero__power,axiom,
    summable_real(aa_real_fun_nat_real(power_power_real,zero_zero_real)) ).

% summable_zero_power
tff(fact_7325_summable__zero__power,axiom,
    summable_int(aa_int_fun_nat_int(power_power_int,zero_zero_int)) ).

% summable_zero_power
tff(fact_7326_summable__zero__power,axiom,
    summable_complex(aa_com8181001469867462503omplex(power_power_complex,zero_zero_complex)) ).

% summable_zero_power
tff(fact_7327_suminf__mult2,axiom,
    ! [F: fun_nat_complex,C: complex] :
      ( summable_complex(F)
     => ( aa_complex_complex(times_times_complex(suminf_complex(F)),C) = suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_fa(F),C)) ) ) ).

% suminf_mult2
tff(fact_7328_suminf__mult2,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(F)
     => ( aa_real_real(times_times_real(suminf_real(F)),C) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_fb(F),C)) ) ) ).

% suminf_mult2
tff(fact_7329_suminf__mult,axiom,
    ! [F: fun_nat_complex,C: complex] :
      ( summable_complex(F)
     => ( suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_ey(F),C)) = aa_complex_complex(times_times_complex(C),suminf_complex(F)) ) ) ).

% suminf_mult
tff(fact_7330_suminf__mult,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(F)
     => ( suminf_real(aa_real_fun_nat_real(aTP_Lamm_ez(F),C)) = aa_real_real(times_times_real(C),suminf_real(F)) ) ) ).

% suminf_mult
tff(fact_7331_suminf__add,axiom,
    ! [F: fun_nat_complex,G: fun_nat_complex] :
      ( summable_complex(F)
     => ( summable_complex(G)
       => ( aa_complex_complex(plus_plus_complex(suminf_complex(F)),suminf_complex(G)) = suminf_complex(aa_fun2955221450537884980omplex(aTP_Lamm_fc(F),G)) ) ) ) ).

% suminf_add
tff(fact_7332_suminf__add,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( summable_real(F)
     => ( summable_real(G)
       => ( aa_real_real(plus_plus_real(suminf_real(F)),suminf_real(G)) = suminf_real(aa_fun1839206346889086768t_real(aTP_Lamm_fd(F),G)) ) ) ) ).

% suminf_add
tff(fact_7333_suminf__add,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat] :
      ( summable_nat(F)
     => ( summable_nat(G)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,suminf_nat(F)),suminf_nat(G)) = suminf_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_fe(F),G)) ) ) ) ).

% suminf_add
tff(fact_7334_suminf__add,axiom,
    ! [F: fun_nat_int,G: fun_nat_int] :
      ( summable_int(F)
     => ( summable_int(G)
       => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,suminf_int(F)),suminf_int(G)) = suminf_int(aa_fun5565043122120104496at_int(aTP_Lamm_ff(F),G)) ) ) ) ).

% suminf_add
tff(fact_7335_suminf__diff,axiom,
    ! [F: fun_nat_complex,G: fun_nat_complex] :
      ( summable_complex(F)
     => ( summable_complex(G)
       => ( aa_complex_complex(minus_minus_complex(suminf_complex(F)),suminf_complex(G)) = suminf_complex(aa_fun2955221450537884980omplex(aTP_Lamm_fg(F),G)) ) ) ) ).

% suminf_diff
tff(fact_7336_suminf__diff,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( summable_real(F)
     => ( summable_real(G)
       => ( aa_real_real(minus_minus_real(suminf_real(F)),suminf_real(G)) = suminf_real(aa_fun1839206346889086768t_real(aTP_Lamm_fh(F),G)) ) ) ) ).

% suminf_diff
tff(fact_7337_suminf__divide,axiom,
    ! [F: fun_nat_complex,C: complex] :
      ( summable_complex(F)
     => ( suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_es(F),C)) = aa_complex_complex(divide1717551699836669952omplex(suminf_complex(F)),C) ) ) ).

% suminf_divide
tff(fact_7338_suminf__divide,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(F)
     => ( suminf_real(aa_real_fun_nat_real(aTP_Lamm_et(F),C)) = aa_real_real(divide_divide_real(suminf_real(F)),C) ) ) ).

% suminf_divide
tff(fact_7339_suminf__minus,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( suminf_real(aTP_Lamm_fi(F)) = aa_real_real(uminus_uminus_real,suminf_real(F)) ) ) ).

% suminf_minus
tff(fact_7340_suminf__minus,axiom,
    ! [F: fun_nat_complex] :
      ( summable_complex(F)
     => ( suminf_complex(aTP_Lamm_fj(F)) = aa_complex_complex(uminus1482373934393186551omplex,suminf_complex(F)) ) ) ).

% suminf_minus
tff(fact_7341_ceiling__ge__round,axiom,
    ! [X: real] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim8280529875227126926d_real,X)),aa_real_int(archim7802044766580827645g_real,X)) ).

% ceiling_ge_round
tff(fact_7342_add__tan__eq,axiom,
    ! [X: complex,Y2: complex] :
      ( ( aa_complex_complex(cos_complex,X) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,Y2) != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(tan_complex,X)),aa_complex_complex(tan_complex,Y2)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(sin_complex,aa_complex_complex(plus_plus_complex(X),Y2))),aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,X)),aa_complex_complex(cos_complex,Y2))) ) ) ) ).

% add_tan_eq
tff(fact_7343_add__tan__eq,axiom,
    ! [X: real,Y2: real] :
      ( ( aa_real_real(cos_real,X) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Y2) != zero_zero_real )
       => ( aa_real_real(plus_plus_real(aa_real_real(tan_real,X)),aa_real_real(tan_real,Y2)) = aa_real_real(divide_divide_real(aa_real_real(sin_real,aa_real_real(plus_plus_real(X),Y2))),aa_real_real(times_times_real(aa_real_real(cos_real,X)),aa_real_real(cos_real,Y2))) ) ) ) ).

% add_tan_eq
tff(fact_7344_suminf__nonneg,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),suminf_real(F)) ) ) ).

% suminf_nonneg
tff(fact_7345_suminf__nonneg,axiom,
    ! [F: fun_nat_nat] :
      ( summable_nat(F)
     => ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),suminf_nat(F)) ) ) ).

% suminf_nonneg
tff(fact_7346_suminf__nonneg,axiom,
    ! [F: fun_nat_int] :
      ( summable_int(F)
     => ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N))
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),suminf_int(F)) ) ) ).

% suminf_nonneg
tff(fact_7347_suminf__eq__zero__iff,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N))
       => ( ( suminf_real(F) = zero_zero_real )
        <=> ! [N4: nat] : aa_nat_real(F,N4) = zero_zero_real ) ) ) ).

% suminf_eq_zero_iff
tff(fact_7348_suminf__eq__zero__iff,axiom,
    ! [F: fun_nat_nat] :
      ( summable_nat(F)
     => ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N))
       => ( ( suminf_nat(F) = zero_zero_nat )
        <=> ! [N4: nat] : aa_nat_nat(F,N4) = zero_zero_nat ) ) ) ).

% suminf_eq_zero_iff
tff(fact_7349_suminf__eq__zero__iff,axiom,
    ! [F: fun_nat_int] :
      ( summable_int(F)
     => ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N))
       => ( ( suminf_int(F) = zero_zero_int )
        <=> ! [N4: nat] : aa_nat_int(F,N4) = zero_zero_int ) ) ) ).

% suminf_eq_zero_iff
tff(fact_7350_suminf__pos,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(F,N))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),suminf_real(F)) ) ) ).

% suminf_pos
tff(fact_7351_suminf__pos,axiom,
    ! [F: fun_nat_nat] :
      ( summable_nat(F)
     => ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,N))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),suminf_nat(F)) ) ) ).

% suminf_pos
tff(fact_7352_suminf__pos,axiom,
    ! [F: fun_nat_int] :
      ( summable_int(F)
     => ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,N))
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),suminf_int(F)) ) ) ).

% suminf_pos
tff(fact_7353_cos__one__sin__zero,axiom,
    ! [X: complex] :
      ( ( aa_complex_complex(cos_complex,X) = one_one_complex )
     => ( aa_complex_complex(sin_complex,X) = zero_zero_complex ) ) ).

% cos_one_sin_zero
tff(fact_7354_cos__one__sin__zero,axiom,
    ! [X: real] :
      ( ( aa_real_real(cos_real,X) = one_one_real )
     => ( aa_real_real(sin_real,X) = zero_zero_real ) ) ).

% cos_one_sin_zero
tff(fact_7355_sin__add,axiom,
    ! [X: real,Y2: real] : aa_real_real(sin_real,aa_real_real(plus_plus_real(X),Y2)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(sin_real,X)),aa_real_real(cos_real,Y2))),aa_real_real(times_times_real(aa_real_real(cos_real,X)),aa_real_real(sin_real,Y2))) ).

% sin_add
tff(fact_7356_sin__diff,axiom,
    ! [X: complex,Y2: complex] : aa_complex_complex(sin_complex,aa_complex_complex(minus_minus_complex(X),Y2)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(aa_complex_complex(sin_complex,X)),aa_complex_complex(cos_complex,Y2))),aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,X)),aa_complex_complex(sin_complex,Y2))) ).

% sin_diff
tff(fact_7357_sin__diff,axiom,
    ! [X: real,Y2: real] : aa_real_real(sin_real,aa_real_real(minus_minus_real(X),Y2)) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(aa_real_real(sin_real,X)),aa_real_real(cos_real,Y2))),aa_real_real(times_times_real(aa_real_real(cos_real,X)),aa_real_real(sin_real,Y2))) ).

% sin_diff
tff(fact_7358_lemma__tan__add1,axiom,
    ! [X: complex,Y2: complex] :
      ( ( aa_complex_complex(cos_complex,X) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,Y2) != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(one_one_complex),aa_complex_complex(times_times_complex(aa_complex_complex(tan_complex,X)),aa_complex_complex(tan_complex,Y2))) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(X),Y2))),aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,X)),aa_complex_complex(cos_complex,Y2))) ) ) ) ).

% lemma_tan_add1
tff(fact_7359_lemma__tan__add1,axiom,
    ! [X: real,Y2: real] :
      ( ( aa_real_real(cos_real,X) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Y2) != zero_zero_real )
       => ( aa_real_real(minus_minus_real(one_one_real),aa_real_real(times_times_real(aa_real_real(tan_real,X)),aa_real_real(tan_real,Y2))) = aa_real_real(divide_divide_real(aa_real_real(cos_real,aa_real_real(plus_plus_real(X),Y2))),aa_real_real(times_times_real(aa_real_real(cos_real,X)),aa_real_real(cos_real,Y2))) ) ) ) ).

% lemma_tan_add1
tff(fact_7360_tan__diff,axiom,
    ! [X: complex,Y2: complex] :
      ( ( aa_complex_complex(cos_complex,X) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,Y2) != zero_zero_complex )
       => ( ( aa_complex_complex(cos_complex,aa_complex_complex(minus_minus_complex(X),Y2)) != zero_zero_complex )
         => ( aa_complex_complex(tan_complex,aa_complex_complex(minus_minus_complex(X),Y2)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(tan_complex,X)),aa_complex_complex(tan_complex,Y2))),aa_complex_complex(plus_plus_complex(one_one_complex),aa_complex_complex(times_times_complex(aa_complex_complex(tan_complex,X)),aa_complex_complex(tan_complex,Y2)))) ) ) ) ) ).

% tan_diff
tff(fact_7361_tan__diff,axiom,
    ! [X: real,Y2: real] :
      ( ( aa_real_real(cos_real,X) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Y2) != zero_zero_real )
       => ( ( aa_real_real(cos_real,aa_real_real(minus_minus_real(X),Y2)) != zero_zero_real )
         => ( aa_real_real(tan_real,aa_real_real(minus_minus_real(X),Y2)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(tan_real,X)),aa_real_real(tan_real,Y2))),aa_real_real(plus_plus_real(one_one_real),aa_real_real(times_times_real(aa_real_real(tan_real,X)),aa_real_real(tan_real,Y2)))) ) ) ) ) ).

% tan_diff
tff(fact_7362_tan__add,axiom,
    ! [X: complex,Y2: complex] :
      ( ( aa_complex_complex(cos_complex,X) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,Y2) != zero_zero_complex )
       => ( ( aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(X),Y2)) != zero_zero_complex )
         => ( aa_complex_complex(tan_complex,aa_complex_complex(plus_plus_complex(X),Y2)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(tan_complex,X)),aa_complex_complex(tan_complex,Y2))),aa_complex_complex(minus_minus_complex(one_one_complex),aa_complex_complex(times_times_complex(aa_complex_complex(tan_complex,X)),aa_complex_complex(tan_complex,Y2)))) ) ) ) ) ).

% tan_add
tff(fact_7363_tan__add,axiom,
    ! [X: real,Y2: real] :
      ( ( aa_real_real(cos_real,X) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Y2) != zero_zero_real )
       => ( ( aa_real_real(cos_real,aa_real_real(plus_plus_real(X),Y2)) != zero_zero_real )
         => ( aa_real_real(tan_real,aa_real_real(plus_plus_real(X),Y2)) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(tan_real,X)),aa_real_real(tan_real,Y2))),aa_real_real(minus_minus_real(one_one_real),aa_real_real(times_times_real(aa_real_real(tan_real,X)),aa_real_real(tan_real,Y2)))) ) ) ) ) ).

% tan_add
tff(fact_7364_ceiling__le,axiom,
    ! [X: real,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,A))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X)),A) ) ).

% ceiling_le
tff(fact_7365_ceiling__le,axiom,
    ! [X: rat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,A))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,X)),A) ) ).

% ceiling_le
tff(fact_7366_ceiling__le__iff,axiom,
    ! [X: real,Z2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,X)),Z2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,Z2)) ) ).

% ceiling_le_iff
tff(fact_7367_ceiling__le__iff,axiom,
    ! [X: rat,Z2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,X)),Z2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,Z2)) ) ).

% ceiling_le_iff
tff(fact_7368_less__ceiling__iff,axiom,
    ! [Z2: int,X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),aa_rat_int(archim2889992004027027881ng_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z2)),X) ) ).

% less_ceiling_iff
tff(fact_7369_less__ceiling__iff,axiom,
    ! [Z2: int,X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),aa_real_int(archim7802044766580827645g_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Z2)),X) ) ).

% less_ceiling_iff
tff(fact_7370_summable__0__powser,axiom,
    ! [F: fun_nat_complex] : summable_complex(aTP_Lamm_ee(F)) ).

% summable_0_powser
tff(fact_7371_summable__0__powser,axiom,
    ! [F: fun_nat_real] : summable_real(aTP_Lamm_ef(F)) ).

% summable_0_powser
tff(fact_7372_summable__zero__power_H,axiom,
    ! [F: fun_nat_complex] : summable_complex(aTP_Lamm_ee(F)) ).

% summable_zero_power'
tff(fact_7373_summable__zero__power_H,axiom,
    ! [F: fun_nat_real] : summable_real(aTP_Lamm_ef(F)) ).

% summable_zero_power'
tff(fact_7374_summable__zero__power_H,axiom,
    ! [F: fun_nat_int] : summable_int(aTP_Lamm_fr(F)) ).

% summable_zero_power'
tff(fact_7375_ceiling__add__le,axiom,
    ! [X: rat,Y2: rat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(plus_plus_rat(X),Y2))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_rat_int(archim2889992004027027881ng_rat,Y2))) ).

% ceiling_add_le
tff(fact_7376_ceiling__add__le,axiom,
    ! [X: real,Y2: real] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(plus_plus_real(X),Y2))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,X)),aa_real_int(archim7802044766580827645g_real,Y2))) ).

% ceiling_add_le
tff(fact_7377_cos__monotone__0__pi__le,axiom,
    ! [Y2: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),pi)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y2)) ) ) ) ).

% cos_monotone_0_pi_le
tff(fact_7378_cos__mono__le__eq,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),pi)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),pi)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y2))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),X) ) ) ) ) ) ).

% cos_mono_le_eq
tff(fact_7379_cos__inj__pi,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),pi)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),pi)
           => ( ( aa_real_real(cos_real,X) = aa_real_real(cos_real,Y2) )
             => ( X = Y2 ) ) ) ) ) ) ).

% cos_inj_pi
tff(fact_7380_cos__ge__minus__one,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(cos_real,X)) ).

% cos_ge_minus_one
tff(fact_7381_powser__split__head_I3_J,axiom,
    ! [F: fun_nat_complex,Z2: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_fp(F),Z2))
     => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_fs(F),Z2)) ) ).

% powser_split_head(3)
tff(fact_7382_powser__split__head_I3_J,axiom,
    ! [F: fun_nat_real,Z2: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_fn(F),Z2))
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_ft(F),Z2)) ) ).

% powser_split_head(3)
tff(fact_7383_summable__powser__split__head,axiom,
    ! [F: fun_nat_complex,Z2: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_fs(F),Z2))
    <=> summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_fp(F),Z2)) ) ).

% summable_powser_split_head
tff(fact_7384_summable__powser__split__head,axiom,
    ! [F: fun_nat_real,Z2: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_ft(F),Z2))
    <=> summable_real(aa_real_fun_nat_real(aTP_Lamm_fn(F),Z2)) ) ).

% summable_powser_split_head
tff(fact_7385_summable__powser__ignore__initial__segment,axiom,
    ! [F: fun_nat_complex,Ma: nat,Z2: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_fu(F),Ma),Z2))
    <=> summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_fp(F),Z2)) ) ).

% summable_powser_ignore_initial_segment
tff(fact_7386_summable__powser__ignore__initial__segment,axiom,
    ! [F: fun_nat_real,Ma: nat,Z2: real] :
      ( summable_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_fv(F),Ma),Z2))
    <=> summable_real(aa_real_fun_nat_real(aTP_Lamm_fn(F),Z2)) ) ).

% summable_powser_ignore_initial_segment
tff(fact_7387_abs__cos__le__one,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(cos_real,X))),one_one_real) ).

% abs_cos_le_one
tff(fact_7388_summable__norm__comparison__test,axiom,
    ! [F: fun_nat_complex,G: fun_nat_real] :
      ( ? [N5: nat] :
        ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N5),N)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,N))),aa_nat_real(G,N)) )
     => ( summable_real(G)
       => summable_real(aTP_Lamm_ev(F)) ) ) ).

% summable_norm_comparison_test
tff(fact_7389_summable__rabs__comparison__test,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ? [N5: nat] :
        ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N5),N)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_nat_real(F,N))),aa_nat_real(G,N)) )
     => ( summable_real(G)
       => summable_real(aTP_Lamm_fm(F)) ) ) ).

% summable_rabs_comparison_test
tff(fact_7390_summable__rabs,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(aTP_Lamm_fm(F))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,suminf_real(F))),suminf_real(aTP_Lamm_fm(F))) ) ).

% summable_rabs
tff(fact_7391_suminf__pos2,axiom,
    ! [F: fun_nat_real,I2: nat] :
      ( summable_real(F)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(F,I2))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),suminf_real(F)) ) ) ) ).

% suminf_pos2
tff(fact_7392_suminf__pos2,axiom,
    ! [F: fun_nat_nat,I2: nat] :
      ( summable_nat(F)
     => ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,I2))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),suminf_nat(F)) ) ) ) ).

% suminf_pos2
tff(fact_7393_suminf__pos2,axiom,
    ! [F: fun_nat_int,I2: nat] :
      ( summable_int(F)
     => ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,I2))
         => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),suminf_int(F)) ) ) ) ).

% suminf_pos2
tff(fact_7394_suminf__pos__iff,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),suminf_real(F))
        <=> ? [I: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(F,I)) ) ) ) ).

% suminf_pos_iff
tff(fact_7395_suminf__pos__iff,axiom,
    ! [F: fun_nat_nat] :
      ( summable_nat(F)
     => ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),suminf_nat(F))
        <=> ? [I: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,I)) ) ) ) ).

% suminf_pos_iff
tff(fact_7396_suminf__pos__iff,axiom,
    ! [F: fun_nat_int] :
      ( summable_int(F)
     => ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),suminf_int(F))
        <=> ? [I: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,I)) ) ) ) ).

% suminf_pos_iff
tff(fact_7397_cos__diff,axiom,
    ! [X: complex,Y2: complex] : aa_complex_complex(cos_complex,aa_complex_complex(minus_minus_complex(X),Y2)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,X)),aa_complex_complex(cos_complex,Y2))),aa_complex_complex(times_times_complex(aa_complex_complex(sin_complex,X)),aa_complex_complex(sin_complex,Y2))) ).

% cos_diff
tff(fact_7398_cos__diff,axiom,
    ! [X: real,Y2: real] : aa_real_real(cos_real,aa_real_real(minus_minus_real(X),Y2)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(cos_real,X)),aa_real_real(cos_real,Y2))),aa_real_real(times_times_real(aa_real_real(sin_real,X)),aa_real_real(sin_real,Y2))) ).

% cos_diff
tff(fact_7399_cos__add,axiom,
    ! [X: complex,Y2: complex] : aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(X),Y2)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,X)),aa_complex_complex(cos_complex,Y2))),aa_complex_complex(times_times_complex(aa_complex_complex(sin_complex,X)),aa_complex_complex(sin_complex,Y2))) ).

% cos_add
tff(fact_7400_cos__add,axiom,
    ! [X: real,Y2: real] : aa_real_real(cos_real,aa_real_real(plus_plus_real(X),Y2)) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(aa_real_real(cos_real,X)),aa_real_real(cos_real,Y2))),aa_real_real(times_times_real(aa_real_real(sin_real,X)),aa_real_real(sin_real,Y2))) ).

% cos_add
tff(fact_7401_sin__zero__norm__cos__one,axiom,
    ! [X: real] :
      ( ( aa_real_real(sin_real,X) = zero_zero_real )
     => ( aa_real_real(real_V7735802525324610683m_real,aa_real_real(cos_real,X)) = one_one_real ) ) ).

% sin_zero_norm_cos_one
tff(fact_7402_sin__zero__norm__cos__one,axiom,
    ! [X: complex] :
      ( ( aa_complex_complex(sin_complex,X) = zero_zero_complex )
     => ( aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(cos_complex,X)) = one_one_real ) ) ).

% sin_zero_norm_cos_one
tff(fact_7403_of__int__ceiling__le__add__one,axiom,
    ! [R: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,R))),aa_real_real(plus_plus_real(R),one_one_real)) ).

% of_int_ceiling_le_add_one
tff(fact_7404_of__int__ceiling__le__add__one,axiom,
    ! [R: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,R))),aa_rat_rat(plus_plus_rat(R),one_one_rat)) ).

% of_int_ceiling_le_add_one
tff(fact_7405_of__int__ceiling__diff__one__le,axiom,
    ! [R: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,R))),one_one_real)),R) ).

% of_int_ceiling_diff_one_le
tff(fact_7406_of__int__ceiling__diff__one__le,axiom,
    ! [R: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,R))),one_one_rat)),R) ).

% of_int_ceiling_diff_one_le
tff(fact_7407_cos__two__neq__zero,axiom,
    aa_real_real(cos_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) != zero_zero_real ).

% cos_two_neq_zero
tff(fact_7408_powser__inside,axiom,
    ! [F: fun_nat_real,X: real,Z2: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_fn(F),X))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Z2)),aa_real_real(real_V7735802525324610683m_real,X))
       => summable_real(aa_real_fun_nat_real(aTP_Lamm_fn(F),Z2)) ) ) ).

% powser_inside
tff(fact_7409_powser__inside,axiom,
    ! [F: fun_nat_complex,X: complex,Z2: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_fp(F),X))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Z2)),aa_complex_real(real_V1022390504157884413omplex,X))
       => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_fp(F),Z2)) ) ) ).

% powser_inside
tff(fact_7410_cos__monotone__0__pi,axiom,
    ! [Y2: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),pi)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y2)) ) ) ) ).

% cos_monotone_0_pi
tff(fact_7411_cos__mono__less__eq,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),pi)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),pi)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,X)),aa_real_real(cos_real,Y2))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),X) ) ) ) ) ) ).

% cos_mono_less_eq
tff(fact_7412_tan__half,axiom,
    ! [X: complex] : aa_complex_complex(tan_complex,X) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(sin_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),X))),aa_complex_complex(plus_plus_complex(aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),X))),one_one_complex)) ).

% tan_half
tff(fact_7413_tan__half,axiom,
    ! [X: real] : aa_real_real(tan_real,X) = aa_real_real(divide_divide_real(aa_real_real(sin_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X))),aa_real_real(plus_plus_real(aa_real_real(cos_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X))),one_one_real)) ).

% tan_half
tff(fact_7414_complete__algebra__summable__geometric,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X)),one_one_real)
     => summable_real(aa_real_fun_nat_real(power_power_real,X)) ) ).

% complete_algebra_summable_geometric
tff(fact_7415_complete__algebra__summable__geometric,axiom,
    ! [X: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X)),one_one_real)
     => summable_complex(aa_com8181001469867462503omplex(power_power_complex,X)) ) ).

% complete_algebra_summable_geometric
tff(fact_7416_summable__geometric,axiom,
    ! [C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,C)),one_one_real)
     => summable_real(aa_real_fun_nat_real(power_power_real,C)) ) ).

% summable_geometric
tff(fact_7417_summable__geometric,axiom,
    ! [C: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,C)),one_one_real)
     => summable_complex(aa_com8181001469867462503omplex(power_power_complex,C)) ) ).

% summable_geometric
tff(fact_7418_cos__monotone__minus__pi__0_H,axiom,
    ! [Y2: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),Y2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,Y2)),aa_real_real(cos_real,X)) ) ) ) ).

% cos_monotone_minus_pi_0'
tff(fact_7419_suminf__split__head,axiom,
    ! [F: fun_nat_complex] :
      ( summable_complex(F)
     => ( suminf_complex(aTP_Lamm_fl(F)) = aa_complex_complex(minus_minus_complex(suminf_complex(F)),aa_nat_complex(F,zero_zero_nat)) ) ) ).

% suminf_split_head
tff(fact_7420_suminf__split__head,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( suminf_real(aTP_Lamm_fk(F)) = aa_real_real(minus_minus_real(suminf_real(F)),aa_nat_real(F,zero_zero_nat)) ) ) ).

% suminf_split_head
tff(fact_7421_ceiling__divide__eq__div,axiom,
    ! [A: int,B: int] : aa_real_int(archim7802044766580827645g_real,aa_real_real(divide_divide_real(aa_int_real(ring_1_of_int_real,A)),aa_int_real(ring_1_of_int_real,B))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A)),B)) ).

% ceiling_divide_eq_div
tff(fact_7422_ceiling__divide__eq__div,axiom,
    ! [A: int,B: int] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(divide_divide_rat(aa_int_rat(ring_1_of_int_rat,A)),aa_int_rat(ring_1_of_int_rat,B))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A)),B)) ).

% ceiling_divide_eq_div
tff(fact_7423_sin__zero__abs__cos__one,axiom,
    ! [X: real] :
      ( ( aa_real_real(sin_real,X) = zero_zero_real )
     => ( aa_real_real(abs_abs_real,aa_real_real(cos_real,X)) = one_one_real ) ) ).

% sin_zero_abs_cos_one
tff(fact_7424_summable__norm,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(aTP_Lamm_eu(F))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,suminf_real(F))),suminf_real(aTP_Lamm_eu(F))) ) ).

% summable_norm
tff(fact_7425_summable__norm,axiom,
    ! [F: fun_nat_complex] :
      ( summable_real(aTP_Lamm_ev(F))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,suminf_complex(F))),suminf_real(aTP_Lamm_ev(F))) ) ).

% summable_norm
tff(fact_7426_sin__double,axiom,
    ! [X: complex] : aa_complex_complex(sin_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),X)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(sin_complex,X))),aa_complex_complex(cos_complex,X)) ).

% sin_double
tff(fact_7427_sin__double,axiom,
    ! [X: real] : aa_real_real(sin_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X)) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(sin_real,X))),aa_real_real(cos_real,X)) ).

% sin_double
tff(fact_7428_ceiling__split,axiom,
    ! [P: fun_int_o,T: real] :
      ( aa_int_o(P,aa_real_int(archim7802044766580827645g_real,T))
    <=> ! [I: int] :
          ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(aa_int_real(ring_1_of_int_real,I)),one_one_real)),T)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T),aa_int_real(ring_1_of_int_real,I)) )
         => aa_int_o(P,I) ) ) ).

% ceiling_split
tff(fact_7429_ceiling__split,axiom,
    ! [P: fun_int_o,T: rat] :
      ( aa_int_o(P,aa_rat_int(archim2889992004027027881ng_rat,T))
    <=> ! [I: int] :
          ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,I)),one_one_rat)),T)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,T),aa_int_rat(ring_1_of_int_rat,I)) )
         => aa_int_o(P,I) ) ) ).

% ceiling_split
tff(fact_7430_ceiling__eq__iff,axiom,
    ! [X: real,A: int] :
      ( ( aa_real_int(archim7802044766580827645g_real,X) = A )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(aa_int_real(ring_1_of_int_real,A)),one_one_real)),X)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,A)) ) ) ).

% ceiling_eq_iff
tff(fact_7431_ceiling__eq__iff,axiom,
    ! [X: rat,A: int] :
      ( ( aa_rat_int(archim2889992004027027881ng_rat,X) = A )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,A)),one_one_rat)),X)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,A)) ) ) ).

% ceiling_eq_iff
tff(fact_7432_ceiling__unique,axiom,
    ! [Z2: int,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(aa_int_real(ring_1_of_int_real,Z2)),one_one_real)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,Z2))
       => ( aa_real_int(archim7802044766580827645g_real,X) = Z2 ) ) ) ).

% ceiling_unique
tff(fact_7433_ceiling__unique,axiom,
    ! [Z2: int,X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,Z2)),one_one_rat)),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,Z2))
       => ( aa_rat_int(archim2889992004027027881ng_rat,X) = Z2 ) ) ) ).

% ceiling_unique
tff(fact_7434_ceiling__correct,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,X))),one_one_real)),X)
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,X))) ) ).

% ceiling_correct
tff(fact_7435_ceiling__correct,axiom,
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,X))),one_one_rat)),X)
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,X))) ) ).

% ceiling_correct
tff(fact_7436_mult__ceiling__le,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(times_times_real(A),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(archim7802044766580827645g_real,A)),aa_real_int(archim7802044766580827645g_real,B))) ) ) ).

% mult_ceiling_le
tff(fact_7437_mult__ceiling__le,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(times_times_rat(A),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_rat_int(archim2889992004027027881ng_rat,A)),aa_rat_int(archim2889992004027027881ng_rat,B))) ) ) ).

% mult_ceiling_le
tff(fact_7438_ceiling__less__iff,axiom,
    ! [X: real,Z2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,X)),Z2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(minus_minus_real(aa_int_real(ring_1_of_int_real,Z2)),one_one_real)) ) ).

% ceiling_less_iff
tff(fact_7439_ceiling__less__iff,axiom,
    ! [X: rat,Z2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,X)),Z2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,Z2)),one_one_rat)) ) ).

% ceiling_less_iff
tff(fact_7440_le__ceiling__iff,axiom,
    ! [Z2: int,X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z2),aa_rat_int(archim2889992004027027881ng_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,Z2)),one_one_rat)),X) ) ).

% le_ceiling_iff
tff(fact_7441_le__ceiling__iff,axiom,
    ! [Z2: int,X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z2),aa_real_int(archim7802044766580827645g_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(aa_int_real(ring_1_of_int_real,Z2)),one_one_real)),X) ) ).

% le_ceiling_iff
tff(fact_7442_cos__two__less__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),zero_zero_real) ).

% cos_two_less_zero
tff(fact_7443_cos__two__le__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),zero_zero_real) ).

% cos_two_le_zero
tff(fact_7444_cos__is__zero,axiom,
    ? [X4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X4)
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
      & ( aa_real_real(cos_real,X4) = zero_zero_real )
      & ! [Y5: real] :
          ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y5)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y5),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
            & ( aa_real_real(cos_real,Y5) = zero_zero_real ) )
         => ( Y5 = X4 ) ) ) ).

% cos_is_zero
tff(fact_7445_tan__double,axiom,
    ! [X: complex] :
      ( ( aa_complex_complex(cos_complex,X) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),X)) != zero_zero_complex )
       => ( aa_complex_complex(tan_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),X)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(tan_complex,X))),aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(tan_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% tan_double
tff(fact_7446_tan__double,axiom,
    ! [X: real] :
      ( ( aa_real_real(cos_real,X) != zero_zero_real )
     => ( ( aa_real_real(cos_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X)) != zero_zero_real )
       => ( aa_real_real(tan_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X)) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(tan_real,X))),aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(tan_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% tan_double
tff(fact_7447_cos__monotone__minus__pi__0,axiom,
    ! [Y2: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),Y2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,Y2)),aa_real_real(cos_real,X)) ) ) ) ).

% cos_monotone_minus_pi_0
tff(fact_7448_cos__total,axiom,
    ! [Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),one_one_real)
       => ? [X4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X4)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),pi)
            & ( aa_real_real(cos_real,X4) = Y2 )
            & ! [Y5: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y5)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y5),pi)
                  & ( aa_real_real(cos_real,Y5) = Y2 ) )
               => ( Y5 = X4 ) ) ) ) ) ).

% cos_total
tff(fact_7449_sincos__principal__value,axiom,
    ! [X: real] :
    ? [Y4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Y4)
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y4),pi)
      & ( aa_real_real(sin_real,Y4) = aa_real_real(sin_real,X) )
      & ( aa_real_real(cos_real,Y4) = aa_real_real(cos_real,X) ) ) ).

% sincos_principal_value
tff(fact_7450_ceiling__divide__upper,axiom,
    ! [Q: real,P2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Q)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,P2),aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,aa_real_real(divide_divide_real(P2),Q)))),Q)) ) ).

% ceiling_divide_upper
tff(fact_7451_ceiling__divide__upper,axiom,
    ! [Q: rat,P2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Q)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,P2),aa_rat_rat(times_times_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(divide_divide_rat(P2),Q)))),Q)) ) ).

% ceiling_divide_upper
tff(fact_7452_powser__split__head_I1_J,axiom,
    ! [F: fun_nat_complex,Z2: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_fp(F),Z2))
     => ( suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_fp(F),Z2)) = aa_complex_complex(plus_plus_complex(aa_nat_complex(F,zero_zero_nat)),aa_complex_complex(times_times_complex(suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_fs(F),Z2))),Z2)) ) ) ).

% powser_split_head(1)
tff(fact_7453_powser__split__head_I1_J,axiom,
    ! [F: fun_nat_real,Z2: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_fn(F),Z2))
     => ( suminf_real(aa_real_fun_nat_real(aTP_Lamm_fn(F),Z2)) = aa_real_real(plus_plus_real(aa_nat_real(F,zero_zero_nat)),aa_real_real(times_times_real(suminf_real(aa_real_fun_nat_real(aTP_Lamm_ft(F),Z2))),Z2)) ) ) ).

% powser_split_head(1)
tff(fact_7454_powser__split__head_I2_J,axiom,
    ! [F: fun_nat_complex,Z2: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_fp(F),Z2))
     => ( aa_complex_complex(times_times_complex(suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_fs(F),Z2))),Z2) = aa_complex_complex(minus_minus_complex(suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_fp(F),Z2))),aa_nat_complex(F,zero_zero_nat)) ) ) ).

% powser_split_head(2)
tff(fact_7455_powser__split__head_I2_J,axiom,
    ! [F: fun_nat_real,Z2: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_fn(F),Z2))
     => ( aa_real_real(times_times_real(suminf_real(aa_real_fun_nat_real(aTP_Lamm_ft(F),Z2))),Z2) = aa_real_real(minus_minus_real(suminf_real(aa_real_fun_nat_real(aTP_Lamm_fn(F),Z2))),aa_nat_real(F,zero_zero_nat)) ) ) ).

% powser_split_head(2)
tff(fact_7456_cos__tan,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,X)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => ( aa_real_real(cos_real,X) = aa_real_real(divide_divide_real(one_one_real),aa_real_real(sqrt,aa_real_real(plus_plus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(tan_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% cos_tan
tff(fact_7457_tan__45,axiom,
    aa_real_real(tan_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one))))) = one_one_real ).

% tan_45
tff(fact_7458_suminf__exist__split,axiom,
    ! [R: real,F: fun_nat_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R)
     => ( summable_real(F)
       => ? [N6: nat] :
          ! [N7: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N6),N7)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,suminf_real(aa_nat_fun_nat_real(aTP_Lamm_eo(F),N7)))),R) ) ) ) ).

% suminf_exist_split
tff(fact_7459_suminf__exist__split,axiom,
    ! [R: real,F: fun_nat_complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R)
     => ( summable_complex(F)
       => ? [N6: nat] :
          ! [N7: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N6),N7)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,suminf_complex(aa_nat6454746962644518025omplex(aTP_Lamm_ep(F),N7)))),R) ) ) ) ).

% suminf_exist_split
tff(fact_7460_tan__60,axiom,
    aa_real_real(tan_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)))) = aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))) ).

% tan_60
tff(fact_7461_summable__power__series,axiom,
    ! [F: fun_nat_real,Z2: real] :
      ( ! [I3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,I3)),one_one_real)
     => ( ! [I3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,I3))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Z2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),one_one_real)
           => summable_real(aa_real_fun_nat_real(aTP_Lamm_fn(F),Z2)) ) ) ) ) ).

% summable_power_series
tff(fact_7462_Abel__lemma,axiom,
    ! [R: real,R0: real,A: fun_nat_complex,M6: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),R)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,R),R0)
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(A,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,R0),N))),M6)
         => summable_real(aa_fun7054779078796324018t_real(aTP_Lamm_fw(R),A)) ) ) ) ).

% Abel_lemma
tff(fact_7463_cos__45,axiom,
    aa_real_real(cos_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one))))) = aa_real_real(divide_divide_real(aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% cos_45
tff(fact_7464_sin__cos__le1,axiom,
    ! [X: real,Y2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(sin_real,X)),aa_real_real(sin_real,Y2))),aa_real_real(times_times_real(aa_real_real(cos_real,X)),aa_real_real(cos_real,Y2))))),one_one_real) ).

% sin_cos_le1
tff(fact_7465_cos__plus__cos,axiom,
    ! [W: complex,Z2: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(cos_complex,W)),aa_complex_complex(cos_complex,Z2)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(W),Z2)),numera6690914467698888265omplex(aa_num_num(bit0,one)))))),aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(W),Z2)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).

% cos_plus_cos
tff(fact_7466_cos__plus__cos,axiom,
    ! [W: real,Z2: real] : aa_real_real(plus_plus_real(aa_real_real(cos_real,W)),aa_real_real(cos_real,Z2)) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(cos_real,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(W),Z2)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))),aa_real_real(cos_real,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(W),Z2)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% cos_plus_cos
tff(fact_7467_cos__times__cos,axiom,
    ! [W: complex,Z2: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,W)),aa_complex_complex(cos_complex,Z2)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(cos_complex,aa_complex_complex(minus_minus_complex(W),Z2))),aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(W),Z2)))),numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% cos_times_cos
tff(fact_7468_cos__times__cos,axiom,
    ! [W: real,Z2: real] : aa_real_real(times_times_real(aa_real_real(cos_real,W)),aa_real_real(cos_real,Z2)) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(cos_real,aa_real_real(minus_minus_real(W),Z2))),aa_real_real(cos_real,aa_real_real(plus_plus_real(W),Z2)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% cos_times_cos
tff(fact_7469_summable__ratio__test,axiom,
    ! [C: real,N2: nat,F: fun_nat_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),one_one_real)
     => ( ! [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,aa_nat_nat(suc,N)))),aa_real_real(times_times_real(C),aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,N)))) )
       => summable_real(F) ) ) ).

% summable_ratio_test
tff(fact_7470_summable__ratio__test,axiom,
    ! [C: real,N2: nat,F: fun_nat_complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),one_one_real)
     => ( ! [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),N)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,aa_nat_nat(suc,N)))),aa_real_real(times_times_real(C),aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,N)))) )
       => summable_complex(F) ) ) ).

% summable_ratio_test
tff(fact_7471_cos__squared__eq,axiom,
    ! [X: complex] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(cos_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(sin_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cos_squared_eq
tff(fact_7472_cos__squared__eq,axiom,
    ! [X: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cos_squared_eq
tff(fact_7473_sin__squared__eq,axiom,
    ! [X: complex] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(sin_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(cos_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% sin_squared_eq
tff(fact_7474_sin__squared__eq,axiom,
    ! [X: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% sin_squared_eq
tff(fact_7475_ceiling__divide__lower,axiom,
    ! [Q: real,P2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Q)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(aa_real_real(minus_minus_real(aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,aa_real_real(divide_divide_real(P2),Q)))),one_one_real)),Q)),P2) ) ).

% ceiling_divide_lower
tff(fact_7476_ceiling__divide__lower,axiom,
    ! [Q: rat,P2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Q)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(divide_divide_rat(P2),Q)))),one_one_rat)),Q)),P2) ) ).

% ceiling_divide_lower
tff(fact_7477_tan__gt__zero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(tan_real,X)) ) ) ).

% tan_gt_zero
tff(fact_7478_lemma__tan__total,axiom,
    ! [Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y2)
     => ? [X4: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X4)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,X4),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),aa_real_real(tan_real,X4)) ) ) ).

% lemma_tan_total
tff(fact_7479_ceiling__eq,axiom,
    ! [Nb: int,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Nb)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(plus_plus_real(aa_int_real(ring_1_of_int_real,Nb)),one_one_real))
       => ( aa_real_int(archim7802044766580827645g_real,X) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Nb),one_one_int) ) ) ) ).

% ceiling_eq
tff(fact_7480_ceiling__eq,axiom,
    ! [Nb: int,X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Nb)),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,Nb)),one_one_rat))
       => ( aa_rat_int(archim2889992004027027881ng_rat,X) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Nb),one_one_int) ) ) ) ).

% ceiling_eq
tff(fact_7481_lemma__tan__total1,axiom,
    ! [Y2: real] :
    ? [X4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X4)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,X4),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
      & ( aa_real_real(tan_real,X4) = Y2 ) ) ).

% lemma_tan_total1
tff(fact_7482_tan__mono__lt__eq,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,X)),aa_real_real(tan_real,Y2))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2) ) ) ) ) ) ).

% tan_mono_lt_eq
tff(fact_7483_tan__monotone_H,axiom,
    ! [Y2: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),X)
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,Y2)),aa_real_real(tan_real,X)) ) ) ) ) ) ).

% tan_monotone'
tff(fact_7484_tan__monotone,axiom,
    ! [Y2: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,Y2)),aa_real_real(tan_real,X)) ) ) ) ).

% tan_monotone
tff(fact_7485_tan__total,axiom,
    ! [Y2: real] :
    ? [X4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X4)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,X4),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
      & ( aa_real_real(tan_real,X4) = Y2 )
      & ! [Y5: real] :
          ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y5)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,Y5),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
            & ( aa_real_real(tan_real,Y5) = Y2 ) )
         => ( Y5 = X4 ) ) ) ).

% tan_total
tff(fact_7486_tan__minus__45,axiom,
    aa_real_real(tan_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))))) = aa_real_real(uminus_uminus_real,one_one_real) ).

% tan_minus_45
tff(fact_7487_cos__double__less__one,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X))),one_one_real) ) ) ).

% cos_double_less_one
tff(fact_7488_tan__inverse,axiom,
    ! [Y2: real] : aa_real_real(divide_divide_real(one_one_real),aa_real_real(tan_real,Y2)) = aa_real_real(tan_real,aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Y2)) ).

% tan_inverse
tff(fact_7489_cos__gt__zero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(cos_real,X)) ) ) ).

% cos_gt_zero
tff(fact_7490_cos__60,axiom,
    aa_real_real(cos_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)))) = aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% cos_60
tff(fact_7491_cos__30,axiom,
    aa_real_real(cos_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,one))))) = aa_real_real(divide_divide_real(aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% cos_30
tff(fact_7492_cos__one__2pi__int,axiom,
    ! [X: real] :
      ( ( aa_real_real(cos_real,X) = one_one_real )
    <=> ? [X3: int] : X = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,X3)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),pi) ) ).

% cos_one_2pi_int
tff(fact_7493_cos__double__cos,axiom,
    ! [W: complex] : aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),W)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(cos_complex,W)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),one_one_complex) ).

% cos_double_cos
tff(fact_7494_cos__double__cos,axiom,
    ! [W: real] : aa_real_real(cos_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),W)) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,W)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),one_one_real) ).

% cos_double_cos
tff(fact_7495_cos__treble__cos,axiom,
    ! [X: complex] : aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit1,one))),X)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(cos_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))))),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit1,one))),aa_complex_complex(cos_complex,X))) ).

% cos_treble_cos
tff(fact_7496_cos__treble__cos,axiom,
    ! [X: real] : aa_real_real(cos_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))),X)) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))))),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))),aa_real_real(cos_real,X))) ).

% cos_treble_cos
tff(fact_7497_cos__diff__cos,axiom,
    ! [W: complex,Z2: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(cos_complex,W)),aa_complex_complex(cos_complex,Z2)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(sin_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(W),Z2)),numera6690914467698888265omplex(aa_num_num(bit0,one)))))),aa_complex_complex(sin_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(Z2),W)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).

% cos_diff_cos
tff(fact_7498_cos__diff__cos,axiom,
    ! [W: real,Z2: real] : aa_real_real(minus_minus_real(aa_real_real(cos_real,W)),aa_real_real(cos_real,Z2)) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(sin_real,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(W),Z2)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))),aa_real_real(sin_real,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(Z2),W)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% cos_diff_cos
tff(fact_7499_sin__diff__sin,axiom,
    ! [W: complex,Z2: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(sin_complex,W)),aa_complex_complex(sin_complex,Z2)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(sin_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(W),Z2)),numera6690914467698888265omplex(aa_num_num(bit0,one)))))),aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(W),Z2)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).

% sin_diff_sin
tff(fact_7500_sin__diff__sin,axiom,
    ! [W: real,Z2: real] : aa_real_real(minus_minus_real(aa_real_real(sin_real,W)),aa_real_real(sin_real,Z2)) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(sin_real,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(W),Z2)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))),aa_real_real(cos_real,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(W),Z2)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% sin_diff_sin
tff(fact_7501_sin__plus__sin,axiom,
    ! [W: complex,Z2: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(sin_complex,W)),aa_complex_complex(sin_complex,Z2)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(sin_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(W),Z2)),numera6690914467698888265omplex(aa_num_num(bit0,one)))))),aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(W),Z2)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).

% sin_plus_sin
tff(fact_7502_sin__plus__sin,axiom,
    ! [W: real,Z2: real] : aa_real_real(plus_plus_real(aa_real_real(sin_real,W)),aa_real_real(sin_real,Z2)) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(sin_real,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(W),Z2)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))),aa_real_real(cos_real,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(W),Z2)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% sin_plus_sin
tff(fact_7503_cos__times__sin,axiom,
    ! [W: complex,Z2: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,W)),aa_complex_complex(sin_complex,Z2)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(sin_complex,aa_complex_complex(plus_plus_complex(W),Z2))),aa_complex_complex(sin_complex,aa_complex_complex(minus_minus_complex(W),Z2)))),numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% cos_times_sin
tff(fact_7504_cos__times__sin,axiom,
    ! [W: real,Z2: real] : aa_real_real(times_times_real(aa_real_real(cos_real,W)),aa_real_real(sin_real,Z2)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(sin_real,aa_real_real(plus_plus_real(W),Z2))),aa_real_real(sin_real,aa_real_real(minus_minus_real(W),Z2)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% cos_times_sin
tff(fact_7505_sin__times__cos,axiom,
    ! [W: complex,Z2: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(sin_complex,W)),aa_complex_complex(cos_complex,Z2)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(sin_complex,aa_complex_complex(plus_plus_complex(W),Z2))),aa_complex_complex(sin_complex,aa_complex_complex(minus_minus_complex(W),Z2)))),numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% sin_times_cos
tff(fact_7506_sin__times__cos,axiom,
    ! [W: real,Z2: real] : aa_real_real(times_times_real(aa_real_real(sin_real,W)),aa_real_real(cos_real,Z2)) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(sin_real,aa_real_real(plus_plus_real(W),Z2))),aa_real_real(sin_real,aa_real_real(minus_minus_real(W),Z2)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% sin_times_cos
tff(fact_7507_sin__times__sin,axiom,
    ! [W: complex,Z2: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(sin_complex,W)),aa_complex_complex(sin_complex,Z2)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(cos_complex,aa_complex_complex(minus_minus_complex(W),Z2))),aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(W),Z2)))),numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% sin_times_sin
tff(fact_7508_sin__times__sin,axiom,
    ! [W: real,Z2: real] : aa_real_real(times_times_real(aa_real_real(sin_real,W)),aa_real_real(sin_real,Z2)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(cos_real,aa_real_real(minus_minus_real(W),Z2))),aa_real_real(cos_real,aa_real_real(plus_plus_real(W),Z2)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% sin_times_sin
tff(fact_7509_cos__double,axiom,
    ! [X: complex] : aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),X)) = aa_complex_complex(minus_minus_complex(aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(cos_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(sin_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cos_double
tff(fact_7510_cos__double,axiom,
    ! [X: real] : aa_real_real(cos_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X)) = aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cos_double
tff(fact_7511_tan__total__pos,axiom,
    ! [Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y2)
     => ? [X4: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X4)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,X4),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
          & ( aa_real_real(tan_real,X4) = Y2 ) ) ) ).

% tan_total_pos
tff(fact_7512_tan__pos__pi2__le,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(tan_real,X)) ) ) ).

% tan_pos_pi2_le
tff(fact_7513_tan__less__zero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,X)),zero_zero_real) ) ) ).

% tan_less_zero
tff(fact_7514_tan__mono__le,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(tan_real,X)),aa_real_real(tan_real,Y2)) ) ) ) ).

% tan_mono_le
tff(fact_7515_tan__mono__le__eq,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(tan_real,X)),aa_real_real(tan_real,Y2))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y2) ) ) ) ) ) ).

% tan_mono_le_eq
tff(fact_7516_tan__bound__pi2,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,X)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(tan_real,X))),one_one_real) ) ).

% tan_bound_pi2
tff(fact_7517_tan__30,axiom,
    aa_real_real(tan_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,one))))) = aa_real_real(divide_divide_real(one_one_real),aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)))) ).

% tan_30
tff(fact_7518_cos__gt__zero__pi,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(cos_real,X)) ) ) ).

% cos_gt_zero_pi
tff(fact_7519_cos__ge__zero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(cos_real,X)) ) ) ).

% cos_ge_zero
tff(fact_7520_arctan__unique,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( ( aa_real_real(tan_real,X) = Y2 )
         => ( aa_real_real(arctan,Y2) = X ) ) ) ) ).

% arctan_unique
tff(fact_7521_arctan__tan,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_real(arctan,aa_real_real(tan_real,X)) = X ) ) ) ).

% arctan_tan
tff(fact_7522_arctan,axiom,
    ! [Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arctan,Y2))
      & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Y2)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
      & ( aa_real_real(tan_real,aa_real_real(arctan,Y2)) = Y2 ) ) ).

% arctan
tff(fact_7523_cos__one__2pi,axiom,
    ! [X: real] :
      ( ( aa_real_real(cos_real,X) = one_one_real )
    <=> ( ? [X3: nat] : X = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,X3)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),pi)
        | ? [X3: nat] : X = aa_real_real(uminus_uminus_real,aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,X3)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),pi)) ) ) ).

% cos_one_2pi
tff(fact_7524_cos__double__sin,axiom,
    ! [W: complex] : aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),W)) = aa_complex_complex(minus_minus_complex(one_one_complex),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(sin_complex,W)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% cos_double_sin
tff(fact_7525_cos__double__sin,axiom,
    ! [W: real] : aa_real_real(cos_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),W)) = aa_real_real(minus_minus_real(one_one_real),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_real,W)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% cos_double_sin
tff(fact_7526_tan__total__pi4,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,X)),one_one_real)
     => ? [Z: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))))),Z)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,Z),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))))
          & ( aa_real_real(tan_real,Z) = X ) ) ) ).

% tan_total_pi4
tff(fact_7527_cos__arctan,axiom,
    ! [X: real] : aa_real_real(cos_real,aa_real_real(arctan,X)) = aa_real_real(divide_divide_real(one_one_real),aa_real_real(sqrt,aa_real_real(plus_plus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% cos_arctan
tff(fact_7528_sincos__total__pi,axiom,
    ! [Y2: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y2)
     => ( ( aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
       => ? [T3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T3)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T3),pi)
            & ( X = aa_real_real(cos_real,T3) )
            & ( Y2 = aa_real_real(sin_real,T3) ) ) ) ) ).

% sincos_total_pi
tff(fact_7529_sin__cos__sqrt,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,X))
     => ( aa_real_real(sin_real,X) = aa_real_real(sqrt,aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% sin_cos_sqrt
tff(fact_7530_sin__expansion__lemma,axiom,
    ! [X: real,Ma: nat] : aa_real_real(sin_real,aa_real_real(plus_plus_real(X),aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Ma))),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) = aa_real_real(cos_real,aa_real_real(plus_plus_real(X),aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Ma)),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% sin_expansion_lemma
tff(fact_7531_cos__zero__iff__int,axiom,
    ! [X: real] :
      ( ( aa_real_real(cos_real,X) = zero_zero_real )
    <=> ? [I: int] :
          ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),I)
          & ( X = aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,I)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).

% cos_zero_iff_int
tff(fact_7532_cos__zero__lemma,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( ( aa_real_real(cos_real,X) = zero_zero_real )
       => ? [N: nat] :
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
            & ( X = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ) ).

% cos_zero_lemma
tff(fact_7533_cos__zero__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(cos_real,X) = zero_zero_real )
    <=> ( ? [N4: nat] :
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4)
            & ( X = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N4)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) )
        | ? [N4: nat] :
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4)
            & ( X = aa_real_real(uminus_uminus_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N4)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ) ) ).

% cos_zero_iff
tff(fact_7534_cos__expansion__lemma,axiom,
    ! [X: real,Ma: nat] : aa_real_real(cos_real,aa_real_real(plus_plus_real(X),aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Ma))),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,aa_real_real(plus_plus_real(X),aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Ma)),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))) ).

% cos_expansion_lemma
tff(fact_7535_sincos__total__pi__half,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y2)
       => ( ( aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
         => ? [T3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T3)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T3),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
              & ( X = aa_real_real(cos_real,T3) )
              & ( Y2 = aa_real_real(sin_real,T3) ) ) ) ) ) ).

% sincos_total_pi_half
tff(fact_7536_sincos__total__2pi__le,axiom,
    ! [X: real,Y2: real] :
      ( ( aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
     => ? [T3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T3)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T3),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))
          & ( X = aa_real_real(cos_real,T3) )
          & ( Y2 = aa_real_real(sin_real,T3) ) ) ) ).

% sincos_total_2pi_le
tff(fact_7537_ceiling__log__nat__eq__if,axiom,
    ! [B: nat,Nb: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)),K2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
         => ( aa_real_int(archim7802044766580827645g_real,aa_real_real(log(aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int) ) ) ) ) ).

% ceiling_log_nat_eq_if
tff(fact_7538_ceiling__log2__div2,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_int(archim7802044766580827645g_real,aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat))))),one_one_int) ) ) ).

% ceiling_log2_div2
tff(fact_7539_complex__unimodular__polar,axiom,
    ! [Z2: complex] :
      ( ( aa_complex_real(real_V1022390504157884413omplex,Z2) = one_one_real )
     => ~ ! [T3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T3)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,T3),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))
             => ( Z2 != aa_real_complex(complex2(aa_real_real(cos_real,T3)),aa_real_real(sin_real,T3)) ) ) ) ) ).

% complex_unimodular_polar
tff(fact_7540_ceiling__log__eq__powr__iff,axiom,
    ! [X: real,B: real,K2: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => ( ( aa_real_int(archim7802044766580827645g_real,aa_real_real(log(B),X)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,K2)),one_one_int) )
        <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(powr_real(B),aa_nat_real(semiri5074537144036343181t_real,K2))),X)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(powr_real(B),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),one_one_nat)))) ) ) ) ) ).

% ceiling_log_eq_powr_iff
tff(fact_7541_cos__arcsin,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),one_one_real)
       => ( aa_real_real(cos_real,aa_real_real(arcsin,X)) = aa_real_real(sqrt,aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% cos_arcsin
tff(fact_7542_sin__arccos__abs,axiom,
    ! [Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y2)),one_one_real)
     => ( aa_real_real(sin_real,aa_real_real(arccos,Y2)) = aa_real_real(sqrt,aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% sin_arccos_abs
tff(fact_7543_sin__arccos,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),one_one_real)
       => ( aa_real_real(sin_real,aa_real_real(arccos,X)) = aa_real_real(sqrt,aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% sin_arccos
tff(fact_7544_powr__one__eq__one,axiom,
    ! [A: real] : aa_real_real(powr_real(one_one_real),A) = one_one_real ).

% powr_one_eq_one
tff(fact_7545_powr__zero__eq__one,axiom,
    ! [X: real] :
      aa_real_real(powr_real(X),zero_zero_real) = $ite(X = zero_zero_real,zero_zero_real,one_one_real) ).

% powr_zero_eq_one
tff(fact_7546_powr__nonneg__iff,axiom,
    ! [A: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(A),X)),zero_zero_real)
    <=> ( A = zero_zero_real ) ) ).

% powr_nonneg_iff
tff(fact_7547_powr__less__cancel__iff,axiom,
    ! [X: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(powr_real(X),A)),aa_real_real(powr_real(X),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% powr_less_cancel_iff
tff(fact_7548_arccos__1,axiom,
    aa_real_real(arccos,one_one_real) = zero_zero_real ).

% arccos_1
tff(fact_7549_powr__eq__one__iff,axiom,
    ! [A: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( ( aa_real_real(powr_real(A),X) = one_one_real )
      <=> ( X = zero_zero_real ) ) ) ).

% powr_eq_one_iff
tff(fact_7550_powr__one__gt__zero__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(powr_real(X),one_one_real) = X )
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X) ) ).

% powr_one_gt_zero_iff
tff(fact_7551_powr__one,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_real(powr_real(X),one_one_real) = X ) ) ).

% powr_one
tff(fact_7552_powr__le__cancel__iff,axiom,
    ! [X: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(X),A)),aa_real_real(powr_real(X),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% powr_le_cancel_iff
tff(fact_7553_numeral__powr__numeral__real,axiom,
    ! [Ma: num,Nb: num] : aa_real_real(powr_real(aa_num_real(numeral_numeral_real,Ma)),aa_num_real(numeral_numeral_real,Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Ma)),aa_num_nat(numeral_numeral_nat,Nb)) ).

% numeral_powr_numeral_real
tff(fact_7554_arccos__minus__1,axiom,
    aa_real_real(arccos,aa_real_real(uminus_uminus_real,one_one_real)) = pi ).

% arccos_minus_1
tff(fact_7555_log__powr__cancel,axiom,
    ! [A: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_real(log(A),aa_real_real(powr_real(A),Y2)) = Y2 ) ) ) ).

% log_powr_cancel
tff(fact_7556_powr__log__cancel,axiom,
    ! [A: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
         => ( aa_real_real(powr_real(A),aa_real_real(log(A),X)) = X ) ) ) ) ).

% powr_log_cancel
tff(fact_7557_cos__arccos,axiom,
    ! [Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),one_one_real)
       => ( aa_real_real(cos_real,aa_real_real(arccos,Y2)) = Y2 ) ) ) ).

% cos_arccos
tff(fact_7558_sin__arcsin,axiom,
    ! [Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),one_one_real)
       => ( aa_real_real(sin_real,aa_real_real(arcsin,Y2)) = Y2 ) ) ) ).

% sin_arcsin
tff(fact_7559_norm__cos__sin,axiom,
    ! [T: real] : aa_complex_real(real_V1022390504157884413omplex,aa_real_complex(complex2(aa_real_real(cos_real,T)),aa_real_real(sin_real,T))) = one_one_real ).

% norm_cos_sin
tff(fact_7560_powr__numeral,axiom,
    ! [X: real,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_real(powr_real(X),aa_num_real(numeral_numeral_real,Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,Nb)) ) ) ).

% powr_numeral
tff(fact_7561_arccos__0,axiom,
    aa_real_real(arccos,zero_zero_real) = aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% arccos_0
tff(fact_7562_arcsin__1,axiom,
    aa_real_real(arcsin,one_one_real) = aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% arcsin_1
tff(fact_7563_arcsin__minus__1,axiom,
    aa_real_real(arcsin,aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% arcsin_minus_1
tff(fact_7564_square__powr__half,axiom,
    ! [X: real] : aa_real_real(powr_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = aa_real_real(abs_abs_real,X) ).

% square_powr_half
tff(fact_7565_powr__powr,axiom,
    ! [X: real,A: real,B: real] : aa_real_real(powr_real(aa_real_real(powr_real(X),A)),B) = aa_real_real(powr_real(X),aa_real_real(times_times_real(A),B)) ).

% powr_powr
tff(fact_7566_powr__ge__pzero,axiom,
    ! [X: real,Y2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(powr_real(X),Y2)) ).

% powr_ge_pzero
tff(fact_7567_powr__mono2,axiom,
    ! [A: real,X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(X),A)),aa_real_real(powr_real(Y2),A)) ) ) ) ).

% powr_mono2
tff(fact_7568_powr__less__mono,axiom,
    ! [A: real,B: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(powr_real(X),A)),aa_real_real(powr_real(X),B)) ) ) ).

% powr_less_mono
tff(fact_7569_powr__less__cancel,axiom,
    ! [X: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(powr_real(X),A)),aa_real_real(powr_real(X),B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% powr_less_cancel
tff(fact_7570_powr__mono,axiom,
    ! [A: real,B: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),X)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(X),A)),aa_real_real(powr_real(X),B)) ) ) ).

% powr_mono
tff(fact_7571_Complex__eq__1,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_complex(complex2(A),B) = one_one_complex )
    <=> ( ( A = one_one_real )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_1
tff(fact_7572_one__complex_Ocode,axiom,
    one_one_complex = aa_real_complex(complex2(one_one_real),zero_zero_real) ).

% one_complex.code
tff(fact_7573_powr__mono2_H,axiom,
    ! [A: real,X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(Y2),A)),aa_real_real(powr_real(X),A)) ) ) ) ).

% powr_mono2'
tff(fact_7574_powr__less__mono2,axiom,
    ! [A: real,X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(powr_real(X),A)),aa_real_real(powr_real(Y2),A)) ) ) ) ).

% powr_less_mono2
tff(fact_7575_powr__inj,axiom,
    ! [A: real,X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( ( aa_real_real(powr_real(A),X) = aa_real_real(powr_real(A),Y2) )
        <=> ( X = Y2 ) ) ) ) ).

% powr_inj
tff(fact_7576_gr__one__powr,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(powr_real(X),Y2)) ) ) ).

% gr_one_powr
tff(fact_7577_ge__one__powr__ge__zero,axiom,
    ! [X: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(powr_real(X),A)) ) ) ).

% ge_one_powr_ge_zero
tff(fact_7578_powr__mono__both,axiom,
    ! [A: real,B: real,X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),X)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(X),A)),aa_real_real(powr_real(Y2),B)) ) ) ) ) ).

% powr_mono_both
tff(fact_7579_powr__le1,axiom,
    ! [A: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(X),A)),one_one_real) ) ) ) ).

% powr_le1
tff(fact_7580_powr__divide,axiom,
    ! [X: real,Y2: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y2)
       => ( aa_real_real(powr_real(aa_real_real(divide_divide_real(X),Y2)),A) = aa_real_real(divide_divide_real(aa_real_real(powr_real(X),A)),aa_real_real(powr_real(Y2),A)) ) ) ) ).

% powr_divide
tff(fact_7581_powr__mult,axiom,
    ! [X: real,Y2: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y2)
       => ( aa_real_real(powr_real(aa_real_real(times_times_real(X),Y2)),A) = aa_real_real(times_times_real(aa_real_real(powr_real(X),A)),aa_real_real(powr_real(Y2),A)) ) ) ) ).

% powr_mult
tff(fact_7582_divide__powr__uminus,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(divide_divide_real(A),aa_real_real(powr_real(B),C)) = aa_real_real(times_times_real(A),aa_real_real(powr_real(B),aa_real_real(uminus_uminus_real,C))) ).

% divide_powr_uminus
tff(fact_7583_ln__powr,axiom,
    ! [X: real,Y2: real] :
      ( ( X != zero_zero_real )
     => ( aa_real_real(ln_ln_real,aa_real_real(powr_real(X),Y2)) = aa_real_real(times_times_real(Y2),aa_real_real(ln_ln_real,X)) ) ) ).

% ln_powr
tff(fact_7584_log__powr,axiom,
    ! [X: real,B: real,Y2: real] :
      ( ( X != zero_zero_real )
     => ( aa_real_real(log(B),aa_real_real(powr_real(X),Y2)) = aa_real_real(times_times_real(Y2),aa_real_real(log(B),X)) ) ) ).

% log_powr
tff(fact_7585_Complex__eq__neg__1,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_complex(complex2(A),B) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) )
    <=> ( ( A = aa_real_real(uminus_uminus_real,one_one_real) )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_neg_1
tff(fact_7586_powr__add,axiom,
    ! [X: real,A: real,B: real] : aa_real_real(powr_real(X),aa_real_real(plus_plus_real(A),B)) = aa_real_real(times_times_real(aa_real_real(powr_real(X),A)),aa_real_real(powr_real(X),B)) ).

% powr_add
tff(fact_7587_powr__diff,axiom,
    ! [W: real,Z1: real,Z22: real] : aa_real_real(powr_real(W),aa_real_real(minus_minus_real(Z1),Z22)) = aa_real_real(divide_divide_real(aa_real_real(powr_real(W),Z1)),aa_real_real(powr_real(W),Z22)) ).

% powr_diff
tff(fact_7588_complex__mult,axiom,
    ! [A: real,B: real,C: real,D: real] : aa_complex_complex(times_times_complex(aa_real_complex(complex2(A),B)),aa_real_complex(complex2(C),D)) = aa_real_complex(complex2(aa_real_real(minus_minus_real(aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),D))),aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),D)),aa_real_real(times_times_real(B),C))) ).

% complex_mult
tff(fact_7589_arccos__le__arccos,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Y2)),aa_real_real(arccos,X)) ) ) ) ).

% arccos_le_arccos
tff(fact_7590_arccos__eq__iff,axiom,
    ! [X: real,Y2: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y2)),one_one_real) )
     => ( ( aa_real_real(arccos,X) = aa_real_real(arccos,Y2) )
      <=> ( X = Y2 ) ) ) ).

% arccos_eq_iff
tff(fact_7591_arccos__le__mono,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y2)),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,X)),aa_real_real(arccos,Y2))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),X) ) ) ) ).

% arccos_le_mono
tff(fact_7592_arcsin__minus,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),one_one_real)
       => ( aa_real_real(arcsin,aa_real_real(uminus_uminus_real,X)) = aa_real_real(uminus_uminus_real,aa_real_real(arcsin,X)) ) ) ) ).

% arcsin_minus
tff(fact_7593_arcsin__le__arcsin,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,X)),aa_real_real(arcsin,Y2)) ) ) ) ).

% arcsin_le_arcsin
tff(fact_7594_arcsin__eq__iff,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y2)),one_one_real)
       => ( ( aa_real_real(arcsin,X) = aa_real_real(arcsin,Y2) )
        <=> ( X = Y2 ) ) ) ) ).

% arcsin_eq_iff
tff(fact_7595_arcsin__le__mono,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y2)),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,X)),aa_real_real(arcsin,Y2))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y2) ) ) ) ).

% arcsin_le_mono
tff(fact_7596_less__log__iff,axiom,
    ! [B: real,X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),aa_real_real(log(B),X))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(powr_real(B),Y2)),X) ) ) ) ).

% less_log_iff
tff(fact_7597_log__less__iff,axiom,
    ! [B: real,X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(log(B),X)),Y2)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(powr_real(B),Y2)) ) ) ) ).

% log_less_iff
tff(fact_7598_less__powr__iff,axiom,
    ! [B: real,X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(powr_real(B),Y2))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(log(B),X)),Y2) ) ) ) ).

% less_powr_iff
tff(fact_7599_powr__less__iff,axiom,
    ! [B: real,X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(powr_real(B),Y2)),X)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),aa_real_real(log(B),X)) ) ) ) ).

% powr_less_iff
tff(fact_7600_powr__minus__divide,axiom,
    ! [X: real,A: real] : aa_real_real(powr_real(X),aa_real_real(uminus_uminus_real,A)) = aa_real_real(divide_divide_real(one_one_real),aa_real_real(powr_real(X),A)) ).

% powr_minus_divide
tff(fact_7601_arccos__lbound,axiom,
    ! [Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Y2)) ) ) ).

% arccos_lbound
tff(fact_7602_arccos__less__arccos,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arccos,Y2)),aa_real_real(arccos,X)) ) ) ) ).

% arccos_less_arccos
tff(fact_7603_arccos__less__mono,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y2)),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arccos,X)),aa_real_real(arccos,Y2))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),X) ) ) ) ).

% arccos_less_mono
tff(fact_7604_arccos__ubound,axiom,
    ! [Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Y2)),pi) ) ) ).

% arccos_ubound
tff(fact_7605_arccos__cos,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),pi)
       => ( aa_real_real(arccos,aa_real_real(cos_real,X)) = X ) ) ) ).

% arccos_cos
tff(fact_7606_powr__neg__one,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_real(powr_real(X),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(divide_divide_real(one_one_real),X) ) ) ).

% powr_neg_one
tff(fact_7607_arcsin__less__arcsin,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arcsin,X)),aa_real_real(arcsin,Y2)) ) ) ) ).

% arcsin_less_arcsin
tff(fact_7608_powr__mult__base,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_real(times_times_real(X),aa_real_real(powr_real(X),Y2)) = aa_real_real(powr_real(X),aa_real_real(plus_plus_real(one_one_real),Y2)) ) ) ).

% powr_mult_base
tff(fact_7609_arcsin__less__mono,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y2)),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arcsin,X)),aa_real_real(arcsin,Y2))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2) ) ) ) ).

% arcsin_less_mono
tff(fact_7610_le__log__iff,axiom,
    ! [B: real,X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),aa_real_real(log(B),X))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(B),Y2)),X) ) ) ) ).

% le_log_iff
tff(fact_7611_log__le__iff,axiom,
    ! [B: real,X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(log(B),X)),Y2)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(powr_real(B),Y2)) ) ) ) ).

% log_le_iff
tff(fact_7612_le__powr__iff,axiom,
    ! [B: real,X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(powr_real(B),Y2))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(log(B),X)),Y2) ) ) ) ).

% le_powr_iff
tff(fact_7613_powr__le__iff,axiom,
    ! [B: real,X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(B),Y2)),X)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),aa_real_real(log(B),X)) ) ) ) ).

% powr_le_iff
tff(fact_7614_cos__arccos__abs,axiom,
    ! [Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y2)),one_one_real)
     => ( aa_real_real(cos_real,aa_real_real(arccos,Y2)) = Y2 ) ) ).

% cos_arccos_abs
tff(fact_7615_arccos__cos__eq__abs,axiom,
    ! [Theta: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Theta)),pi)
     => ( aa_real_real(arccos,aa_real_real(cos_real,Theta)) = aa_real_real(abs_abs_real,Theta) ) ) ).

% arccos_cos_eq_abs
tff(fact_7616_ln__powr__bound,axiom,
    ! [X: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,X)),aa_real_real(divide_divide_real(aa_real_real(powr_real(X),A)),A)) ) ) ).

% ln_powr_bound
tff(fact_7617_ln__powr__bound2,axiom,
    ! [X: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(aa_real_real(ln_ln_real,X)),A)),aa_real_real(times_times_real(aa_real_real(powr_real(A),A)),X)) ) ) ).

% ln_powr_bound2
tff(fact_7618_add__log__eq__powr,axiom,
    ! [B: real,X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( ( B != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
         => ( aa_real_real(plus_plus_real(Y2),aa_real_real(log(B),X)) = aa_real_real(log(B),aa_real_real(times_times_real(aa_real_real(powr_real(B),Y2)),X)) ) ) ) ) ).

% add_log_eq_powr
tff(fact_7619_log__add__eq__powr,axiom,
    ! [B: real,X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( ( B != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
         => ( aa_real_real(plus_plus_real(aa_real_real(log(B),X)),Y2) = aa_real_real(log(B),aa_real_real(times_times_real(X),aa_real_real(powr_real(B),Y2))) ) ) ) ) ).

% log_add_eq_powr
tff(fact_7620_minus__log__eq__powr,axiom,
    ! [B: real,X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( ( B != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
         => ( aa_real_real(minus_minus_real(Y2),aa_real_real(log(B),X)) = aa_real_real(log(B),aa_real_real(divide_divide_real(aa_real_real(powr_real(B),Y2)),X)) ) ) ) ) ).

% minus_log_eq_powr
tff(fact_7621_arccos__lt__bounded,axiom,
    ! [Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Y2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(arccos,Y2))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arccos,Y2)),pi) ) ) ) ).

% arccos_lt_bounded
tff(fact_7622_arccos__bounded,axiom,
    ! [Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Y2))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Y2)),pi) ) ) ) ).

% arccos_bounded
tff(fact_7623_sin__arccos__nonzero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),one_one_real)
       => ( aa_real_real(sin_real,aa_real_real(arccos,X)) != zero_zero_real ) ) ) ).

% sin_arccos_nonzero
tff(fact_7624_arccos__cos2,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),X)
       => ( aa_real_real(arccos,aa_real_real(cos_real,X)) = aa_real_real(uminus_uminus_real,X) ) ) ) ).

% arccos_cos2
tff(fact_7625_arccos__minus,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),one_one_real)
       => ( aa_real_real(arccos,aa_real_real(uminus_uminus_real,X)) = aa_real_real(minus_minus_real(pi),aa_real_real(arccos,X)) ) ) ) ).

% arccos_minus
tff(fact_7626_cos__arcsin__nonzero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),one_one_real)
       => ( aa_real_real(cos_real,aa_real_real(arcsin,X)) != zero_zero_real ) ) ) ).

% cos_arcsin_nonzero
tff(fact_7627_powr__def,axiom,
    ! [X: real,A: real] :
      aa_real_real(powr_real(X),A) = $ite(X = zero_zero_real,zero_zero_real,aa_real_real(exp_real,aa_real_real(times_times_real(A),aa_real_real(ln_ln_real,X)))) ).

% powr_def
tff(fact_7628_log__minus__eq__powr,axiom,
    ! [B: real,X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
     => ( ( B != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
         => ( aa_real_real(minus_minus_real(aa_real_real(log(B),X)),Y2) = aa_real_real(log(B),aa_real_real(times_times_real(X),aa_real_real(powr_real(B),aa_real_real(uminus_uminus_real,Y2)))) ) ) ) ) ).

% log_minus_eq_powr
tff(fact_7629_arccos,axiom,
    ! [Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Y2))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Y2)),pi)
          & ( aa_real_real(cos_real,aa_real_real(arccos,Y2)) = Y2 ) ) ) ) ).

% arccos
tff(fact_7630_arccos__minus__abs,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real)
     => ( aa_real_real(arccos,aa_real_real(uminus_uminus_real,X)) = aa_real_real(minus_minus_real(pi),aa_real_real(arccos,X)) ) ) ).

% arccos_minus_abs
tff(fact_7631_complex__norm,axiom,
    ! [X: real,Y2: real] : aa_complex_real(real_V1022390504157884413omplex,aa_real_complex(complex2(X),Y2)) = aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% complex_norm
tff(fact_7632_powr__half__sqrt,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_real(powr_real(X),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = aa_real_real(sqrt,X) ) ) ).

% powr_half_sqrt
tff(fact_7633_powr__neg__numeral,axiom,
    ! [X: real,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_real(powr_real(X),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_real_real(divide_divide_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,Nb))) ) ) ).

% powr_neg_numeral
tff(fact_7634_arccos__le__pi2,axiom,
    ! [Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Y2)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ).

% arccos_le_pi2
tff(fact_7635_arcsin__lt__bounded,axiom,
    ! [Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Y2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Y2))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arcsin,Y2)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).

% arcsin_lt_bounded
tff(fact_7636_arcsin__bounded,axiom,
    ! [Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Y2))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Y2)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).

% arcsin_bounded
tff(fact_7637_arcsin__ubound,axiom,
    ! [Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Y2)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ).

% arcsin_ubound
tff(fact_7638_arcsin__lbound,axiom,
    ! [Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Y2)) ) ) ).

% arcsin_lbound
tff(fact_7639_arcsin__sin,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_real(arcsin,aa_real_real(sin_real,X)) = X ) ) ) ).

% arcsin_sin
tff(fact_7640_le__arcsin__iff,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Y2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),aa_real_real(arcsin,X))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Y2)),X) ) ) ) ) ) ).

% le_arcsin_iff
tff(fact_7641_arcsin__le__iff,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Y2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,X)),Y2)
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(sin_real,Y2)) ) ) ) ) ) ).

% arcsin_le_iff
tff(fact_7642_arcsin__pi,axiom,
    ! [Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Y2))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Y2)),pi)
          & ( aa_real_real(sin_real,aa_real_real(arcsin,Y2)) = Y2 ) ) ) ) ).

% arcsin_pi
tff(fact_7643_arcsin,axiom,
    ! [Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Y2))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Y2)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
          & ( aa_real_real(sin_real,aa_real_real(arcsin,Y2)) = Y2 ) ) ) ) ).

% arcsin
tff(fact_7644_arccos__cos__eq__abs__2pi,axiom,
    ! [Theta: real] :
      ~ ! [K: int] : aa_real_real(arccos,aa_real_real(cos_real,Theta)) != aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(Theta),aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,K)),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)))) ).

% arccos_cos_eq_abs_2pi
tff(fact_7645_arcosh__def,axiom,
    ! [X: real] : aa_real_real(arcosh_real,X) = aa_real_real(ln_ln_real,aa_real_real(plus_plus_real(X),aa_real_real(powr_real(aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)),aa_real_real(real_V1803761363581548252l_real,aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))))) ).

% arcosh_def
tff(fact_7646_geometric__deriv__sums,axiom,
    ! [Z2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Z2)),one_one_real)
     => aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_fx,Z2)),aa_real_real(divide_divide_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(one_one_real),Z2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% geometric_deriv_sums
tff(fact_7647_geometric__deriv__sums,axiom,
    ! [Z2: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Z2)),one_one_real)
     => aa_complex_o(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_fy,Z2)),aa_complex_complex(divide1717551699836669952omplex(one_one_complex),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(minus_minus_complex(one_one_complex),Z2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% geometric_deriv_sums
tff(fact_7648_arsinh__def,axiom,
    ! [X: real] : aa_real_real(arsinh_real,X) = aa_real_real(ln_ln_real,aa_real_real(plus_plus_real(X),aa_real_real(powr_real(aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)),aa_real_real(real_V1803761363581548252l_real,aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))))) ).

% arsinh_def
tff(fact_7649_floor__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
       => ( ( aa_real_int(archim6058952711729229775r_real,aa_real_real(log(aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K2))) = aa_nat_int(semiri1314217659103216013at_int,Nb) )
        <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)),K2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
tff(fact_7650_monoseq__def,axiom,
    ! [X7: fun_nat_real] :
      ( topolo6980174941875973593q_real(X7)
    <=> ( ! [M3: nat,N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M3),N4)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X7,M3)),aa_nat_real(X7,N4)) )
        | ! [M3: nat,N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M3),N4)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X7,N4)),aa_nat_real(X7,M3)) ) ) ) ).

% monoseq_def
tff(fact_7651_monoseq__def,axiom,
    ! [X7: fun_nat_set_int] :
      ( topolo3100542954746470799et_int(X7)
    <=> ( ! [M3: nat,N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M3),N4)
           => ord_less_eq_set_int(aa_nat_set_int(X7,M3),aa_nat_set_int(X7,N4)) )
        | ! [M3: nat,N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M3),N4)
           => ord_less_eq_set_int(aa_nat_set_int(X7,N4),aa_nat_set_int(X7,M3)) ) ) ) ).

% monoseq_def
tff(fact_7652_monoseq__def,axiom,
    ! [X7: fun_nat_rat] :
      ( topolo4267028734544971653eq_rat(X7)
    <=> ( ! [M3: nat,N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M3),N4)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X7,M3)),aa_nat_rat(X7,N4)) )
        | ! [M3: nat,N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M3),N4)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X7,N4)),aa_nat_rat(X7,M3)) ) ) ) ).

% monoseq_def
tff(fact_7653_monoseq__def,axiom,
    ! [X7: fun_nat_num] :
      ( topolo1459490580787246023eq_num(X7)
    <=> ( ! [M3: nat,N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M3),N4)
           => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(X7,M3)),aa_nat_num(X7,N4)) )
        | ! [M3: nat,N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M3),N4)
           => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(X7,N4)),aa_nat_num(X7,M3)) ) ) ) ).

% monoseq_def
tff(fact_7654_monoseq__def,axiom,
    ! [X7: fun_nat_nat] :
      ( topolo4902158794631467389eq_nat(X7)
    <=> ( ! [M3: nat,N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M3),N4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(X7,M3)),aa_nat_nat(X7,N4)) )
        | ! [M3: nat,N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M3),N4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(X7,N4)),aa_nat_nat(X7,M3)) ) ) ) ).

% monoseq_def
tff(fact_7655_monoseq__def,axiom,
    ! [X7: fun_nat_int] :
      ( topolo4899668324122417113eq_int(X7)
    <=> ( ! [M3: nat,N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M3),N4)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(X7,M3)),aa_nat_int(X7,N4)) )
        | ! [M3: nat,N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M3),N4)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(X7,N4)),aa_nat_int(X7,M3)) ) ) ) ).

% monoseq_def
tff(fact_7656_summable__complex__of__real,axiom,
    ! [F: fun_nat_real] :
      ( summable_complex(aTP_Lamm_fz(F))
    <=> summable_real(F) ) ).

% summable_complex_of_real
tff(fact_7657_of__int__floor__cancel,axiom,
    ! [X: real] :
      ( ( aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,X)) = X )
    <=> ? [N4: int] : X = aa_int_real(ring_1_of_int_real,N4) ) ).

% of_int_floor_cancel
tff(fact_7658_of__int__floor__cancel,axiom,
    ! [X: rat] :
      ( ( aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,X)) = X )
    <=> ? [N4: int] : X = aa_int_rat(ring_1_of_int_rat,N4) ) ).

% of_int_floor_cancel
tff(fact_7659_sums__zero,axiom,
    aa_complex_o(sums_complex(aTP_Lamm_eh),zero_zero_complex) ).

% sums_zero
tff(fact_7660_sums__zero,axiom,
    aa_real_o(sums_real(aTP_Lamm_ei),zero_zero_real) ).

% sums_zero
tff(fact_7661_sums__zero,axiom,
    aa_nat_o(sums_nat(aTP_Lamm_av),zero_zero_nat) ).

% sums_zero
tff(fact_7662_sums__zero,axiom,
    aa_int_o(sums_int(aTP_Lamm_ej),zero_zero_int) ).

% sums_zero
tff(fact_7663_of__real__1,axiom,
    aa_real_real(real_V1803761363581548252l_real,one_one_real) = one_one_real ).

% of_real_1
tff(fact_7664_of__real__1,axiom,
    aa_real_complex(real_V4546457046886955230omplex,one_one_real) = one_one_complex ).

% of_real_1
tff(fact_7665_of__real__eq__1__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(real_V1803761363581548252l_real,X) = one_one_real )
    <=> ( X = one_one_real ) ) ).

% of_real_eq_1_iff
tff(fact_7666_of__real__eq__1__iff,axiom,
    ! [X: real] :
      ( ( aa_real_complex(real_V4546457046886955230omplex,X) = one_one_complex )
    <=> ( X = one_one_real ) ) ).

% of_real_eq_1_iff
tff(fact_7667_of__real__mult,axiom,
    ! [X: real,Y2: real] : aa_real_real(real_V1803761363581548252l_real,aa_real_real(times_times_real(X),Y2)) = aa_real_real(times_times_real(aa_real_real(real_V1803761363581548252l_real,X)),aa_real_real(real_V1803761363581548252l_real,Y2)) ).

% of_real_mult
tff(fact_7668_of__real__mult,axiom,
    ! [X: real,Y2: real] : aa_real_complex(real_V4546457046886955230omplex,aa_real_real(times_times_real(X),Y2)) = aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,X)),aa_real_complex(real_V4546457046886955230omplex,Y2)) ).

% of_real_mult
tff(fact_7669_of__real__numeral,axiom,
    ! [W: num] : aa_real_real(real_V1803761363581548252l_real,aa_num_real(numeral_numeral_real,W)) = aa_num_real(numeral_numeral_real,W) ).

% of_real_numeral
tff(fact_7670_of__real__numeral,axiom,
    ! [W: num] : aa_real_complex(real_V4546457046886955230omplex,aa_num_real(numeral_numeral_real,W)) = numera6690914467698888265omplex(W) ).

% of_real_numeral
tff(fact_7671_floor__numeral,axiom,
    ! [V: num] : aa_real_int(archim6058952711729229775r_real,aa_num_real(numeral_numeral_real,V)) = aa_num_int(numeral_numeral_int,V) ).

% floor_numeral
tff(fact_7672_floor__numeral,axiom,
    ! [V: num] : aa_rat_int(archim3151403230148437115or_rat,aa_num_rat(numeral_numeral_rat,V)) = aa_num_int(numeral_numeral_int,V) ).

% floor_numeral
tff(fact_7673_of__real__divide,axiom,
    ! [X: real,Y2: real] : aa_real_real(real_V1803761363581548252l_real,aa_real_real(divide_divide_real(X),Y2)) = aa_real_real(divide_divide_real(aa_real_real(real_V1803761363581548252l_real,X)),aa_real_real(real_V1803761363581548252l_real,Y2)) ).

% of_real_divide
tff(fact_7674_of__real__divide,axiom,
    ! [X: real,Y2: real] : aa_real_complex(real_V4546457046886955230omplex,aa_real_real(divide_divide_real(X),Y2)) = aa_complex_complex(divide1717551699836669952omplex(aa_real_complex(real_V4546457046886955230omplex,X)),aa_real_complex(real_V4546457046886955230omplex,Y2)) ).

% of_real_divide
tff(fact_7675_floor__one,axiom,
    aa_real_int(archim6058952711729229775r_real,one_one_real) = one_one_int ).

% floor_one
tff(fact_7676_floor__one,axiom,
    aa_rat_int(archim3151403230148437115or_rat,one_one_rat) = one_one_int ).

% floor_one
tff(fact_7677_of__real__add,axiom,
    ! [X: real,Y2: real] : aa_real_real(real_V1803761363581548252l_real,aa_real_real(plus_plus_real(X),Y2)) = aa_real_real(plus_plus_real(aa_real_real(real_V1803761363581548252l_real,X)),aa_real_real(real_V1803761363581548252l_real,Y2)) ).

% of_real_add
tff(fact_7678_of__real__add,axiom,
    ! [X: real,Y2: real] : aa_real_complex(real_V4546457046886955230omplex,aa_real_real(plus_plus_real(X),Y2)) = aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,X)),aa_real_complex(real_V4546457046886955230omplex,Y2)) ).

% of_real_add
tff(fact_7679_of__real__power,axiom,
    ! [X: real,Nb: nat] : aa_real_real(real_V1803761363581548252l_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(real_V1803761363581548252l_real,X)),Nb) ).

% of_real_power
tff(fact_7680_of__real__power,axiom,
    ! [X: real,Nb: nat] : aa_real_complex(real_V4546457046886955230omplex,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Nb)) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_real_complex(real_V4546457046886955230omplex,X)),Nb) ).

% of_real_power
tff(fact_7681_zero__le__floor,axiom,
    ! [X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(archim6058952711729229775r_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X) ) ).

% zero_le_floor
tff(fact_7682_zero__le__floor,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_rat_int(archim3151403230148437115or_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X) ) ).

% zero_le_floor
tff(fact_7683_floor__less__zero,axiom,
    ! [X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim6058952711729229775r_real,X)),zero_zero_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),zero_zero_real) ) ).

% floor_less_zero
tff(fact_7684_floor__less__zero,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,X)),zero_zero_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),zero_zero_rat) ) ).

% floor_less_zero
tff(fact_7685_numeral__le__floor,axiom,
    ! [V: num,X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,V)),aa_real_int(archim6058952711729229775r_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,V)),X) ) ).

% numeral_le_floor
tff(fact_7686_numeral__le__floor,axiom,
    ! [V: num,X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,V)),aa_rat_int(archim3151403230148437115or_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,V)),X) ) ).

% numeral_le_floor
tff(fact_7687_zero__less__floor,axiom,
    ! [X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_real_int(archim6058952711729229775r_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),X) ) ).

% zero_less_floor
tff(fact_7688_zero__less__floor,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_rat_int(archim3151403230148437115or_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),X) ) ).

% zero_less_floor
tff(fact_7689_floor__le__zero,axiom,
    ! [X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X)),zero_zero_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),one_one_real) ) ).

% floor_le_zero
tff(fact_7690_floor__le__zero,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X)),zero_zero_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),one_one_rat) ) ).

% floor_le_zero
tff(fact_7691_floor__less__numeral,axiom,
    ! [X: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim6058952711729229775r_real,X)),aa_num_int(numeral_numeral_int,V))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_num_real(numeral_numeral_real,V)) ) ).

% floor_less_numeral
tff(fact_7692_floor__less__numeral,axiom,
    ! [X: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_num_int(numeral_numeral_int,V))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_num_rat(numeral_numeral_rat,V)) ) ).

% floor_less_numeral
tff(fact_7693_one__le__floor,axiom,
    ! [X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_real_int(archim6058952711729229775r_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),X) ) ).

% one_le_floor
tff(fact_7694_one__le__floor,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_rat_int(archim3151403230148437115or_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),X) ) ).

% one_le_floor
tff(fact_7695_floor__less__one,axiom,
    ! [X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim6058952711729229775r_real,X)),one_one_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),one_one_real) ) ).

% floor_less_one
tff(fact_7696_floor__less__one,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,X)),one_one_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),one_one_rat) ) ).

% floor_less_one
tff(fact_7697_floor__neg__numeral,axiom,
    ! [V: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)) ).

% floor_neg_numeral
tff(fact_7698_floor__neg__numeral,axiom,
    ! [V: num] : aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)) ).

% floor_neg_numeral
tff(fact_7699_of__real__neg__numeral,axiom,
    ! [W: num] : aa_real_real(real_V1803761363581548252l_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) ).

% of_real_neg_numeral
tff(fact_7700_of__real__neg__numeral,axiom,
    ! [W: num] : aa_real_complex(real_V4546457046886955230omplex,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) ).

% of_real_neg_numeral
tff(fact_7701_floor__diff__numeral,axiom,
    ! [X: real,V: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(minus_minus_real(X),aa_num_real(numeral_numeral_real,V))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(archim6058952711729229775r_real,X)),aa_num_int(numeral_numeral_int,V)) ).

% floor_diff_numeral
tff(fact_7702_floor__diff__numeral,axiom,
    ! [X: rat,V: num] : aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(minus_minus_rat(X),aa_num_rat(numeral_numeral_rat,V))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_num_int(numeral_numeral_int,V)) ).

% floor_diff_numeral
tff(fact_7703_cos__of__real__pi,axiom,
    aa_real_real(cos_real,aa_real_real(real_V1803761363581548252l_real,pi)) = aa_real_real(uminus_uminus_real,one_one_real) ).

% cos_of_real_pi
tff(fact_7704_cos__of__real__pi,axiom,
    aa_complex_complex(cos_complex,aa_real_complex(real_V4546457046886955230omplex,pi)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% cos_of_real_pi
tff(fact_7705_floor__diff__one,axiom,
    ! [X: real] : aa_real_int(archim6058952711729229775r_real,aa_real_real(minus_minus_real(X),one_one_real)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(archim6058952711729229775r_real,X)),one_one_int) ).

% floor_diff_one
tff(fact_7706_floor__diff__one,axiom,
    ! [X: rat] : aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(minus_minus_rat(X),one_one_rat)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_rat_int(archim3151403230148437115or_rat,X)),one_one_int) ).

% floor_diff_one
tff(fact_7707_floor__numeral__power,axiom,
    ! [X: num,Nb: nat] : aa_real_int(archim6058952711729229775r_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,X)),Nb)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X)),Nb) ).

% floor_numeral_power
tff(fact_7708_floor__numeral__power,axiom,
    ! [X: num,Nb: nat] : aa_rat_int(archim3151403230148437115or_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,X)),Nb)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X)),Nb) ).

% floor_numeral_power
tff(fact_7709_powser__sums__zero__iff,axiom,
    ! [A: fun_nat_complex,X: complex] :
      ( aa_complex_o(sums_complex(aTP_Lamm_ee(A)),X)
    <=> ( aa_nat_complex(A,zero_zero_nat) = X ) ) ).

% powser_sums_zero_iff
tff(fact_7710_powser__sums__zero__iff,axiom,
    ! [A: fun_nat_real,X: real] :
      ( aa_real_o(sums_real(aTP_Lamm_ef(A)),X)
    <=> ( aa_nat_real(A,zero_zero_nat) = X ) ) ).

% powser_sums_zero_iff
tff(fact_7711_floor__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(divide_divide_real(aa_num_real(numeral_numeral_real,A)),aa_num_real(numeral_numeral_real,B))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,A)),aa_num_int(numeral_numeral_int,B)) ).

% floor_divide_eq_div_numeral
tff(fact_7712_numeral__less__floor,axiom,
    ! [V: num,X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,V)),aa_real_int(archim6058952711729229775r_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_num_real(numeral_numeral_real,V)),one_one_real)),X) ) ).

% numeral_less_floor
tff(fact_7713_numeral__less__floor,axiom,
    ! [V: num,X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,V)),aa_rat_int(archim3151403230148437115or_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,V)),one_one_rat)),X) ) ).

% numeral_less_floor
tff(fact_7714_floor__le__numeral,axiom,
    ! [X: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X)),aa_num_int(numeral_numeral_int,V))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(plus_plus_real(aa_num_real(numeral_numeral_real,V)),one_one_real)) ) ).

% floor_le_numeral
tff(fact_7715_floor__le__numeral,axiom,
    ! [X: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_num_int(numeral_numeral_int,V))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,V)),one_one_rat)) ) ).

% floor_le_numeral
tff(fact_7716_one__less__floor,axiom,
    ! [X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_real_int(archim6058952711729229775r_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X) ) ).

% one_less_floor
tff(fact_7717_one__less__floor,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_rat_int(archim3151403230148437115or_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),X) ) ).

% one_less_floor
tff(fact_7718_floor__le__one,axiom,
    ! [X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X)),one_one_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ).

% floor_le_one
tff(fact_7719_floor__le__one,axiom,
    ! [X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X)),one_one_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))) ) ).

% floor_le_one
tff(fact_7720_neg__numeral__le__floor,axiom,
    ! [V: num,X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_real_int(archim6058952711729229775r_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),X) ) ).

% neg_numeral_le_floor
tff(fact_7721_neg__numeral__le__floor,axiom,
    ! [V: num,X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_rat_int(archim3151403230148437115or_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),X) ) ).

% neg_numeral_le_floor
tff(fact_7722_floor__less__neg__numeral,axiom,
    ! [X: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim6058952711729229775r_real,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))) ) ).

% floor_less_neg_numeral
tff(fact_7723_floor__less__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))) ) ).

% floor_less_neg_numeral
tff(fact_7724_norm__of__real__add1,axiom,
    ! [X: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(plus_plus_real(aa_real_real(real_V1803761363581548252l_real,X)),one_one_real)) = aa_real_real(abs_abs_real,aa_real_real(plus_plus_real(X),one_one_real)) ).

% norm_of_real_add1
tff(fact_7725_norm__of__real__add1,axiom,
    ! [X: real] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,X)),one_one_complex)) = aa_real_real(abs_abs_real,aa_real_real(plus_plus_real(X),one_one_real)) ).

% norm_of_real_add1
tff(fact_7726_norm__of__real__addn,axiom,
    ! [X: real,B: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(plus_plus_real(aa_real_real(real_V1803761363581548252l_real,X)),aa_num_real(numeral_numeral_real,B))) = aa_real_real(abs_abs_real,aa_real_real(plus_plus_real(X),aa_num_real(numeral_numeral_real,B))) ).

% norm_of_real_addn
tff(fact_7727_norm__of__real__addn,axiom,
    ! [X: real,B: num] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,X)),numera6690914467698888265omplex(B))) = aa_real_real(abs_abs_real,aa_real_real(plus_plus_real(X),aa_num_real(numeral_numeral_real,B))) ).

% norm_of_real_addn
tff(fact_7728_floor__one__divide__eq__div__numeral,axiom,
    ! [B: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,B))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_num_int(numeral_numeral_int,B)) ).

% floor_one_divide_eq_div_numeral
tff(fact_7729_floor__minus__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(aa_num_real(numeral_numeral_real,A)),aa_num_real(numeral_numeral_real,B)))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,A))),aa_num_int(numeral_numeral_int,B)) ).

% floor_minus_divide_eq_div_numeral
tff(fact_7730_cos__of__real__pi__half,axiom,
    aa_real_real(cos_real,aa_real_real(divide_divide_real(aa_real_real(real_V1803761363581548252l_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = zero_zero_real ).

% cos_of_real_pi_half
tff(fact_7731_cos__of__real__pi__half,axiom,
    aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(aa_real_complex(real_V4546457046886955230omplex,pi)),numera6690914467698888265omplex(aa_num_num(bit0,one)))) = zero_zero_complex ).

% cos_of_real_pi_half
tff(fact_7732_sin__of__real__pi__half,axiom,
    aa_real_real(sin_real,aa_real_real(divide_divide_real(aa_real_real(real_V1803761363581548252l_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = one_one_real ).

% sin_of_real_pi_half
tff(fact_7733_sin__of__real__pi__half,axiom,
    aa_complex_complex(sin_complex,aa_complex_complex(divide1717551699836669952omplex(aa_real_complex(real_V4546457046886955230omplex,pi)),numera6690914467698888265omplex(aa_num_num(bit0,one)))) = one_one_complex ).

% sin_of_real_pi_half
tff(fact_7734_neg__numeral__less__floor,axiom,
    ! [V: num,X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_real_int(archim6058952711729229775r_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),one_one_real)),X) ) ).

% neg_numeral_less_floor
tff(fact_7735_neg__numeral__less__floor,axiom,
    ! [V: num,X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_rat_int(archim3151403230148437115or_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),one_one_rat)),X) ) ).

% neg_numeral_less_floor
tff(fact_7736_floor__le__neg__numeral,axiom,
    ! [X: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),one_one_real)) ) ).

% floor_le_neg_numeral
tff(fact_7737_floor__le__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),one_one_rat)) ) ).

% floor_le_neg_numeral
tff(fact_7738_floor__minus__one__divide__eq__div__numeral,axiom,
    ! [B: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,B)))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,B)) ).

% floor_minus_one_divide_eq_div_numeral
tff(fact_7739_sums__le,axiom,
    ! [F: fun_nat_real,G: fun_nat_real,S: real,T: real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N)),aa_nat_real(G,N))
     => ( aa_real_o(sums_real(F),S)
       => ( aa_real_o(sums_real(G),T)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,S),T) ) ) ) ).

% sums_le
tff(fact_7740_sums__le,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat,S: nat,T: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,N)),aa_nat_nat(G,N))
     => ( aa_nat_o(sums_nat(F),S)
       => ( aa_nat_o(sums_nat(G),T)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,S),T) ) ) ) ).

% sums_le
tff(fact_7741_sums__le,axiom,
    ! [F: fun_nat_int,G: fun_nat_int,S: int,T: int] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,N)),aa_nat_int(G,N))
     => ( aa_int_o(sums_int(F),S)
       => ( aa_int_o(sums_int(G),T)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,S),T) ) ) ) ).

% sums_le
tff(fact_7742_sums__of__real,axiom,
    ! [X7: fun_nat_real,A: real] :
      ( aa_real_o(sums_real(X7),A)
     => aa_real_o(sums_real(aTP_Lamm_ga(X7)),aa_real_real(real_V1803761363581548252l_real,A)) ) ).

% sums_of_real
tff(fact_7743_sums__of__real,axiom,
    ! [X7: fun_nat_real,A: real] :
      ( aa_real_o(sums_real(X7),A)
     => aa_complex_o(sums_complex(aTP_Lamm_fz(X7)),aa_real_complex(real_V4546457046886955230omplex,A)) ) ).

% sums_of_real
tff(fact_7744_sums__of__real__iff,axiom,
    ! [F: fun_nat_real,C: real] :
      ( aa_real_o(sums_real(aTP_Lamm_ga(F)),aa_real_real(real_V1803761363581548252l_real,C))
    <=> aa_real_o(sums_real(F),C) ) ).

% sums_of_real_iff
tff(fact_7745_sums__of__real__iff,axiom,
    ! [F: fun_nat_real,C: real] :
      ( aa_complex_o(sums_complex(aTP_Lamm_fz(F)),aa_real_complex(real_V4546457046886955230omplex,C))
    <=> aa_real_o(sums_real(F),C) ) ).

% sums_of_real_iff
tff(fact_7746_complex__exp__exists,axiom,
    ! [Z2: complex] :
    ? [A3: complex,R3: real] : Z2 = aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,R3)),aa_complex_complex(exp_complex,A3)) ).

% complex_exp_exists
tff(fact_7747_sums__single,axiom,
    ! [I2: nat,F: fun_nat_complex] : aa_complex_o(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ek(I2),F)),aa_nat_complex(F,I2)) ).

% sums_single
tff(fact_7748_sums__single,axiom,
    ! [I2: nat,F: fun_nat_real] : aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_el(I2),F)),aa_nat_real(F,I2)) ).

% sums_single
tff(fact_7749_sums__single,axiom,
    ! [I2: nat,F: fun_nat_nat] : aa_nat_o(sums_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_em(I2),F)),aa_nat_nat(F,I2)) ).

% sums_single
tff(fact_7750_sums__single,axiom,
    ! [I2: nat,F: fun_nat_int] : aa_int_o(sums_int(aa_fun5565043122120104496at_int(aTP_Lamm_en(I2),F)),aa_nat_int(F,I2)) ).

% sums_single
tff(fact_7751_sums__mult,axiom,
    ! [F: fun_nat_complex,A: complex,C: complex] :
      ( aa_complex_o(sums_complex(F),A)
     => aa_complex_o(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_ey(F),C)),aa_complex_complex(times_times_complex(C),A)) ) ).

% sums_mult
tff(fact_7752_sums__mult,axiom,
    ! [F: fun_nat_real,A: real,C: real] :
      ( aa_real_o(sums_real(F),A)
     => aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_ez(F),C)),aa_real_real(times_times_real(C),A)) ) ).

% sums_mult
tff(fact_7753_sums__mult2,axiom,
    ! [F: fun_nat_complex,A: complex,C: complex] :
      ( aa_complex_o(sums_complex(F),A)
     => aa_complex_o(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_fa(F),C)),aa_complex_complex(times_times_complex(A),C)) ) ).

% sums_mult2
tff(fact_7754_sums__mult2,axiom,
    ! [F: fun_nat_real,A: real,C: real] :
      ( aa_real_o(sums_real(F),A)
     => aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_fb(F),C)),aa_real_real(times_times_real(A),C)) ) ).

% sums_mult2
tff(fact_7755_sums__add,axiom,
    ! [F: fun_nat_complex,A: complex,G: fun_nat_complex,B: complex] :
      ( aa_complex_o(sums_complex(F),A)
     => ( aa_complex_o(sums_complex(G),B)
       => aa_complex_o(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_fc(F),G)),aa_complex_complex(plus_plus_complex(A),B)) ) ) ).

% sums_add
tff(fact_7756_sums__add,axiom,
    ! [F: fun_nat_real,A: real,G: fun_nat_real,B: real] :
      ( aa_real_o(sums_real(F),A)
     => ( aa_real_o(sums_real(G),B)
       => aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_fd(F),G)),aa_real_real(plus_plus_real(A),B)) ) ) ).

% sums_add
tff(fact_7757_sums__add,axiom,
    ! [F: fun_nat_nat,A: nat,G: fun_nat_nat,B: nat] :
      ( aa_nat_o(sums_nat(F),A)
     => ( aa_nat_o(sums_nat(G),B)
       => aa_nat_o(sums_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_fe(F),G)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% sums_add
tff(fact_7758_sums__add,axiom,
    ! [F: fun_nat_int,A: int,G: fun_nat_int,B: int] :
      ( aa_int_o(sums_int(F),A)
     => ( aa_int_o(sums_int(G),B)
       => aa_int_o(sums_int(aa_fun5565043122120104496at_int(aTP_Lamm_ff(F),G)),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% sums_add
tff(fact_7759_sums__diff,axiom,
    ! [F: fun_nat_complex,A: complex,G: fun_nat_complex,B: complex] :
      ( aa_complex_o(sums_complex(F),A)
     => ( aa_complex_o(sums_complex(G),B)
       => aa_complex_o(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_fg(F),G)),aa_complex_complex(minus_minus_complex(A),B)) ) ) ).

% sums_diff
tff(fact_7760_sums__diff,axiom,
    ! [F: fun_nat_real,A: real,G: fun_nat_real,B: real] :
      ( aa_real_o(sums_real(F),A)
     => ( aa_real_o(sums_real(G),B)
       => aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_fh(F),G)),aa_real_real(minus_minus_real(A),B)) ) ) ).

% sums_diff
tff(fact_7761_sums__divide,axiom,
    ! [F: fun_nat_complex,A: complex,C: complex] :
      ( aa_complex_o(sums_complex(F),A)
     => aa_complex_o(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_es(F),C)),aa_complex_complex(divide1717551699836669952omplex(A),C)) ) ).

% sums_divide
tff(fact_7762_sums__divide,axiom,
    ! [F: fun_nat_real,A: real,C: real] :
      ( aa_real_o(sums_real(F),A)
     => aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_et(F),C)),aa_real_real(divide_divide_real(A),C)) ) ).

% sums_divide
tff(fact_7763_sums__minus,axiom,
    ! [F: fun_nat_real,A: real] :
      ( aa_real_o(sums_real(F),A)
     => aa_real_o(sums_real(aTP_Lamm_fi(F)),aa_real_real(uminus_uminus_real,A)) ) ).

% sums_minus
tff(fact_7764_sums__minus,axiom,
    ! [F: fun_nat_complex,A: complex] :
      ( aa_complex_o(sums_complex(F),A)
     => aa_complex_o(sums_complex(aTP_Lamm_fj(F)),aa_complex_complex(uminus1482373934393186551omplex,A)) ) ).

% sums_minus
tff(fact_7765_floor__mono,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X)),aa_real_int(archim6058952711729229775r_real,Y2)) ) ).

% floor_mono
tff(fact_7766_floor__mono,axiom,
    ! [X: rat,Y2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_rat_int(archim3151403230148437115or_rat,Y2)) ) ).

% floor_mono
tff(fact_7767_of__int__floor__le,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,X))),X) ).

% of_int_floor_le
tff(fact_7768_of__int__floor__le,axiom,
    ! [X: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,X))),X) ).

% of_int_floor_le
tff(fact_7769_floor__less__cancel,axiom,
    ! [X: real,Y2: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim6058952711729229775r_real,X)),aa_real_int(archim6058952711729229775r_real,Y2))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2) ) ).

% floor_less_cancel
tff(fact_7770_floor__less__cancel,axiom,
    ! [X: rat,Y2: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_rat_int(archim3151403230148437115or_rat,Y2))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y2) ) ).

% floor_less_cancel
tff(fact_7771_floor__le__ceiling,axiom,
    ! [X: real] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X)),aa_real_int(archim7802044766580827645g_real,X)) ).

% floor_le_ceiling
tff(fact_7772_floor__le__ceiling,axiom,
    ! [X: rat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_rat_int(archim2889992004027027881ng_rat,X)) ).

% floor_le_ceiling
tff(fact_7773_sums__mult__iff,axiom,
    ! [C: complex,F: fun_nat_complex,D: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_o(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_eq(C),F)),aa_complex_complex(times_times_complex(C),D))
      <=> aa_complex_o(sums_complex(F),D) ) ) ).

% sums_mult_iff
tff(fact_7774_sums__mult__iff,axiom,
    ! [C: real,F: fun_nat_real,D: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_er(C),F)),aa_real_real(times_times_real(C),D))
      <=> aa_real_o(sums_real(F),D) ) ) ).

% sums_mult_iff
tff(fact_7775_sums__mult2__iff,axiom,
    ! [C: complex,F: fun_nat_complex,D: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_o(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_gb(C),F)),aa_complex_complex(times_times_complex(D),C))
      <=> aa_complex_o(sums_complex(F),D) ) ) ).

% sums_mult2_iff
tff(fact_7776_sums__mult2__iff,axiom,
    ! [C: real,F: fun_nat_real,D: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_gc(C),F)),aa_real_real(times_times_real(D),C))
      <=> aa_real_o(sums_real(F),D) ) ) ).

% sums_mult2_iff
tff(fact_7777_complex__of__real__mult__Complex,axiom,
    ! [R: real,X: real,Y2: real] : aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,R)),aa_real_complex(complex2(X),Y2)) = aa_real_complex(complex2(aa_real_real(times_times_real(R),X)),aa_real_real(times_times_real(R),Y2)) ).

% complex_of_real_mult_Complex
tff(fact_7778_Complex__mult__complex__of__real,axiom,
    ! [X: real,Y2: real,R: real] : aa_complex_complex(times_times_complex(aa_real_complex(complex2(X),Y2)),aa_real_complex(real_V4546457046886955230omplex,R)) = aa_real_complex(complex2(aa_real_real(times_times_real(X),R)),aa_real_real(times_times_real(Y2),R)) ).

% Complex_mult_complex_of_real
tff(fact_7779_floor__le__round,axiom,
    ! [X: real] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X)),aa_real_int(archim8280529875227126926d_real,X)) ).

% floor_le_round
tff(fact_7780_floor__le__round,axiom,
    ! [X: rat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_rat_int(archim7778729529865785530nd_rat,X)) ).

% floor_le_round
tff(fact_7781_summable__of__real,axiom,
    ! [X7: fun_nat_real] :
      ( summable_real(X7)
     => summable_real(aTP_Lamm_ga(X7)) ) ).

% summable_of_real
tff(fact_7782_summable__of__real,axiom,
    ! [X7: fun_nat_real] :
      ( summable_real(X7)
     => summable_complex(aTP_Lamm_fz(X7)) ) ).

% summable_of_real
tff(fact_7783_le__floor__iff,axiom,
    ! [Z2: int,X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z2),aa_real_int(archim6058952711729229775r_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z2)),X) ) ).

% le_floor_iff
tff(fact_7784_le__floor__iff,axiom,
    ! [Z2: int,X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z2),aa_rat_int(archim3151403230148437115or_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z2)),X) ) ).

% le_floor_iff
tff(fact_7785_floor__less__iff,axiom,
    ! [X: real,Z2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim6058952711729229775r_real,X)),Z2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_int_real(ring_1_of_int_real,Z2)) ) ).

% floor_less_iff
tff(fact_7786_floor__less__iff,axiom,
    ! [X: rat,Z2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,X)),Z2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,Z2)) ) ).

% floor_less_iff
tff(fact_7787_nonzero__of__real__divide,axiom,
    ! [Y2: real,X: real] :
      ( ( Y2 != zero_zero_real )
     => ( aa_real_real(real_V1803761363581548252l_real,aa_real_real(divide_divide_real(X),Y2)) = aa_real_real(divide_divide_real(aa_real_real(real_V1803761363581548252l_real,X)),aa_real_real(real_V1803761363581548252l_real,Y2)) ) ) ).

% nonzero_of_real_divide
tff(fact_7788_nonzero__of__real__divide,axiom,
    ! [Y2: real,X: real] :
      ( ( Y2 != zero_zero_real )
     => ( aa_real_complex(real_V4546457046886955230omplex,aa_real_real(divide_divide_real(X),Y2)) = aa_complex_complex(divide1717551699836669952omplex(aa_real_complex(real_V4546457046886955230omplex,X)),aa_real_complex(real_V4546457046886955230omplex,Y2)) ) ) ).

% nonzero_of_real_divide
tff(fact_7789_le__floor__add,axiom,
    ! [X: real,Y2: real] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,X)),aa_real_int(archim6058952711729229775r_real,Y2))),aa_real_int(archim6058952711729229775r_real,aa_real_real(plus_plus_real(X),Y2))) ).

% le_floor_add
tff(fact_7790_le__floor__add,axiom,
    ! [X: rat,Y2: rat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_rat_int(archim3151403230148437115or_rat,Y2))),aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(plus_plus_rat(X),Y2))) ).

% le_floor_add
tff(fact_7791_int__add__floor,axiom,
    ! [Z2: int,X: real] : aa_int_int(aa_int_fun_int_int(plus_plus_int,Z2),aa_real_int(archim6058952711729229775r_real,X)) = aa_real_int(archim6058952711729229775r_real,aa_real_real(plus_plus_real(aa_int_real(ring_1_of_int_real,Z2)),X)) ).

% int_add_floor
tff(fact_7792_int__add__floor,axiom,
    ! [Z2: int,X: rat] : aa_int_int(aa_int_fun_int_int(plus_plus_int,Z2),aa_rat_int(archim3151403230148437115or_rat,X)) = aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,Z2)),X)) ).

% int_add_floor
tff(fact_7793_floor__add__int,axiom,
    ! [X: real,Z2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,X)),Z2) = aa_real_int(archim6058952711729229775r_real,aa_real_real(plus_plus_real(X),aa_int_real(ring_1_of_int_real,Z2))) ).

% floor_add_int
tff(fact_7794_floor__add__int,axiom,
    ! [X: rat,Z2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X)),Z2) = aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(plus_plus_rat(X),aa_int_rat(ring_1_of_int_rat,Z2))) ).

% floor_add_int
tff(fact_7795_floor__divide__of__int__eq,axiom,
    ! [K2: int,L: int] : aa_real_int(archim6058952711729229775r_real,aa_real_real(divide_divide_real(aa_int_real(ring_1_of_int_real,K2)),aa_int_real(ring_1_of_int_real,L))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),L) ).

% floor_divide_of_int_eq
tff(fact_7796_floor__divide__of__int__eq,axiom,
    ! [K2: int,L: int] : aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(divide_divide_rat(aa_int_rat(ring_1_of_int_rat,K2)),aa_int_rat(ring_1_of_int_rat,L))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),L) ).

% floor_divide_of_int_eq
tff(fact_7797_sums__mult__D,axiom,
    ! [C: complex,F: fun_nat_complex,A: complex] :
      ( aa_complex_o(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_eq(C),F)),A)
     => ( ( C != zero_zero_complex )
       => aa_complex_o(sums_complex(F),aa_complex_complex(divide1717551699836669952omplex(A),C)) ) ) ).

% sums_mult_D
tff(fact_7798_sums__mult__D,axiom,
    ! [C: real,F: fun_nat_real,A: real] :
      ( aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_er(C),F)),A)
     => ( ( C != zero_zero_real )
       => aa_real_o(sums_real(F),aa_real_real(divide_divide_real(A),C)) ) ) ).

% sums_mult_D
tff(fact_7799_sums__Suc__imp,axiom,
    ! [F: fun_nat_complex,S: complex] :
      ( ( aa_nat_complex(F,zero_zero_nat) = zero_zero_complex )
     => ( aa_complex_o(sums_complex(aTP_Lamm_fl(F)),S)
       => aa_complex_o(sums_complex(F),S) ) ) ).

% sums_Suc_imp
tff(fact_7800_sums__Suc__imp,axiom,
    ! [F: fun_nat_real,S: real] :
      ( ( aa_nat_real(F,zero_zero_nat) = zero_zero_real )
     => ( aa_real_o(sums_real(aTP_Lamm_fk(F)),S)
       => aa_real_o(sums_real(F),S) ) ) ).

% sums_Suc_imp
tff(fact_7801_floor__power,axiom,
    ! [X: real,Nb: nat] :
      ( ( X = aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,X)) )
     => ( aa_real_int(archim6058952711729229775r_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Nb)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_real_int(archim6058952711729229775r_real,X)),Nb) ) ) ).

% floor_power
tff(fact_7802_floor__power,axiom,
    ! [X: rat,Nb: nat] :
      ( ( X = aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,X)) )
     => ( aa_rat_int(archim3151403230148437115or_rat,aa_nat_rat(power_power_rat(X),Nb)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_rat_int(archim3151403230148437115or_rat,X)),Nb) ) ) ).

% floor_power
tff(fact_7803_sums__Suc,axiom,
    ! [F: fun_nat_complex,L: complex] :
      ( aa_complex_o(sums_complex(aTP_Lamm_fl(F)),L)
     => aa_complex_o(sums_complex(F),aa_complex_complex(plus_plus_complex(L),aa_nat_complex(F,zero_zero_nat))) ) ).

% sums_Suc
tff(fact_7804_sums__Suc,axiom,
    ! [F: fun_nat_real,L: real] :
      ( aa_real_o(sums_real(aTP_Lamm_fk(F)),L)
     => aa_real_o(sums_real(F),aa_real_real(plus_plus_real(L),aa_nat_real(F,zero_zero_nat))) ) ).

% sums_Suc
tff(fact_7805_sums__Suc,axiom,
    ! [F: fun_nat_nat,L: nat] :
      ( aa_nat_o(sums_nat(aTP_Lamm_gd(F)),L)
     => aa_nat_o(sums_nat(F),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,L),aa_nat_nat(F,zero_zero_nat))) ) ).

% sums_Suc
tff(fact_7806_sums__Suc,axiom,
    ! [F: fun_nat_int,L: int] :
      ( aa_int_o(sums_int(aTP_Lamm_ge(F)),L)
     => aa_int_o(sums_int(F),aa_int_int(aa_int_fun_int_int(plus_plus_int,L),aa_nat_int(F,zero_zero_nat))) ) ).

% sums_Suc
tff(fact_7807_sums__Suc__iff,axiom,
    ! [F: fun_nat_complex,S: complex] :
      ( aa_complex_o(sums_complex(aTP_Lamm_fl(F)),S)
    <=> aa_complex_o(sums_complex(F),aa_complex_complex(plus_plus_complex(S),aa_nat_complex(F,zero_zero_nat))) ) ).

% sums_Suc_iff
tff(fact_7808_sums__Suc__iff,axiom,
    ! [F: fun_nat_real,S: real] :
      ( aa_real_o(sums_real(aTP_Lamm_fk(F)),S)
    <=> aa_real_o(sums_real(F),aa_real_real(plus_plus_real(S),aa_nat_real(F,zero_zero_nat))) ) ).

% sums_Suc_iff
tff(fact_7809_sums__zero__iff__shift,axiom,
    ! [Nb: nat,F: fun_nat_complex,S: complex] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Nb)
         => ( aa_nat_complex(F,I3) = zero_zero_complex ) )
     => ( aa_complex_o(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_gf(Nb),F)),S)
      <=> aa_complex_o(sums_complex(F),S) ) ) ).

% sums_zero_iff_shift
tff(fact_7810_sums__zero__iff__shift,axiom,
    ! [Nb: nat,F: fun_nat_real,S: real] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Nb)
         => ( aa_nat_real(F,I3) = zero_zero_real ) )
     => ( aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_gg(Nb),F)),S)
      <=> aa_real_o(sums_real(F),S) ) ) ).

% sums_zero_iff_shift
tff(fact_7811_suminf__of__real,axiom,
    ! [X7: fun_nat_real] :
      ( summable_real(X7)
     => ( aa_real_real(real_V1803761363581548252l_real,suminf_real(X7)) = suminf_real(aTP_Lamm_ga(X7)) ) ) ).

% suminf_of_real
tff(fact_7812_suminf__of__real,axiom,
    ! [X7: fun_nat_real] :
      ( summable_real(X7)
     => ( aa_real_complex(real_V4546457046886955230omplex,suminf_real(X7)) = suminf_complex(aTP_Lamm_fz(X7)) ) ) ).

% suminf_of_real
tff(fact_7813_norm__less__p1,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X)),aa_real_real(real_V7735802525324610683m_real,aa_real_real(plus_plus_real(aa_real_real(real_V1803761363581548252l_real,aa_real_real(real_V7735802525324610683m_real,X))),one_one_real))) ).

% norm_less_p1
tff(fact_7814_norm__less__p1,axiom,
    ! [X: complex] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X)),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,aa_complex_real(real_V1022390504157884413omplex,X))),one_one_complex))) ).

% norm_less_p1
tff(fact_7815_one__add__floor,axiom,
    ! [X: real] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,X)),one_one_int) = aa_real_int(archim6058952711729229775r_real,aa_real_real(plus_plus_real(X),one_one_real)) ).

% one_add_floor
tff(fact_7816_one__add__floor,axiom,
    ! [X: rat] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X)),one_one_int) = aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(plus_plus_rat(X),one_one_rat)) ).

% one_add_floor
tff(fact_7817_powser__sums__if,axiom,
    ! [Ma: nat,Z2: complex] : aa_complex_o(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_gh(Ma),Z2)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Z2),Ma)) ).

% powser_sums_if
tff(fact_7818_powser__sums__if,axiom,
    ! [Ma: nat,Z2: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_gi(Ma),Z2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z2),Ma)) ).

% powser_sums_if
tff(fact_7819_powser__sums__if,axiom,
    ! [Ma: nat,Z2: int] : aa_int_o(sums_int(aa_int_fun_nat_int(aTP_Lamm_gj(Ma),Z2)),aa_nat_int(aa_int_fun_nat_int(power_power_int,Z2),Ma)) ).

% powser_sums_if
tff(fact_7820_powser__sums__zero,axiom,
    ! [A: fun_nat_complex] : aa_complex_o(sums_complex(aTP_Lamm_ee(A)),aa_nat_complex(A,zero_zero_nat)) ).

% powser_sums_zero
tff(fact_7821_powser__sums__zero,axiom,
    ! [A: fun_nat_real] : aa_real_o(sums_real(aTP_Lamm_ef(A)),aa_nat_real(A,zero_zero_nat)) ).

% powser_sums_zero
tff(fact_7822_floor__divide__of__nat__eq,axiom,
    ! [Ma: nat,Nb: nat] : aa_real_int(archim6058952711729229775r_real,aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,Ma)),aa_nat_real(semiri5074537144036343181t_real,Nb))) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),Nb)) ).

% floor_divide_of_nat_eq
tff(fact_7823_floor__divide__of__nat__eq,axiom,
    ! [Ma: nat,Nb: nat] : aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(divide_divide_rat(aa_nat_rat(semiri681578069525770553at_rat,Ma)),aa_nat_rat(semiri681578069525770553at_rat,Nb))) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),Nb)) ).

% floor_divide_of_nat_eq
tff(fact_7824_ceiling__altdef,axiom,
    ! [X: real] :
      aa_real_int(archim7802044766580827645g_real,X) = $ite(X = aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,X)),aa_real_int(archim6058952711729229775r_real,X),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,X)),one_one_int)) ).

% ceiling_altdef
tff(fact_7825_ceiling__altdef,axiom,
    ! [X: rat] :
      aa_rat_int(archim2889992004027027881ng_rat,X) = $ite(X = aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,X)),aa_rat_int(archim3151403230148437115or_rat,X),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X)),one_one_int)) ).

% ceiling_altdef
tff(fact_7826_ceiling__diff__floor__le__1,axiom,
    ! [X: real] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(archim7802044766580827645g_real,X)),aa_real_int(archim6058952711729229775r_real,X))),one_one_int) ).

% ceiling_diff_floor_le_1
tff(fact_7827_ceiling__diff__floor__le__1,axiom,
    ! [X: rat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_rat_int(archim2889992004027027881ng_rat,X)),aa_rat_int(archim3151403230148437115or_rat,X))),one_one_int) ).

% ceiling_diff_floor_le_1
tff(fact_7828_floor__eq,axiom,
    ! [Nb: int,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Nb)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(plus_plus_real(aa_int_real(ring_1_of_int_real,Nb)),one_one_real))
       => ( aa_real_int(archim6058952711729229775r_real,X) = Nb ) ) ) ).

% floor_eq
tff(fact_7829_real__of__int__floor__add__one__gt,axiom,
    ! [R: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,R),aa_real_real(plus_plus_real(aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R))),one_one_real)) ).

% real_of_int_floor_add_one_gt
tff(fact_7830_real__of__int__floor__add__one__ge,axiom,
    ! [R: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,R),aa_real_real(plus_plus_real(aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R))),one_one_real)) ).

% real_of_int_floor_add_one_ge
tff(fact_7831_real__of__int__floor__gt__diff__one,axiom,
    ! [R: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(R),one_one_real)),aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R))) ).

% real_of_int_floor_gt_diff_one
tff(fact_7832_real__of__int__floor__ge__diff__one,axiom,
    ! [R: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(R),one_one_real)),aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R))) ).

% real_of_int_floor_ge_diff_one
tff(fact_7833_floor__split,axiom,
    ! [P: fun_int_o,T: real] :
      ( aa_int_o(P,aa_real_int(archim6058952711729229775r_real,T))
    <=> ! [I: int] :
          ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,I)),T)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,T),aa_real_real(plus_plus_real(aa_int_real(ring_1_of_int_real,I)),one_one_real)) )
         => aa_int_o(P,I) ) ) ).

% floor_split
tff(fact_7834_floor__split,axiom,
    ! [P: fun_int_o,T: rat] :
      ( aa_int_o(P,aa_rat_int(archim3151403230148437115or_rat,T))
    <=> ! [I: int] :
          ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,I)),T)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,T),aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,I)),one_one_rat)) )
         => aa_int_o(P,I) ) ) ).

% floor_split
tff(fact_7835_floor__eq__iff,axiom,
    ! [X: real,A: int] :
      ( ( aa_real_int(archim6058952711729229775r_real,X) = A )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,A)),X)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(plus_plus_real(aa_int_real(ring_1_of_int_real,A)),one_one_real)) ) ) ).

% floor_eq_iff
tff(fact_7836_floor__eq__iff,axiom,
    ! [X: rat,A: int] :
      ( ( aa_rat_int(archim3151403230148437115or_rat,X) = A )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A)),X)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,A)),one_one_rat)) ) ) ).

% floor_eq_iff
tff(fact_7837_floor__unique,axiom,
    ! [Z2: int,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z2)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(plus_plus_real(aa_int_real(ring_1_of_int_real,Z2)),one_one_real))
       => ( aa_real_int(archim6058952711729229775r_real,X) = Z2 ) ) ) ).

% floor_unique
tff(fact_7838_floor__unique,axiom,
    ! [Z2: int,X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z2)),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,Z2)),one_one_rat))
       => ( aa_rat_int(archim3151403230148437115or_rat,X) = Z2 ) ) ) ).

% floor_unique
tff(fact_7839_le__mult__floor,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(archim6058952711729229775r_real,A)),aa_real_int(archim6058952711729229775r_real,B))),aa_real_int(archim6058952711729229775r_real,aa_real_real(times_times_real(A),B))) ) ) ).

% le_mult_floor
tff(fact_7840_le__mult__floor,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_rat_int(archim3151403230148437115or_rat,A)),aa_rat_int(archim3151403230148437115or_rat,B))),aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(times_times_rat(A),B))) ) ) ).

% le_mult_floor
tff(fact_7841_less__floor__iff,axiom,
    ! [Z2: int,X: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),aa_real_int(archim6058952711729229775r_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_int_real(ring_1_of_int_real,Z2)),one_one_real)),X) ) ).

% less_floor_iff
tff(fact_7842_less__floor__iff,axiom,
    ! [Z2: int,X: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),aa_rat_int(archim3151403230148437115or_rat,X))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,Z2)),one_one_rat)),X) ) ).

% less_floor_iff
tff(fact_7843_floor__le__iff,axiom,
    ! [X: real,Z2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,X)),Z2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(plus_plus_real(aa_int_real(ring_1_of_int_real,Z2)),one_one_real)) ) ).

% floor_le_iff
tff(fact_7844_floor__le__iff,axiom,
    ! [X: rat,Z2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,X)),Z2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,Z2)),one_one_rat)) ) ).

% floor_le_iff
tff(fact_7845_floor__correct,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,X))),X)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,X)),one_one_int))) ) ).

% floor_correct
tff(fact_7846_floor__correct,axiom,
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,X))),X)
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X)),one_one_int))) ) ).

% floor_correct
tff(fact_7847_norm__of__real__diff,axiom,
    ! [B: real,A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(aa_real_real(real_V1803761363581548252l_real,B)),aa_real_real(real_V1803761363581548252l_real,A)))),aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(B),A))) ).

% norm_of_real_diff
tff(fact_7848_norm__of__real__diff,axiom,
    ! [B: real,A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_real_complex(real_V4546457046886955230omplex,B)),aa_real_complex(real_V4546457046886955230omplex,A)))),aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(B),A))) ).

% norm_of_real_diff
tff(fact_7849_floor__eq2,axiom,
    ! [Nb: int,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Nb)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(plus_plus_real(aa_int_real(ring_1_of_int_real,Nb)),one_one_real))
       => ( aa_real_int(archim6058952711729229775r_real,X) = Nb ) ) ) ).

% floor_eq2
tff(fact_7850_floor__divide__real__eq__div,axiom,
    ! [B: int,A: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
     => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(divide_divide_real(A),aa_int_real(ring_1_of_int_real,B))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_real_int(archim6058952711729229775r_real,A)),B) ) ) ).

% floor_divide_real_eq_div
tff(fact_7851_floor__divide__lower,axiom,
    ! [Q: real,P2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Q)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,aa_real_real(divide_divide_real(P2),Q)))),Q)),P2) ) ).

% floor_divide_lower
tff(fact_7852_floor__divide__lower,axiom,
    ! [Q: rat,P2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Q)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(divide_divide_rat(P2),Q)))),Q)),P2) ) ).

% floor_divide_lower
tff(fact_7853_cos__int__times__real,axiom,
    ! [Ma: int,X: real] : aa_real_real(cos_real,aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,Ma)),aa_real_real(real_V1803761363581548252l_real,X))) = aa_real_real(real_V1803761363581548252l_real,aa_real_real(cos_real,aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,Ma)),X))) ).

% cos_int_times_real
tff(fact_7854_cos__int__times__real,axiom,
    ! [Ma: int,X: real] : aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(aa_int_complex(ring_17405671764205052669omplex,Ma)),aa_real_complex(real_V4546457046886955230omplex,X))) = aa_real_complex(real_V4546457046886955230omplex,aa_real_real(cos_real,aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,Ma)),X))) ).

% cos_int_times_real
tff(fact_7855_sin__int__times__real,axiom,
    ! [Ma: int,X: real] : aa_real_real(sin_real,aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,Ma)),aa_real_real(real_V1803761363581548252l_real,X))) = aa_real_real(real_V1803761363581548252l_real,aa_real_real(sin_real,aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,Ma)),X))) ).

% sin_int_times_real
tff(fact_7856_sin__int__times__real,axiom,
    ! [Ma: int,X: real] : aa_complex_complex(sin_complex,aa_complex_complex(times_times_complex(aa_int_complex(ring_17405671764205052669omplex,Ma)),aa_real_complex(real_V4546457046886955230omplex,X))) = aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sin_real,aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,Ma)),X))) ).

% sin_int_times_real
tff(fact_7857_floor__divide__upper,axiom,
    ! [Q: real,P2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Q)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,P2),aa_real_real(times_times_real(aa_real_real(plus_plus_real(aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,aa_real_real(divide_divide_real(P2),Q)))),one_one_real)),Q)) ) ).

% floor_divide_upper
tff(fact_7858_floor__divide__upper,axiom,
    ! [Q: rat,P2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Q)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,P2),aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(divide_divide_rat(P2),Q)))),one_one_rat)),Q)) ) ).

% floor_divide_upper
tff(fact_7859_geometric__sums,axiom,
    ! [C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,C)),one_one_real)
     => aa_real_o(sums_real(aa_real_fun_nat_real(power_power_real,C)),aa_real_real(divide_divide_real(one_one_real),aa_real_real(minus_minus_real(one_one_real),C))) ) ).

% geometric_sums
tff(fact_7860_geometric__sums,axiom,
    ! [C: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,C)),one_one_real)
     => aa_complex_o(sums_complex(aa_com8181001469867462503omplex(power_power_complex,C)),aa_complex_complex(divide1717551699836669952omplex(one_one_complex),aa_complex_complex(minus_minus_complex(one_one_complex),C))) ) ).

% geometric_sums
tff(fact_7861_power__half__series,axiom,
    aa_real_o(sums_real(aTP_Lamm_gk),one_one_real) ).

% power_half_series
tff(fact_7862_round__def,axiom,
    ! [X: real] : aa_real_int(archim8280529875227126926d_real,X) = aa_real_int(archim6058952711729229775r_real,aa_real_real(plus_plus_real(X),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% round_def
tff(fact_7863_round__def,axiom,
    ! [X: rat] : aa_rat_int(archim7778729529865785530nd_rat,X) = aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(plus_plus_rat(X),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))) ).

% round_def
tff(fact_7864_sums__if_H,axiom,
    ! [G: fun_nat_real,X: real] :
      ( aa_real_o(sums_real(G),X)
     => aa_real_o(sums_real(aTP_Lamm_gl(G)),X) ) ).

% sums_if'
tff(fact_7865_cos__sin__eq,axiom,
    ! [X: real] : aa_real_real(cos_real,X) = aa_real_real(sin_real,aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_real_real(real_V1803761363581548252l_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),X)) ).

% cos_sin_eq
tff(fact_7866_cos__sin__eq,axiom,
    ! [X: complex] : aa_complex_complex(cos_complex,X) = aa_complex_complex(sin_complex,aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(aa_real_complex(real_V4546457046886955230omplex,pi)),numera6690914467698888265omplex(aa_num_num(bit0,one)))),X)) ).

% cos_sin_eq
tff(fact_7867_sin__cos__eq,axiom,
    ! [X: real] : aa_real_real(sin_real,X) = aa_real_real(cos_real,aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_real_real(real_V1803761363581548252l_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),X)) ).

% sin_cos_eq
tff(fact_7868_sin__cos__eq,axiom,
    ! [X: complex] : aa_complex_complex(sin_complex,X) = aa_complex_complex(cos_complex,aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(aa_real_complex(real_V4546457046886955230omplex,pi)),numera6690914467698888265omplex(aa_num_num(bit0,one)))),X)) ).

% sin_cos_eq
tff(fact_7869_sums__if,axiom,
    ! [G: fun_nat_real,X: real,F: fun_nat_real,Y2: real] :
      ( aa_real_o(sums_real(G),X)
     => ( aa_real_o(sums_real(F),Y2)
       => aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_gm(G),F)),aa_real_real(plus_plus_real(X),Y2)) ) ) ).

% sums_if
tff(fact_7870_minus__sin__cos__eq,axiom,
    ! [X: real] : aa_real_real(uminus_uminus_real,aa_real_real(sin_real,X)) = aa_real_real(cos_real,aa_real_real(plus_plus_real(X),aa_real_real(divide_divide_real(aa_real_real(real_V1803761363581548252l_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% minus_sin_cos_eq
tff(fact_7871_minus__sin__cos__eq,axiom,
    ! [X: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(sin_complex,X)) = aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(X),aa_complex_complex(divide1717551699836669952omplex(aa_real_complex(real_V4546457046886955230omplex,pi)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).

% minus_sin_cos_eq
tff(fact_7872_floor__log__eq__powr__iff,axiom,
    ! [X: real,B: real,K2: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => ( ( aa_real_int(archim6058952711729229775r_real,aa_real_real(log(B),X)) = K2 )
        <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(B),aa_int_real(ring_1_of_int_real,K2))),X)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(powr_real(B),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),one_one_int)))) ) ) ) ) ).

% floor_log_eq_powr_iff
tff(fact_7873_monoseq__minus,axiom,
    ! [A: fun_nat_int] :
      ( topolo4899668324122417113eq_int(A)
     => topolo4899668324122417113eq_int(aTP_Lamm_gn(A)) ) ).

% monoseq_minus
tff(fact_7874_monoseq__minus,axiom,
    ! [A: fun_nat_rat] :
      ( topolo4267028734544971653eq_rat(A)
     => topolo4267028734544971653eq_rat(aTP_Lamm_go(A)) ) ).

% monoseq_minus
tff(fact_7875_monoseq__minus,axiom,
    ! [A: fun_nat_Code_integer] :
      ( topolo2919662092509805066nteger(A)
     => topolo2919662092509805066nteger(aTP_Lamm_gp(A)) ) ).

% monoseq_minus
tff(fact_7876_monoseq__minus,axiom,
    ! [A: fun_nat_real] :
      ( topolo6980174941875973593q_real(A)
     => topolo6980174941875973593q_real(aTP_Lamm_fi(A)) ) ).

% monoseq_minus
tff(fact_7877_floor__log2__div2,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),one_one_int) ) ) ).

% floor_log2_div2
tff(fact_7878_floor__log__nat__eq__if,axiom,
    ! [B: nat,Nb: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)),K2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B)
         => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(log(aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K2))) = aa_nat_int(semiri1314217659103216013at_int,Nb) ) ) ) ) ).

% floor_log_nat_eq_if
tff(fact_7879_mono__SucI1,axiom,
    ! [X7: fun_nat_real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X7,N)),aa_nat_real(X7,aa_nat_nat(suc,N)))
     => topolo6980174941875973593q_real(X7) ) ).

% mono_SucI1
tff(fact_7880_mono__SucI1,axiom,
    ! [X7: fun_nat_set_int] :
      ( ! [N: nat] : ord_less_eq_set_int(aa_nat_set_int(X7,N),aa_nat_set_int(X7,aa_nat_nat(suc,N)))
     => topolo3100542954746470799et_int(X7) ) ).

% mono_SucI1
tff(fact_7881_mono__SucI1,axiom,
    ! [X7: fun_nat_rat] :
      ( ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X7,N)),aa_nat_rat(X7,aa_nat_nat(suc,N)))
     => topolo4267028734544971653eq_rat(X7) ) ).

% mono_SucI1
tff(fact_7882_mono__SucI1,axiom,
    ! [X7: fun_nat_num] :
      ( ! [N: nat] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(X7,N)),aa_nat_num(X7,aa_nat_nat(suc,N)))
     => topolo1459490580787246023eq_num(X7) ) ).

% mono_SucI1
tff(fact_7883_mono__SucI1,axiom,
    ! [X7: fun_nat_nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(X7,N)),aa_nat_nat(X7,aa_nat_nat(suc,N)))
     => topolo4902158794631467389eq_nat(X7) ) ).

% mono_SucI1
tff(fact_7884_mono__SucI1,axiom,
    ! [X7: fun_nat_int] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(X7,N)),aa_nat_int(X7,aa_nat_nat(suc,N)))
     => topolo4899668324122417113eq_int(X7) ) ).

% mono_SucI1
tff(fact_7885_mono__SucI2,axiom,
    ! [X7: fun_nat_real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X7,aa_nat_nat(suc,N))),aa_nat_real(X7,N))
     => topolo6980174941875973593q_real(X7) ) ).

% mono_SucI2
tff(fact_7886_mono__SucI2,axiom,
    ! [X7: fun_nat_set_int] :
      ( ! [N: nat] : ord_less_eq_set_int(aa_nat_set_int(X7,aa_nat_nat(suc,N)),aa_nat_set_int(X7,N))
     => topolo3100542954746470799et_int(X7) ) ).

% mono_SucI2
tff(fact_7887_mono__SucI2,axiom,
    ! [X7: fun_nat_rat] :
      ( ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X7,aa_nat_nat(suc,N))),aa_nat_rat(X7,N))
     => topolo4267028734544971653eq_rat(X7) ) ).

% mono_SucI2
tff(fact_7888_mono__SucI2,axiom,
    ! [X7: fun_nat_num] :
      ( ! [N: nat] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(X7,aa_nat_nat(suc,N))),aa_nat_num(X7,N))
     => topolo1459490580787246023eq_num(X7) ) ).

% mono_SucI2
tff(fact_7889_mono__SucI2,axiom,
    ! [X7: fun_nat_nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(X7,aa_nat_nat(suc,N))),aa_nat_nat(X7,N))
     => topolo4902158794631467389eq_nat(X7) ) ).

% mono_SucI2
tff(fact_7890_mono__SucI2,axiom,
    ! [X7: fun_nat_int] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(X7,aa_nat_nat(suc,N))),aa_nat_int(X7,N))
     => topolo4899668324122417113eq_int(X7) ) ).

% mono_SucI2
tff(fact_7891_monoseq__Suc,axiom,
    ! [X7: fun_nat_real] :
      ( topolo6980174941875973593q_real(X7)
    <=> ( ! [N4: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X7,N4)),aa_nat_real(X7,aa_nat_nat(suc,N4)))
        | ! [N4: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X7,aa_nat_nat(suc,N4))),aa_nat_real(X7,N4)) ) ) ).

% monoseq_Suc
tff(fact_7892_monoseq__Suc,axiom,
    ! [X7: fun_nat_set_int] :
      ( topolo3100542954746470799et_int(X7)
    <=> ( ! [N4: nat] : ord_less_eq_set_int(aa_nat_set_int(X7,N4),aa_nat_set_int(X7,aa_nat_nat(suc,N4)))
        | ! [N4: nat] : ord_less_eq_set_int(aa_nat_set_int(X7,aa_nat_nat(suc,N4)),aa_nat_set_int(X7,N4)) ) ) ).

% monoseq_Suc
tff(fact_7893_monoseq__Suc,axiom,
    ! [X7: fun_nat_rat] :
      ( topolo4267028734544971653eq_rat(X7)
    <=> ( ! [N4: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X7,N4)),aa_nat_rat(X7,aa_nat_nat(suc,N4)))
        | ! [N4: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X7,aa_nat_nat(suc,N4))),aa_nat_rat(X7,N4)) ) ) ).

% monoseq_Suc
tff(fact_7894_monoseq__Suc,axiom,
    ! [X7: fun_nat_num] :
      ( topolo1459490580787246023eq_num(X7)
    <=> ( ! [N4: nat] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(X7,N4)),aa_nat_num(X7,aa_nat_nat(suc,N4)))
        | ! [N4: nat] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(X7,aa_nat_nat(suc,N4))),aa_nat_num(X7,N4)) ) ) ).

% monoseq_Suc
tff(fact_7895_monoseq__Suc,axiom,
    ! [X7: fun_nat_nat] :
      ( topolo4902158794631467389eq_nat(X7)
    <=> ( ! [N4: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(X7,N4)),aa_nat_nat(X7,aa_nat_nat(suc,N4)))
        | ! [N4: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(X7,aa_nat_nat(suc,N4))),aa_nat_nat(X7,N4)) ) ) ).

% monoseq_Suc
tff(fact_7896_monoseq__Suc,axiom,
    ! [X7: fun_nat_int] :
      ( topolo4899668324122417113eq_int(X7)
    <=> ( ! [N4: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(X7,N4)),aa_nat_int(X7,aa_nat_nat(suc,N4)))
        | ! [N4: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(X7,aa_nat_nat(suc,N4))),aa_nat_int(X7,N4)) ) ) ).

% monoseq_Suc
tff(fact_7897_monoI1,axiom,
    ! [X7: fun_nat_real] :
      ( ! [M2: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X7,M2)),aa_nat_real(X7,N)) )
     => topolo6980174941875973593q_real(X7) ) ).

% monoI1
tff(fact_7898_monoI1,axiom,
    ! [X7: fun_nat_set_int] :
      ( ! [M2: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
         => ord_less_eq_set_int(aa_nat_set_int(X7,M2),aa_nat_set_int(X7,N)) )
     => topolo3100542954746470799et_int(X7) ) ).

% monoI1
tff(fact_7899_monoI1,axiom,
    ! [X7: fun_nat_rat] :
      ( ! [M2: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X7,M2)),aa_nat_rat(X7,N)) )
     => topolo4267028734544971653eq_rat(X7) ) ).

% monoI1
tff(fact_7900_monoI1,axiom,
    ! [X7: fun_nat_num] :
      ( ! [M2: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(X7,M2)),aa_nat_num(X7,N)) )
     => topolo1459490580787246023eq_num(X7) ) ).

% monoI1
tff(fact_7901_monoI1,axiom,
    ! [X7: fun_nat_nat] :
      ( ! [M2: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(X7,M2)),aa_nat_nat(X7,N)) )
     => topolo4902158794631467389eq_nat(X7) ) ).

% monoI1
tff(fact_7902_monoI1,axiom,
    ! [X7: fun_nat_int] :
      ( ! [M2: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(X7,M2)),aa_nat_int(X7,N)) )
     => topolo4899668324122417113eq_int(X7) ) ).

% monoI1
tff(fact_7903_monoI2,axiom,
    ! [X7: fun_nat_real] :
      ( ! [M2: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X7,N)),aa_nat_real(X7,M2)) )
     => topolo6980174941875973593q_real(X7) ) ).

% monoI2
tff(fact_7904_monoI2,axiom,
    ! [X7: fun_nat_set_int] :
      ( ! [M2: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
         => ord_less_eq_set_int(aa_nat_set_int(X7,N),aa_nat_set_int(X7,M2)) )
     => topolo3100542954746470799et_int(X7) ) ).

% monoI2
tff(fact_7905_monoI2,axiom,
    ! [X7: fun_nat_rat] :
      ( ! [M2: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X7,N)),aa_nat_rat(X7,M2)) )
     => topolo4267028734544971653eq_rat(X7) ) ).

% monoI2
tff(fact_7906_monoI2,axiom,
    ! [X7: fun_nat_num] :
      ( ! [M2: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(X7,N)),aa_nat_num(X7,M2)) )
     => topolo1459490580787246023eq_num(X7) ) ).

% monoI2
tff(fact_7907_monoI2,axiom,
    ! [X7: fun_nat_nat] :
      ( ! [M2: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(X7,N)),aa_nat_nat(X7,M2)) )
     => topolo4902158794631467389eq_nat(X7) ) ).

% monoI2
tff(fact_7908_monoI2,axiom,
    ! [X7: fun_nat_int] :
      ( ! [M2: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(X7,N)),aa_nat_int(X7,M2)) )
     => topolo4899668324122417113eq_int(X7) ) ).

% monoI2
tff(fact_7909_diffs__equiv,axiom,
    ! [C: fun_nat_complex,X: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_gq(C),X))
     => aa_complex_o(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_gr(C),X)),suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_gq(C),X))) ) ).

% diffs_equiv
tff(fact_7910_diffs__equiv,axiom,
    ! [C: fun_nat_real,X: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_gs(C),X))
     => aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_gt(C),X)),suminf_real(aa_real_fun_nat_real(aTP_Lamm_gs(C),X))) ) ).

% diffs_equiv
tff(fact_7911_round__altdef,axiom,
    ! [X: real] :
      aa_real_int(archim8280529875227126926d_real,X) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_real_real(archim2898591450579166408c_real,X)),aa_real_int(archim7802044766580827645g_real,X),aa_real_int(archim6058952711729229775r_real,X)) ).

% round_altdef
tff(fact_7912_round__altdef,axiom,
    ! [X: rat] :
      aa_rat_int(archim7778729529865785530nd_rat,X) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_rat_rat(archimedean_frac_rat,X)),aa_rat_int(archim2889992004027027881ng_rat,X),aa_rat_int(archim3151403230148437115or_rat,X)) ).

% round_altdef
tff(fact_7913_sin__paired,axiom,
    ! [X: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_gu,X)),aa_real_real(sin_real,X)) ).

% sin_paired
tff(fact_7914_pochhammer__double,axiom,
    ! [Z2: complex,Nb: nat] : aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Z2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,Z2),Nb))),aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,aa_complex_complex(plus_plus_complex(Z2),aa_complex_complex(divide1717551699836669952omplex(one_one_complex),numera6690914467698888265omplex(aa_num_num(bit0,one))))),Nb)) ).

% pochhammer_double
tff(fact_7915_pochhammer__double,axiom,
    ! [Z2: rat,Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Z2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),aa_nat_rat(comm_s4028243227959126397er_rat(Z2),Nb))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(Z2),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),Nb)) ).

% pochhammer_double
tff(fact_7916_pochhammer__double,axiom,
    ! [Z2: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Z2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z2),Nb))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(plus_plus_real(Z2),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Nb)) ).

% pochhammer_double
tff(fact_7917_of__nat__code,axiom,
    ! [Nb: nat] : aa_nat_complex(semiri8010041392384452111omplex,Nb) = aa_complex_complex(semiri2816024913162550771omplex(aTP_Lamm_gv,Nb),zero_zero_complex) ).

% of_nat_code
tff(fact_7918_of__nat__code,axiom,
    ! [Nb: nat] : aa_nat_rat(semiri681578069525770553at_rat,Nb) = aa_rat_rat(semiri7787848453975740701ux_rat(aTP_Lamm_gw,Nb),zero_zero_rat) ).

% of_nat_code
tff(fact_7919_of__nat__code,axiom,
    ! [Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,Nb) = aa_real_real(aa_nat_fun_real_real(semiri7260567687927622513x_real(aTP_Lamm_gx),Nb),zero_zero_real) ).

% of_nat_code
tff(fact_7920_of__nat__code,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,Nb) = aa_int_int(semiri8420488043553186161ux_int(aTP_Lamm_gy,Nb),zero_zero_int) ).

% of_nat_code
tff(fact_7921_of__nat__code,axiom,
    ! [Nb: nat] : aa_nat_Extended_enat(semiri4216267220026989637d_enat,Nb) = semiri8563196900006977889d_enat(aTP_Lamm_gz,Nb,zero_z5237406670263579293d_enat) ).

% of_nat_code
tff(fact_7922_of__nat__code,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(semiri8422978514062236437ux_nat(aTP_Lamm_ha),Nb),zero_zero_nat) ).

% of_nat_code
tff(fact_7923_fact__0,axiom,
    aa_nat_complex(semiri5044797733671781792omplex,zero_zero_nat) = one_one_complex ).

% fact_0
tff(fact_7924_fact__0,axiom,
    aa_nat_rat(semiri773545260158071498ct_rat,zero_zero_nat) = one_one_rat ).

% fact_0
tff(fact_7925_fact__0,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,zero_zero_nat) = one_one_int ).

% fact_0
tff(fact_7926_fact__0,axiom,
    aa_nat_real(semiri2265585572941072030t_real,zero_zero_nat) = one_one_real ).

% fact_0
tff(fact_7927_fact__0,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,zero_zero_nat) = one_one_nat ).

% fact_0
tff(fact_7928_fact__1,axiom,
    aa_nat_complex(semiri5044797733671781792omplex,one_one_nat) = one_one_complex ).

% fact_1
tff(fact_7929_fact__1,axiom,
    aa_nat_rat(semiri773545260158071498ct_rat,one_one_nat) = one_one_rat ).

% fact_1
tff(fact_7930_fact__1,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,one_one_nat) = one_one_int ).

% fact_1
tff(fact_7931_fact__1,axiom,
    aa_nat_real(semiri2265585572941072030t_real,one_one_nat) = one_one_real ).

% fact_1
tff(fact_7932_fact__1,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,one_one_nat) = one_one_nat ).

% fact_1
tff(fact_7933_pochhammer__0,axiom,
    ! [A: complex] : aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,A),zero_zero_nat) = one_one_complex ).

% pochhammer_0
tff(fact_7934_pochhammer__0,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),zero_zero_nat) = one_one_real ).

% pochhammer_0
tff(fact_7935_pochhammer__0,axiom,
    ! [A: rat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),zero_zero_nat) = one_one_rat ).

% pochhammer_0
tff(fact_7936_pochhammer__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),zero_zero_nat) = one_one_nat ).

% pochhammer_0
tff(fact_7937_pochhammer__0,axiom,
    ! [A: int] : aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,A),zero_zero_nat) = one_one_int ).

% pochhammer_0
tff(fact_7938_fact__Suc__0,axiom,
    aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(suc,zero_zero_nat)) = one_one_complex ).

% fact_Suc_0
tff(fact_7939_fact__Suc__0,axiom,
    aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(suc,zero_zero_nat)) = one_one_rat ).

% fact_Suc_0
tff(fact_7940_fact__Suc__0,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,zero_zero_nat)) = one_one_int ).

% fact_Suc_0
tff(fact_7941_fact__Suc__0,axiom,
    aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(suc,zero_zero_nat)) = one_one_real ).

% fact_Suc_0
tff(fact_7942_fact__Suc__0,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% fact_Suc_0
tff(fact_7943_fact__Suc,axiom,
    ! [Nb: nat] : aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(suc,Nb)) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,Nb))),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ).

% fact_Suc
tff(fact_7944_fact__Suc,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb))),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ).

% fact_Suc
tff(fact_7945_fact__Suc,axiom,
    ! [Nb: nat] : aa_nat_Extended_enat(semiri4449623510593786356d_enat,aa_nat_nat(suc,Nb)) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,aa_nat_nat(suc,Nb))),aa_nat_Extended_enat(semiri4449623510593786356d_enat,Nb)) ).

% fact_Suc
tff(fact_7946_fact__Suc,axiom,
    ! [Nb: nat] : aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(suc,Nb)) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb))),aa_nat_real(semiri2265585572941072030t_real,Nb)) ).

% fact_Suc
tff(fact_7947_fact__Suc,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,Nb))),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ).

% fact_Suc
tff(fact_7948_fact__2,axiom,
    aa_nat_complex(semiri5044797733671781792omplex,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = numera6690914467698888265omplex(aa_num_num(bit0,one)) ).

% fact_2
tff(fact_7949_fact__2,axiom,
    aa_nat_rat(semiri773545260158071498ct_rat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)) ).

% fact_2
tff(fact_7950_fact__2,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)) ).

% fact_2
tff(fact_7951_fact__2,axiom,
    aa_nat_real(semiri2265585572941072030t_real,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)) ).

% fact_2
tff(fact_7952_fact__2,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% fact_2
tff(fact_7953_pochhammer__fact,axiom,
    ! [Nb: nat] : aa_nat_complex(semiri5044797733671781792omplex,Nb) = aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,one_one_complex),Nb) ).

% pochhammer_fact
tff(fact_7954_pochhammer__fact,axiom,
    ! [Nb: nat] : aa_nat_rat(semiri773545260158071498ct_rat,Nb) = aa_nat_rat(comm_s4028243227959126397er_rat(one_one_rat),Nb) ).

% pochhammer_fact
tff(fact_7955_pochhammer__fact,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1406184849735516958ct_int,Nb) = aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,one_one_int),Nb) ).

% pochhammer_fact
tff(fact_7956_pochhammer__fact,axiom,
    ! [Nb: nat] : aa_nat_real(semiri2265585572941072030t_real,Nb) = aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,one_one_real),Nb) ).

% pochhammer_fact
tff(fact_7957_pochhammer__fact,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,one_one_nat),Nb) ).

% pochhammer_fact
tff(fact_7958_diffs__of__real,axiom,
    ! [F: fun_nat_real,X2: nat] : aa_nat_complex(diffs_complex(aTP_Lamm_fz(F)),X2) = aa_real_complex(real_V4546457046886955230omplex,aa_nat_real(diffs_real(F),X2)) ).

% diffs_of_real
tff(fact_7959_fact__ge__zero,axiom,
    ! [Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ).

% fact_ge_zero
tff(fact_7960_fact__ge__zero,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ).

% fact_ge_zero
tff(fact_7961_fact__ge__zero,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(semiri2265585572941072030t_real,Nb)) ).

% fact_ge_zero
tff(fact_7962_fact__ge__zero,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ).

% fact_ge_zero
tff(fact_7963_fact__gt__zero,axiom,
    ! [Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ).

% fact_gt_zero
tff(fact_7964_fact__gt__zero,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ).

% fact_gt_zero
tff(fact_7965_fact__gt__zero,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(semiri2265585572941072030t_real,Nb)) ).

% fact_gt_zero
tff(fact_7966_fact__gt__zero,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ).

% fact_gt_zero
tff(fact_7967_fact__not__neg,axiom,
    ! [Nb: nat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri773545260158071498ct_rat,Nb)),zero_zero_rat) ).

% fact_not_neg
tff(fact_7968_fact__not__neg,axiom,
    ! [Nb: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1406184849735516958ct_int,Nb)),zero_zero_int) ).

% fact_not_neg
tff(fact_7969_fact__not__neg,axiom,
    ! [Nb: nat] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri2265585572941072030t_real,Nb)),zero_zero_real) ).

% fact_not_neg
tff(fact_7970_fact__not__neg,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),zero_zero_nat) ).

% fact_not_neg
tff(fact_7971_fact__ge__1,axiom,
    ! [Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ).

% fact_ge_1
tff(fact_7972_fact__ge__1,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ).

% fact_ge_1
tff(fact_7973_fact__ge__1,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(semiri2265585572941072030t_real,Nb)) ).

% fact_ge_1
tff(fact_7974_fact__ge__1,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ).

% fact_ge_1
tff(fact_7975_diffs__minus,axiom,
    ! [C: fun_nat_real,X2: nat] : aa_nat_real(diffs_real(aTP_Lamm_fi(C)),X2) = aa_real_real(uminus_uminus_real,aa_nat_real(diffs_real(C),X2)) ).

% diffs_minus
tff(fact_7976_diffs__minus,axiom,
    ! [C: fun_nat_int,X2: nat] : aa_nat_int(diffs_int(aTP_Lamm_gn(C)),X2) = aa_int_int(uminus_uminus_int,aa_nat_int(diffs_int(C),X2)) ).

% diffs_minus
tff(fact_7977_diffs__minus,axiom,
    ! [C: fun_nat_complex,X2: nat] : aa_nat_complex(diffs_complex(aTP_Lamm_fj(C)),X2) = aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(diffs_complex(C),X2)) ).

% diffs_minus
tff(fact_7978_diffs__minus,axiom,
    ! [C: fun_nat_rat,X2: nat] : aa_nat_rat(diffs_rat(aTP_Lamm_go(C)),X2) = aa_rat_rat(uminus_uminus_rat,aa_nat_rat(diffs_rat(C),X2)) ).

% diffs_minus
tff(fact_7979_diffs__minus,axiom,
    ! [C: fun_nat_Code_integer,X2: nat] : aa_nat_Code_integer(diffs_Code_integer(aTP_Lamm_gp(C)),X2) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(diffs_Code_integer(C),X2)) ).

% diffs_minus
tff(fact_7980_fact__mono,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri773545260158071498ct_rat,Ma)),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ) ).

% fact_mono
tff(fact_7981_fact__mono,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1406184849735516958ct_int,Ma)),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ) ).

% fact_mono
tff(fact_7982_fact__mono,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri2265585572941072030t_real,Ma)),aa_nat_real(semiri2265585572941072030t_real,Nb)) ) ).

% fact_mono
tff(fact_7983_fact__mono,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Ma)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ).

% fact_mono
tff(fact_7984_pochhammer__pos,axiom,
    ! [X: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,X),Nb)) ) ).

% pochhammer_pos
tff(fact_7985_pochhammer__pos,axiom,
    ! [X: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),X)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(comm_s4028243227959126397er_rat(X),Nb)) ) ).

% pochhammer_pos
tff(fact_7986_pochhammer__pos,axiom,
    ! [X: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),X)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,X),Nb)) ) ).

% pochhammer_pos
tff(fact_7987_pochhammer__pos,axiom,
    ! [X: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),X)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,X),Nb)) ) ).

% pochhammer_pos
tff(fact_7988_fact__dvd,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(semiri1406184849735516958ct_int,Nb)),aa_nat_int(semiri1406184849735516958ct_int,Ma)) ) ).

% fact_dvd
tff(fact_7989_fact__dvd,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(semiri3624122377584611663nteger,Nb)),aa_nat_Code_integer(semiri3624122377584611663nteger,Ma)) ) ).

% fact_dvd
tff(fact_7990_fact__dvd,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(semiri2265585572941072030t_real,Nb)),aa_nat_real(semiri2265585572941072030t_real,Ma)) ) ).

% fact_dvd
tff(fact_7991_fact__dvd,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),aa_nat_nat(semiri1408675320244567234ct_nat,Ma)) ) ).

% fact_dvd
tff(fact_7992_pochhammer__neq__0__mono,axiom,
    ! [A: complex,Ma: nat,Nb: nat] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,A),Ma) != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma)
       => ( aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,A),Nb) != zero_zero_complex ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_7993_pochhammer__neq__0__mono,axiom,
    ! [A: real,Ma: nat,Nb: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Ma) != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma)
       => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Nb) != zero_zero_real ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_7994_pochhammer__neq__0__mono,axiom,
    ! [A: rat,Ma: nat,Nb: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),Ma) != zero_zero_rat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma)
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(A),Nb) != zero_zero_rat ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_7995_pochhammer__eq__0__mono,axiom,
    ! [A: complex,Nb: nat,Ma: nat] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,A),Nb) = zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma)
       => ( aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,A),Ma) = zero_zero_complex ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_7996_pochhammer__eq__0__mono,axiom,
    ! [A: real,Nb: nat,Ma: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Nb) = zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma)
       => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Ma) = zero_zero_real ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_7997_pochhammer__eq__0__mono,axiom,
    ! [A: rat,Nb: nat,Ma: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),Nb) = zero_zero_rat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma)
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(A),Ma) = zero_zero_rat ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_7998_pochhammer__same,axiom,
    ! [Nb: nat] : aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,Nb))),Nb) = aa_complex_complex(times_times_complex(aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb)),aa_nat_complex(semiri5044797733671781792omplex,Nb)) ).

% pochhammer_same
tff(fact_7999_pochhammer__same,axiom,
    ! [Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb))),Nb) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb)),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ).

% pochhammer_same
tff(fact_8000_pochhammer__same,axiom,
    ! [Nb: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))),Nb) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb)),aa_nat_Code_integer(semiri3624122377584611663nteger,Nb)) ).

% pochhammer_same
tff(fact_8001_pochhammer__same,axiom,
    ! [Nb: nat] : aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Nb))),Nb) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),Nb)),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ).

% pochhammer_same
tff(fact_8002_pochhammer__same,axiom,
    ! [Nb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Nb))),Nb) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb)),aa_nat_real(semiri2265585572941072030t_real,Nb)) ).

% pochhammer_same
tff(fact_8003_frac__ge__0,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(archim2898591450579166408c_real,X)) ).

% frac_ge_0
tff(fact_8004_frac__ge__0,axiom,
    ! [X: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(archimedean_frac_rat,X)) ).

% frac_ge_0
tff(fact_8005_frac__lt__1,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(archim2898591450579166408c_real,X)),one_one_real) ).

% frac_lt_1
tff(fact_8006_frac__lt__1,axiom,
    ! [X: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(archimedean_frac_rat,X)),one_one_rat) ).

% frac_lt_1
tff(fact_8007_frac__1__eq,axiom,
    ! [X: real] : aa_real_real(archim2898591450579166408c_real,aa_real_real(plus_plus_real(X),one_one_real)) = aa_real_real(archim2898591450579166408c_real,X) ).

% frac_1_eq
tff(fact_8008_frac__1__eq,axiom,
    ! [X: rat] : aa_rat_rat(archimedean_frac_rat,aa_rat_rat(plus_plus_rat(X),one_one_rat)) = aa_rat_rat(archimedean_frac_rat,X) ).

% frac_1_eq
tff(fact_8009_fact__less__mono,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ma)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri773545260158071498ct_rat,Ma)),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ) ) ).

% fact_less_mono
tff(fact_8010_fact__less__mono,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ma)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1406184849735516958ct_int,Ma)),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ) ) ).

% fact_less_mono
tff(fact_8011_fact__less__mono,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ma)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri2265585572941072030t_real,Ma)),aa_nat_real(semiri2265585572941072030t_real,Nb)) ) ) ).

% fact_less_mono
tff(fact_8012_fact__less__mono,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ma)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Ma)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ) ).

% fact_less_mono
tff(fact_8013_fact__mod,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(semiri1406184849735516958ct_int,Nb)),aa_nat_int(semiri1406184849735516958ct_int,Ma)) = zero_zero_int ) ) ).

% fact_mod
tff(fact_8014_fact__mod,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => ( modulo364778990260209775nteger(aa_nat_Code_integer(semiri3624122377584611663nteger,Nb),aa_nat_Code_integer(semiri3624122377584611663nteger,Ma)) = zero_z3403309356797280102nteger ) ) ).

% fact_mod
tff(fact_8015_fact__mod,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),aa_nat_nat(semiri1408675320244567234ct_nat,Ma)) = zero_zero_nat ) ) ).

% fact_mod
tff(fact_8016_pochhammer__nonneg,axiom,
    ! [X: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,X),Nb)) ) ).

% pochhammer_nonneg
tff(fact_8017_pochhammer__nonneg,axiom,
    ! [X: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),X)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(comm_s4028243227959126397er_rat(X),Nb)) ) ).

% pochhammer_nonneg
tff(fact_8018_pochhammer__nonneg,axiom,
    ! [X: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),X)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,X),Nb)) ) ).

% pochhammer_nonneg
tff(fact_8019_pochhammer__nonneg,axiom,
    ! [X: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),X)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,X),Nb)) ) ).

% pochhammer_nonneg
tff(fact_8020_fact__le__power,axiom,
    ! [Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri773545260158071498ct_rat,Nb)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),Nb))) ).

% fact_le_power
tff(fact_8021_fact__le__power,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1406184849735516958ct_int,Nb)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),Nb))) ).

% fact_le_power
tff(fact_8022_fact__le__power,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri2265585572941072030t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),Nb))) ).

% fact_le_power
tff(fact_8023_fact__le__power,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),Nb))) ).

% fact_le_power
tff(fact_8024_pochhammer__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,zero_zero_complex),Nb) = $ite(Nb = zero_zero_nat,one_one_complex,zero_zero_complex) ).

% pochhammer_0_left
tff(fact_8025_pochhammer__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,zero_zero_real),Nb) = $ite(Nb = zero_zero_nat,one_one_real,zero_zero_real) ).

% pochhammer_0_left
tff(fact_8026_pochhammer__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_rat(comm_s4028243227959126397er_rat(zero_zero_rat),Nb) = $ite(Nb = zero_zero_nat,one_one_rat,zero_zero_rat) ).

% pochhammer_0_left
tff(fact_8027_pochhammer__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,zero_zero_nat),Nb) = $ite(Nb = zero_zero_nat,one_one_nat,zero_zero_nat) ).

% pochhammer_0_left
tff(fact_8028_pochhammer__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,zero_zero_int),Nb) = $ite(Nb = zero_zero_nat,one_one_int,zero_zero_int) ).

% pochhammer_0_left
tff(fact_8029_pochhammer__rec,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,A),aa_nat_nat(suc,Nb)) = aa_complex_complex(times_times_complex(A),aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,aa_complex_complex(plus_plus_complex(A),one_one_complex)),Nb)) ).

% pochhammer_rec
tff(fact_8030_pochhammer__rec,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),aa_nat_nat(suc,Nb)) = aa_real_real(times_times_real(A),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(plus_plus_real(A),one_one_real)),Nb)) ).

% pochhammer_rec
tff(fact_8031_pochhammer__rec,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,Nb)) = aa_rat_rat(times_times_rat(A),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(A),one_one_rat)),Nb)) ).

% pochhammer_rec
tff(fact_8032_pochhammer__rec,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),Nb)) ).

% pochhammer_rec
tff(fact_8033_pochhammer__rec,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,A),aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),Nb)) ).

% pochhammer_rec
tff(fact_8034_diffs__def,axiom,
    ! [C: fun_nat_rat,X2: nat] : aa_nat_rat(diffs_rat(C),X2) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,X2))),aa_nat_rat(C,aa_nat_nat(suc,X2))) ).

% diffs_def
tff(fact_8035_diffs__def,axiom,
    ! [C: fun_nat_real,X2: nat] : aa_nat_real(diffs_real(C),X2) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,X2))),aa_nat_real(C,aa_nat_nat(suc,X2))) ).

% diffs_def
tff(fact_8036_diffs__def,axiom,
    ! [C: fun_nat_int,X2: nat] : aa_nat_int(diffs_int(C),X2) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,X2))),aa_nat_int(C,aa_nat_nat(suc,X2))) ).

% diffs_def
tff(fact_8037_pochhammer__Suc,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,Nb)) = aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(A),Nb)),aa_rat_rat(plus_plus_rat(A),aa_nat_rat(semiri681578069525770553at_rat,Nb))) ).

% pochhammer_Suc
tff(fact_8038_pochhammer__Suc,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),aa_nat_nat(suc,Nb)) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Nb)),aa_real_real(plus_plus_real(A),aa_nat_real(semiri5074537144036343181t_real,Nb))) ).

% pochhammer_Suc
tff(fact_8039_pochhammer__Suc,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,A),aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,A),Nb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_nat_int(semiri1314217659103216013at_int,Nb))) ).

% pochhammer_Suc
tff(fact_8040_pochhammer__Suc,axiom,
    ! [A: extended_enat,Nb: nat] : aa_nat_Extended_enat(comm_s3181272606743183617d_enat(A),aa_nat_nat(suc,Nb)) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(comm_s3181272606743183617d_enat(A),Nb)),plus_p3455044024723400733d_enat(A,aa_nat_Extended_enat(semiri4216267220026989637d_enat,Nb))) ).

% pochhammer_Suc
tff(fact_8041_pochhammer__Suc,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,Nb))) ).

% pochhammer_Suc
tff(fact_8042_pochhammer__rec_H,axiom,
    ! [Z2: rat,Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(Z2),aa_nat_nat(suc,Nb)) = aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(Z2),aa_nat_rat(semiri681578069525770553at_rat,Nb))),aa_nat_rat(comm_s4028243227959126397er_rat(Z2),Nb)) ).

% pochhammer_rec'
tff(fact_8043_pochhammer__rec_H,axiom,
    ! [Z2: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z2),aa_nat_nat(suc,Nb)) = aa_real_real(times_times_real(aa_real_real(plus_plus_real(Z2),aa_nat_real(semiri5074537144036343181t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z2),Nb)) ).

% pochhammer_rec'
tff(fact_8044_pochhammer__rec_H,axiom,
    ! [Z2: int,Nb: nat] : aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,Z2),aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z2),aa_nat_int(semiri1314217659103216013at_int,Nb))),aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,Z2),Nb)) ).

% pochhammer_rec'
tff(fact_8045_pochhammer__rec_H,axiom,
    ! [Z2: extended_enat,Nb: nat] : aa_nat_Extended_enat(comm_s3181272606743183617d_enat(Z2),aa_nat_nat(suc,Nb)) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(plus_p3455044024723400733d_enat(Z2,aa_nat_Extended_enat(semiri4216267220026989637d_enat,Nb))),aa_nat_Extended_enat(comm_s3181272606743183617d_enat(Z2),Nb)) ).

% pochhammer_rec'
tff(fact_8046_pochhammer__rec_H,axiom,
    ! [Z2: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z2),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z2),aa_nat_nat(semiri1316708129612266289at_nat,Nb))),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z2),Nb)) ).

% pochhammer_rec'
tff(fact_8047_pochhammer__eq__0__iff,axiom,
    ! [A: complex,Nb: nat] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,A),Nb) = zero_zero_complex )
    <=> ? [K3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K3),Nb)
          & ( A = aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,K3)) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_8048_pochhammer__eq__0__iff,axiom,
    ! [A: rat,Nb: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),Nb) = zero_zero_rat )
    <=> ? [K3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K3),Nb)
          & ( A = aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,K3)) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_8049_pochhammer__eq__0__iff,axiom,
    ! [A: real,Nb: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Nb) = zero_zero_real )
    <=> ? [K3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K3),Nb)
          & ( A = aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,K3)) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_8050_pochhammer__of__nat__eq__0__iff,axiom,
    ! [Nb: nat,K2: nat] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,Nb))),K2) = zero_zero_complex )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K2) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_8051_pochhammer__of__nat__eq__0__iff,axiom,
    ! [Nb: nat,K2: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb))),K2) = zero_zero_rat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K2) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_8052_pochhammer__of__nat__eq__0__iff,axiom,
    ! [Nb: nat,K2: nat] :
      ( ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))),K2) = zero_z3403309356797280102nteger )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K2) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_8053_pochhammer__of__nat__eq__0__iff,axiom,
    ! [Nb: nat,K2: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Nb))),K2) = zero_zero_real )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K2) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_8054_pochhammer__of__nat__eq__0__iff,axiom,
    ! [Nb: nat,K2: nat] :
      ( ( aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Nb))),K2) = zero_zero_int )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K2) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_8055_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [Nb: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K2)
     => ( aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,Nb))),K2) = zero_zero_complex ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_8056_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [Nb: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K2)
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb))),K2) = zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_8057_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [Nb: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K2)
     => ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))),K2) = zero_z3403309356797280102nteger ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_8058_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [Nb: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K2)
     => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Nb))),K2) = zero_zero_real ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_8059_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [Nb: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K2)
     => ( aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Nb))),K2) = zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_8060_fact__numeral,axiom,
    ! [K2: num] : aa_nat_complex(semiri5044797733671781792omplex,aa_num_nat(numeral_numeral_nat,K2)) = aa_complex_complex(times_times_complex(numera6690914467698888265omplex(K2)),aa_nat_complex(semiri5044797733671781792omplex,aa_num_nat(pred_numeral,K2))) ).

% fact_numeral
tff(fact_8061_fact__numeral,axiom,
    ! [K2: num] : aa_nat_rat(semiri773545260158071498ct_rat,aa_num_nat(numeral_numeral_nat,K2)) = aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,K2)),aa_nat_rat(semiri773545260158071498ct_rat,aa_num_nat(pred_numeral,K2))) ).

% fact_numeral
tff(fact_8062_fact__numeral,axiom,
    ! [K2: num] : aa_nat_int(semiri1406184849735516958ct_int,aa_num_nat(numeral_numeral_nat,K2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,K2)),aa_nat_int(semiri1406184849735516958ct_int,aa_num_nat(pred_numeral,K2))) ).

% fact_numeral
tff(fact_8063_fact__numeral,axiom,
    ! [K2: num] : aa_nat_real(semiri2265585572941072030t_real,aa_num_nat(numeral_numeral_nat,K2)) = aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,K2)),aa_nat_real(semiri2265585572941072030t_real,aa_num_nat(pred_numeral,K2))) ).

% fact_numeral
tff(fact_8064_fact__numeral,axiom,
    ! [K2: num] : aa_nat_nat(semiri1408675320244567234ct_nat,aa_num_nat(numeral_numeral_nat,K2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,K2)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_num_nat(pred_numeral,K2))) ).

% fact_numeral
tff(fact_8065_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,Nb))),K2) != zero_zero_complex ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_8066_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb))),K2) != zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_8067_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))),K2) != zero_z3403309356797280102nteger ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_8068_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Nb))),K2) != zero_zero_real ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_8069_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Nb))),K2) != zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_8070_pochhammer__product_H,axiom,
    ! [Z2: rat,Nb: nat,Ma: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(Z2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Ma)) = aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(Z2),Nb)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(Z2),aa_nat_rat(semiri681578069525770553at_rat,Nb))),Ma)) ).

% pochhammer_product'
tff(fact_8071_pochhammer__product_H,axiom,
    ! [Z2: real,Nb: nat,Ma: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Ma)) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z2),Nb)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(plus_plus_real(Z2),aa_nat_real(semiri5074537144036343181t_real,Nb))),Ma)) ).

% pochhammer_product'
tff(fact_8072_pochhammer__product_H,axiom,
    ! [Z2: int,Nb: nat,Ma: nat] : aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,Z2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Ma)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,Z2),Nb)),aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z2),aa_nat_int(semiri1314217659103216013at_int,Nb))),Ma)) ).

% pochhammer_product'
tff(fact_8073_pochhammer__product_H,axiom,
    ! [Z2: extended_enat,Nb: nat,Ma: nat] : aa_nat_Extended_enat(comm_s3181272606743183617d_enat(Z2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Ma)) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(comm_s3181272606743183617d_enat(Z2),Nb)),aa_nat_Extended_enat(comm_s3181272606743183617d_enat(plus_p3455044024723400733d_enat(Z2,aa_nat_Extended_enat(semiri4216267220026989637d_enat,Nb))),Ma)) ).

% pochhammer_product'
tff(fact_8074_pochhammer__product_H,axiom,
    ! [Z2: nat,Nb: nat,Ma: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Ma)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z2),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z2),aa_nat_nat(semiri1316708129612266289at_nat,Nb))),Ma)) ).

% pochhammer_product'
tff(fact_8075_fact__double,axiom,
    ! [Nb: nat] : aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,aa_complex_complex(divide1717551699836669952omplex(one_one_complex),numera6690914467698888265omplex(aa_num_num(bit0,one)))),Nb))),aa_nat_complex(semiri5044797733671781792omplex,Nb)) ).

% fact_double
tff(fact_8076_fact__double,axiom,
    ! [Nb: nat] : aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),Nb))),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ).

% fact_double
tff(fact_8077_fact__double,axiom,
    ! [Nb: nat] : aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Nb))),aa_nat_real(semiri2265585572941072030t_real,Nb)) ).

% fact_double
tff(fact_8078_termdiff__converges__all,axiom,
    ! [C: fun_nat_complex,X: complex] :
      ( ! [X4: complex] : summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_fp(C),X4))
     => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_gq(C),X)) ) ).

% termdiff_converges_all
tff(fact_8079_termdiff__converges__all,axiom,
    ! [C: fun_nat_real,X: real] :
      ( ! [X4: real] : summable_real(aa_real_fun_nat_real(aTP_Lamm_fn(C),X4))
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_gs(C),X)) ) ).

% termdiff_converges_all
tff(fact_8080_frac__eq,axiom,
    ! [X: real] :
      ( ( aa_real_real(archim2898591450579166408c_real,X) = X )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,X),one_one_real) ) ) ).

% frac_eq
tff(fact_8081_frac__eq,axiom,
    ! [X: rat] :
      ( ( aa_rat_rat(archimedean_frac_rat,X) = X )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),one_one_rat) ) ) ).

% frac_eq
tff(fact_8082_frac__add,axiom,
    ! [X: real,Y2: real] :
      aa_real_real(archim2898591450579166408c_real,aa_real_real(plus_plus_real(X),Y2)) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(aa_real_real(archim2898591450579166408c_real,X)),aa_real_real(archim2898591450579166408c_real,Y2))),one_one_real),aa_real_real(plus_plus_real(aa_real_real(archim2898591450579166408c_real,X)),aa_real_real(archim2898591450579166408c_real,Y2)),aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(aa_real_real(archim2898591450579166408c_real,X)),aa_real_real(archim2898591450579166408c_real,Y2))),one_one_real)) ).

% frac_add
tff(fact_8083_frac__add,axiom,
    ! [X: rat,Y2: rat] :
      aa_rat_rat(archimedean_frac_rat,aa_rat_rat(plus_plus_rat(X),Y2)) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(archimedean_frac_rat,X)),aa_rat_rat(archimedean_frac_rat,Y2))),one_one_rat),aa_rat_rat(plus_plus_rat(aa_rat_rat(archimedean_frac_rat,X)),aa_rat_rat(archimedean_frac_rat,Y2)),aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(archimedean_frac_rat,X)),aa_rat_rat(archimedean_frac_rat,Y2))),one_one_rat)) ).

% frac_add
tff(fact_8084_pochhammer__product,axiom,
    ! [Ma: nat,Nb: nat,Z2: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(Z2),Nb) = aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(Z2),Ma)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(Z2),aa_nat_rat(semiri681578069525770553at_rat,Ma))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Ma))) ) ) ).

% pochhammer_product
tff(fact_8085_pochhammer__product,axiom,
    ! [Ma: nat,Nb: nat,Z2: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z2),Nb) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z2),Ma)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(plus_plus_real(Z2),aa_nat_real(semiri5074537144036343181t_real,Ma))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Ma))) ) ) ).

% pochhammer_product
tff(fact_8086_pochhammer__product,axiom,
    ! [Ma: nat,Nb: nat,Z2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => ( aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,Z2),Nb) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,Z2),Ma)),aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z2),aa_nat_int(semiri1314217659103216013at_int,Ma))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Ma))) ) ) ).

% pochhammer_product
tff(fact_8087_pochhammer__product,axiom,
    ! [Ma: nat,Nb: nat,Z2: extended_enat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => ( aa_nat_Extended_enat(comm_s3181272606743183617d_enat(Z2),Nb) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(comm_s3181272606743183617d_enat(Z2),Ma)),aa_nat_Extended_enat(comm_s3181272606743183617d_enat(plus_p3455044024723400733d_enat(Z2,aa_nat_Extended_enat(semiri4216267220026989637d_enat,Ma))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Ma))) ) ) ).

% pochhammer_product
tff(fact_8088_pochhammer__product,axiom,
    ! [Ma: nat,Nb: nat,Z2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z2),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z2),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z2),aa_nat_nat(semiri1316708129612266289at_nat,Ma))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Ma))) ) ) ).

% pochhammer_product
tff(fact_8089_square__fact__le__2__fact,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(aa_nat_real(semiri2265585572941072030t_real,Nb)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% square_fact_le_2_fact
tff(fact_8090_fact__num__eq__if,axiom,
    ! [Ma: nat] :
      aa_nat_complex(semiri5044797733671781792omplex,Ma) = $ite(Ma = zero_zero_nat,one_one_complex,aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Ma)),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_8091_fact__num__eq__if,axiom,
    ! [Ma: nat] :
      aa_nat_rat(semiri773545260158071498ct_rat,Ma) = $ite(Ma = zero_zero_nat,one_one_rat,aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Ma)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_8092_fact__num__eq__if,axiom,
    ! [Ma: nat] :
      aa_nat_int(semiri1406184849735516958ct_int,Ma) = $ite(Ma = zero_zero_nat,one_one_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Ma)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_8093_fact__num__eq__if,axiom,
    ! [Ma: nat] :
      aa_nat_Extended_enat(semiri4449623510593786356d_enat,Ma) = $ite(Ma = zero_zero_nat,one_on7984719198319812577d_enat,aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,Ma)),aa_nat_Extended_enat(semiri4449623510593786356d_enat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_8094_fact__num__eq__if,axiom,
    ! [Ma: nat] :
      aa_nat_real(semiri2265585572941072030t_real,Ma) = $ite(Ma = zero_zero_nat,one_one_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Ma)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_8095_fact__num__eq__if,axiom,
    ! [Ma: nat] :
      aa_nat_nat(semiri1408675320244567234ct_nat,Ma) = $ite(Ma = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Ma)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_8096_fact__code,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1406184849735516958ct_int,Nb) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),one_one_nat)) ).

% fact_code
tff(fact_8097_fact__code,axiom,
    ! [Nb: nat] : aa_nat_Extended_enat(semiri4449623510593786356d_enat,Nb) = aa_nat_Extended_enat(semiri4216267220026989637d_enat,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),one_one_nat)) ).

% fact_code
tff(fact_8098_fact__code,axiom,
    ! [Nb: nat] : aa_nat_real(semiri2265585572941072030t_real,Nb) = aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),one_one_nat)) ).

% fact_code
tff(fact_8099_fact__code,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,Nb) = aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),one_one_nat)) ).

% fact_code
tff(fact_8100_fact__reduce,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_rat(semiri773545260158071498ct_rat,Nb) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_8101_fact__reduce,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_int(semiri1406184849735516958ct_int,Nb) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_8102_fact__reduce,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_Extended_enat(semiri4449623510593786356d_enat,Nb) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,Nb)),aa_nat_Extended_enat(semiri4449623510593786356d_enat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_8103_fact__reduce,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_real(semiri2265585572941072030t_real,Nb) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_8104_fact__reduce,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_8105_pochhammer__absorb__comp,axiom,
    ! [R: complex,K2: nat] : aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(R),aa_nat_complex(semiri8010041392384452111omplex,K2))),aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,aa_complex_complex(uminus1482373934393186551omplex,R)),K2)) = aa_complex_complex(times_times_complex(R),aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,R)),one_one_complex)),K2)) ).

% pochhammer_absorb_comp
tff(fact_8106_pochhammer__absorb__comp,axiom,
    ! [R: rat,K2: nat] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(R),aa_nat_rat(semiri681578069525770553at_rat,K2))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,R)),K2)) = aa_rat_rat(times_times_rat(R),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,R)),one_one_rat)),K2)) ).

% pochhammer_absorb_comp
tff(fact_8107_pochhammer__absorb__comp,axiom,
    ! [R: code_integer,K2: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,minus_8373710615458151222nteger(R,aa_nat_Code_integer(semiri4939895301339042750nteger,K2))),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,R)),K2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,R),aa_nat_Code_integer(comm_s8582702949713902594nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,R),one_one_Code_integer)),K2)) ).

% pochhammer_absorb_comp
tff(fact_8108_pochhammer__absorb__comp,axiom,
    ! [R: real,K2: nat] : aa_real_real(times_times_real(aa_real_real(minus_minus_real(R),aa_nat_real(semiri5074537144036343181t_real,K2))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,R)),K2)) = aa_real_real(times_times_real(R),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,R)),one_one_real)),K2)) ).

% pochhammer_absorb_comp
tff(fact_8109_pochhammer__absorb__comp,axiom,
    ! [R: int,K2: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,R),aa_nat_int(semiri1314217659103216013at_int,K2))),aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(uminus_uminus_int,R)),K2)) = aa_int_int(aa_int_fun_int_int(times_times_int,R),aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,R)),one_one_int)),K2)) ).

% pochhammer_absorb_comp
tff(fact_8110_pochhammer__minus,axiom,
    ! [B: complex,K2: nat] : aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,aa_complex_complex(uminus1482373934393186551omplex,B)),K2) = aa_complex_complex(times_times_complex(aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K2)),aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,aa_complex_complex(plus_plus_complex(aa_complex_complex(minus_minus_complex(B),aa_nat_complex(semiri8010041392384452111omplex,K2))),one_one_complex)),K2)) ).

% pochhammer_minus
tff(fact_8111_pochhammer__minus,axiom,
    ! [B: rat,K2: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,B)),K2) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K2)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(B),aa_nat_rat(semiri681578069525770553at_rat,K2))),one_one_rat)),K2)) ).

% pochhammer_minus
tff(fact_8112_pochhammer__minus,axiom,
    ! [B: code_integer,K2: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),K2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),K2)),aa_nat_Code_integer(comm_s8582702949713902594nteger(plus_p5714425477246183910nteger(minus_8373710615458151222nteger(B,aa_nat_Code_integer(semiri4939895301339042750nteger,K2)),one_one_Code_integer)),K2)) ).

% pochhammer_minus
tff(fact_8113_pochhammer__minus,axiom,
    ! [B: real,K2: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,B)),K2) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),K2)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(plus_plus_real(aa_real_real(minus_minus_real(B),aa_nat_real(semiri5074537144036343181t_real,K2))),one_one_real)),K2)) ).

% pochhammer_minus
tff(fact_8114_pochhammer__minus,axiom,
    ! [B: int,K2: nat] : aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(uminus_uminus_int,B)),K2) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),K2)),aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),aa_nat_int(semiri1314217659103216013at_int,K2))),one_one_int)),K2)) ).

% pochhammer_minus
tff(fact_8115_pochhammer__minus_H,axiom,
    ! [B: complex,K2: nat] : aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,aa_complex_complex(plus_plus_complex(aa_complex_complex(minus_minus_complex(B),aa_nat_complex(semiri8010041392384452111omplex,K2))),one_one_complex)),K2) = aa_complex_complex(times_times_complex(aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K2)),aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,aa_complex_complex(uminus1482373934393186551omplex,B)),K2)) ).

% pochhammer_minus'
tff(fact_8116_pochhammer__minus_H,axiom,
    ! [B: rat,K2: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(B),aa_nat_rat(semiri681578069525770553at_rat,K2))),one_one_rat)),K2) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K2)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,B)),K2)) ).

% pochhammer_minus'
tff(fact_8117_pochhammer__minus_H,axiom,
    ! [B: code_integer,K2: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(plus_p5714425477246183910nteger(minus_8373710615458151222nteger(B,aa_nat_Code_integer(semiri4939895301339042750nteger,K2)),one_one_Code_integer)),K2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),K2)),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),K2)) ).

% pochhammer_minus'
tff(fact_8118_pochhammer__minus_H,axiom,
    ! [B: real,K2: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(plus_plus_real(aa_real_real(minus_minus_real(B),aa_nat_real(semiri5074537144036343181t_real,K2))),one_one_real)),K2) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),K2)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,B)),K2)) ).

% pochhammer_minus'
tff(fact_8119_pochhammer__minus_H,axiom,
    ! [B: int,K2: nat] : aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),aa_nat_int(semiri1314217659103216013at_int,K2))),one_one_int)),K2) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),K2)),aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(uminus_uminus_int,B)),K2)) ).

% pochhammer_minus'
tff(fact_8120_termdiff__converges,axiom,
    ! [X: real,K5: real,C: fun_nat_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X)),K5)
     => ( ! [X4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X4)),K5)
           => summable_real(aa_real_fun_nat_real(aTP_Lamm_fn(C),X4)) )
       => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_hb(X),C)) ) ) ).

% termdiff_converges
tff(fact_8121_termdiff__converges,axiom,
    ! [X: complex,K5: real,C: fun_nat_complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X)),K5)
     => ( ! [X4: complex] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X4)),K5)
           => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_fp(C),X4)) )
       => summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_hc(X),C)) ) ) ).

% termdiff_converges
tff(fact_8122_floor__add,axiom,
    ! [X: real,Y2: real] :
      aa_real_int(archim6058952711729229775r_real,aa_real_real(plus_plus_real(X),Y2)) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(aa_real_real(archim2898591450579166408c_real,X)),aa_real_real(archim2898591450579166408c_real,Y2))),one_one_real),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,X)),aa_real_int(archim6058952711729229775r_real,Y2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,X)),aa_real_int(archim6058952711729229775r_real,Y2))),one_one_int)) ).

% floor_add
tff(fact_8123_floor__add,axiom,
    ! [X: rat,Y2: rat] :
      aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(plus_plus_rat(X),Y2)) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(archimedean_frac_rat,X)),aa_rat_rat(archimedean_frac_rat,Y2))),one_one_rat),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_rat_int(archim3151403230148437115or_rat,Y2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_rat_int(archim3151403230148437115or_rat,Y2))),one_one_int)) ).

% floor_add
tff(fact_8124_pochhammer__code,axiom,
    ! [A: complex,Nb: nat] :
      aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,A),Nb) = $ite(Nb = zero_zero_nat,one_one_complex,aa_complex_complex(set_fo1517530859248394432omplex(aTP_Lamm_hd(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),one_one_complex)) ).

% pochhammer_code
tff(fact_8125_pochhammer__code,axiom,
    ! [A: rat,Nb: nat] :
      aa_nat_rat(comm_s4028243227959126397er_rat(A),Nb) = $ite(Nb = zero_zero_nat,one_one_rat,aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_he(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),one_one_rat)) ).

% pochhammer_code
tff(fact_8126_pochhammer__code,axiom,
    ! [A: real,Nb: nat] :
      aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Nb) = $ite(Nb = zero_zero_nat,one_one_real,aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_hf(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),one_one_real)) ).

% pochhammer_code
tff(fact_8127_pochhammer__code,axiom,
    ! [A: int,Nb: nat] :
      aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,A),Nb) = $ite(Nb = zero_zero_nat,one_one_int,aa_int_int(set_fo2581907887559384638at_int(aTP_Lamm_hg(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),one_one_int)) ).

% pochhammer_code
tff(fact_8128_pochhammer__code,axiom,
    ! [A: extended_enat,Nb: nat] :
      aa_nat_Extended_enat(comm_s3181272606743183617d_enat(A),Nb) = $ite(Nb = zero_zero_nat,one_on7984719198319812577d_enat,set_fo2538466533108834004d_enat(aTP_Lamm_hh(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat),one_on7984719198319812577d_enat)) ).

% pochhammer_code
tff(fact_8129_pochhammer__code,axiom,
    ! [A: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),Nb) = $ite(Nb = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_hi(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),one_one_nat)) ).

% pochhammer_code
tff(fact_8130_cos__paired,axiom,
    ! [X: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_hj,X)),aa_real_real(cos_real,X)) ).

% cos_paired
tff(fact_8131_sin__coeff__def,axiom,
    ! [X2: nat] :
      aa_nat_real(sin_coeff,X2) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X2),zero_zero_real,aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X2),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(semiri2265585572941072030t_real,X2))) ).

% sin_coeff_def
tff(fact_8132_choose__dvd,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri3624122377584611663nteger,K2)),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)))),aa_nat_Code_integer(semiri3624122377584611663nteger,Nb)) ) ).

% choose_dvd
tff(fact_8133_choose__dvd,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(aa_nat_rat(semiri773545260158071498ct_rat,K2)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)))),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ) ).

% choose_dvd
tff(fact_8134_choose__dvd,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1406184849735516958ct_int,K2)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)))),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ) ).

% choose_dvd
tff(fact_8135_choose__dvd,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(times_times_real(aa_nat_real(semiri2265585572941072030t_real,K2)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)))),aa_nat_real(semiri2265585572941072030t_real,Nb)) ) ).

% choose_dvd
tff(fact_8136_choose__dvd,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K2)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)))),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ).

% choose_dvd
tff(fact_8137_cos__coeff__def,axiom,
    ! [X2: nat] :
      aa_nat_real(cos_coeff,X2) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X2),aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(semiri2265585572941072030t_real,X2)),zero_zero_real) ).

% cos_coeff_def
tff(fact_8138_fact__fact__dvd__fact,axiom,
    ! [K2: nat,Nb: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri3624122377584611663nteger,K2)),aa_nat_Code_integer(semiri3624122377584611663nteger,Nb))),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),Nb))) ).

% fact_fact_dvd_fact
tff(fact_8139_fact__fact__dvd__fact,axiom,
    ! [K2: nat,Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(aa_nat_rat(semiri773545260158071498ct_rat,K2)),aa_nat_rat(semiri773545260158071498ct_rat,Nb))),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),Nb))) ).

% fact_fact_dvd_fact
tff(fact_8140_fact__fact__dvd__fact,axiom,
    ! [K2: nat,Nb: nat] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1406184849735516958ct_int,K2)),aa_nat_int(semiri1406184849735516958ct_int,Nb))),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),Nb))) ).

% fact_fact_dvd_fact
tff(fact_8141_fact__fact__dvd__fact,axiom,
    ! [K2: nat,Nb: nat] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(times_times_real(aa_nat_real(semiri2265585572941072030t_real,K2)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),Nb))) ).

% fact_fact_dvd_fact
tff(fact_8142_fact__fact__dvd__fact,axiom,
    ! [K2: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K2)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb))),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),Nb))) ).

% fact_fact_dvd_fact
tff(fact_8143_central__binomial__lower__bound,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))),Nb)),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Nb)))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),Nb))) ) ).

% central_binomial_lower_bound
tff(fact_8144_binomial__n__n,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),Nb) = one_one_nat ).

% binomial_n_n
tff(fact_8145_binomial__eq__0__iff,axiom,
    ! [Nb: nat,K2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2) = zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K2) ) ).

% binomial_eq_0_iff
tff(fact_8146_binomial__Suc__Suc,axiom,
    ! [Nb: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),aa_nat_nat(suc,K2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(suc,K2))) ).

% binomial_Suc_Suc
tff(fact_8147_binomial__n__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),zero_zero_nat) = one_one_nat ).

% binomial_n_0
tff(fact_8148_cos__coeff__0,axiom,
    aa_nat_real(cos_coeff,zero_zero_nat) = one_one_real ).

% cos_coeff_0
tff(fact_8149_zero__less__binomial__iff,axiom,
    ! [Nb: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb) ) ).

% zero_less_binomial_iff
tff(fact_8150_binomial__fact__lemma,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K2)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)) = aa_nat_nat(semiri1408675320244567234ct_nat,Nb) ) ) ).

% binomial_fact_lemma
tff(fact_8151_fact__mono__nat,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Ma)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ).

% fact_mono_nat
tff(fact_8152_fact__ge__self,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ).

% fact_ge_self
tff(fact_8153_binomial__eq__0,axiom,
    ! [Nb: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2) = zero_zero_nat ) ) ).

% binomial_eq_0
tff(fact_8154_choose__one,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),one_one_nat) = Nb ).

% choose_one
tff(fact_8155_Suc__times__binomial__eq,axiom,
    ! [Nb: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),aa_nat_nat(suc,K2))),aa_nat_nat(suc,K2)) ).

% Suc_times_binomial_eq
tff(fact_8156_Suc__times__binomial,axiom,
    ! [K2: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),aa_nat_nat(suc,K2))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)) ).

% Suc_times_binomial
tff(fact_8157_binomial__symmetric,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)) ) ) ).

% binomial_symmetric
tff(fact_8158_choose__mult__lemma,axiom,
    ! [Ma: nat,R: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),R)),K2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),K2))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),K2)),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),R)),K2)),K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),R)),Ma)) ).

% choose_mult_lemma
tff(fact_8159_binomial__le__pow,axiom,
    ! [R: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,R),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),R)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),R)) ) ).

% binomial_le_pow
tff(fact_8160_binomial__altdef__nat,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K2)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)))) ) ) ).

% binomial_altdef_nat
tff(fact_8161_zero__less__binomial,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)) ) ).

% zero_less_binomial
tff(fact_8162_fact__less__mono__nat,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ma)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Ma)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ) ).

% fact_less_mono_nat
tff(fact_8163_Suc__times__binomial__add,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,A)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))),aa_nat_nat(suc,A))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,B)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))),A)) ).

% Suc_times_binomial_add
tff(fact_8164_choose__mult,axiom,
    ! [K2: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Ma)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Ma),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),K2))) ) ) ) ).

% choose_mult
tff(fact_8165_binomial__Suc__Suc__eq__times,axiom,
    ! [Nb: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),aa_nat_nat(suc,K2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2))),aa_nat_nat(suc,K2)) ).

% binomial_Suc_Suc_eq_times
tff(fact_8166_binomial__absorb__comp,axiom,
    ! [Nb: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),K2)) ).

% binomial_absorb_comp
tff(fact_8167_binomial__absorption,axiom,
    ! [K2: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(suc,K2))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),K2)) ).

% binomial_absorption
tff(fact_8168_fact__ge__Suc__0__nat,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ).

% fact_ge_Suc_0_nat
tff(fact_8169_dvd__fact,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Ma)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ma),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ) ).

% dvd_fact
tff(fact_8170_binomial__ge__n__over__k__pow__k,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,K2))),K2)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2))) ) ).

% binomial_ge_n_over_k_pow_k
tff(fact_8171_binomial__ge__n__over__k__pow__k,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_rat_rat(divide_divide_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),aa_nat_rat(semiri681578069525770553at_rat,K2))),K2)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2))) ) ).

% binomial_ge_n_over_k_pow_k
tff(fact_8172_binomial__le__pow2,axiom,
    ! [Nb: nat,K2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% binomial_le_pow2
tff(fact_8173_choose__reduce__nat,axiom,
    ! [Nb: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),K2)) ) ) ) ).

% choose_reduce_nat
tff(fact_8174_times__binomial__minus1__eq,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),one_one_nat))) ) ) ).

% times_binomial_minus1_eq
tff(fact_8175_fact__diff__Suc,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Ma))
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Ma)),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Ma)),Nb)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb))) ) ) ).

% fact_diff_Suc
tff(fact_8176_binomial__code,axiom,
    ! [Nb: nat,K2: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K2),
        zero_zero_nat,
        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)),one_one_nat)),Nb),one_one_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,K2))) ) ).

% binomial_code
tff(fact_8177_fact__div__fact__le__pow,axiom,
    ! [R: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,R),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),R)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),R)) ) ).

% fact_div_fact_le_pow
tff(fact_8178_binomial__addition__formula,axiom,
    ! [Nb: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(suc,K2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),aa_nat_nat(suc,K2))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),K2)) ) ) ).

% binomial_addition_formula
tff(fact_8179_fact__binomial,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(semiri5044797733671781792omplex,K2)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2))) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(semiri5044797733671781792omplex,Nb)),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2))) ) ) ).

% fact_binomial
tff(fact_8180_fact__binomial,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_rat_rat(times_times_rat(aa_nat_rat(semiri773545260158071498ct_rat,K2)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2))) = aa_rat_rat(divide_divide_rat(aa_nat_rat(semiri773545260158071498ct_rat,Nb)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2))) ) ) ).

% fact_binomial
tff(fact_8181_fact__binomial,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_real_real(times_times_real(aa_nat_real(semiri2265585572941072030t_real,K2)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2))) = aa_real_real(divide_divide_real(aa_nat_real(semiri2265585572941072030t_real,Nb)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2))) ) ) ).

% fact_binomial
tff(fact_8182_binomial__fact,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(semiri5044797733671781792omplex,Nb)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri5044797733671781792omplex,K2)),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)))) ) ) ).

% binomial_fact
tff(fact_8183_binomial__fact,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)) = aa_rat_rat(divide_divide_rat(aa_nat_rat(semiri773545260158071498ct_rat,Nb)),aa_rat_rat(times_times_rat(aa_nat_rat(semiri773545260158071498ct_rat,K2)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)))) ) ) ).

% binomial_fact
tff(fact_8184_binomial__fact,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)) = aa_real_real(divide_divide_real(aa_nat_real(semiri2265585572941072030t_real,Nb)),aa_real_real(times_times_real(aa_nat_real(semiri2265585572941072030t_real,K2)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)))) ) ) ).

% binomial_fact
tff(fact_8185_binomial__mono,axiom,
    ! [K2: nat,K6: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),K6)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K6)),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K6)) ) ) ).

% binomial_mono
tff(fact_8186_binomial__maximum_H,axiom,
    ! [Nb: nat,K2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),Nb)) ).

% binomial_maximum'
tff(fact_8187_binomial__maximum,axiom,
    ! [Nb: nat,K2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% binomial_maximum
tff(fact_8188_binomial__antimono,axiom,
    ! [K2: nat,K6: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),K6)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),K2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K6),Nb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K6)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)) ) ) ) ).

% binomial_antimono
tff(fact_8189_choose__two,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% choose_two
tff(fact_8190_binomial__strict__mono,axiom,
    ! [K2: nat,K6: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),K6)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K6)),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K6)) ) ) ).

% binomial_strict_mono
tff(fact_8191_binomial__strict__antimono,axiom,
    ! [K2: nat,K6: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),K6)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K2))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K6),Nb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K6)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)) ) ) ) ).

% binomial_strict_antimono
tff(fact_8192_binomial__less__binomial__Suc,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(suc,K2))) ) ).

% binomial_less_binomial_Suc
tff(fact_8193_central__binomial__odd,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% central_binomial_odd
tff(fact_8194_exp__two__pi__i_H,axiom,
    aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(imaginary_unit),aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,pi)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) = one_one_complex ).

% exp_two_pi_i'
tff(fact_8195_exp__two__pi__i,axiom,
    aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_real_complex(real_V4546457046886955230omplex,pi))),imaginary_unit)) = one_one_complex ).

% exp_two_pi_i
tff(fact_8196_cot__less__zero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cot_real,X)),zero_zero_real) ) ) ).

% cot_less_zero
tff(fact_8197_gbinomial__code,axiom,
    ! [A: complex,K2: nat] :
      aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,A),K2) = $ite(K2 = zero_zero_nat,one_one_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(set_fo1517530859248394432omplex(aTP_Lamm_hk(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),one_one_nat)),one_one_complex)),aa_nat_complex(semiri5044797733671781792omplex,K2))) ).

% gbinomial_code
tff(fact_8198_gbinomial__code,axiom,
    ! [A: rat,K2: nat] :
      aa_nat_rat(gbinomial_rat(A),K2) = $ite(K2 = zero_zero_nat,one_one_rat,aa_rat_rat(divide_divide_rat(aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_hl(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),one_one_nat)),one_one_rat)),aa_nat_rat(semiri773545260158071498ct_rat,K2))) ).

% gbinomial_code
tff(fact_8199_gbinomial__code,axiom,
    ! [A: real,K2: nat] :
      aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2) = $ite(K2 = zero_zero_nat,one_one_real,aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_hm(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),one_one_nat)),one_one_real)),aa_nat_real(semiri2265585572941072030t_real,K2))) ).

% gbinomial_code
tff(fact_8200_cot__periodic,axiom,
    ! [X: real] : aa_real_real(cot_real,aa_real_real(plus_plus_real(X),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) = aa_real_real(cot_real,X) ).

% cot_periodic
tff(fact_8201_gbinomial__0_I1_J,axiom,
    ! [A: complex] : aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,A),zero_zero_nat) = one_one_complex ).

% gbinomial_0(1)
tff(fact_8202_gbinomial__0_I1_J,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),zero_zero_nat) = one_one_real ).

% gbinomial_0(1)
tff(fact_8203_gbinomial__0_I1_J,axiom,
    ! [A: rat] : aa_nat_rat(gbinomial_rat(A),zero_zero_nat) = one_one_rat ).

% gbinomial_0(1)
tff(fact_8204_gbinomial__0_I1_J,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gbinomial_nat,A),zero_zero_nat) = one_one_nat ).

% gbinomial_0(1)
tff(fact_8205_gbinomial__0_I1_J,axiom,
    ! [A: int] : aa_nat_int(aa_int_fun_nat_int(gbinomial_int,A),zero_zero_nat) = one_one_int ).

% gbinomial_0(1)
tff(fact_8206_norm__ii,axiom,
    aa_complex_real(real_V1022390504157884413omplex,imaginary_unit) = one_one_real ).

% norm_ii
tff(fact_8207_divide__i,axiom,
    ! [X: complex] : aa_complex_complex(divide1717551699836669952omplex(X),imaginary_unit) = aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,imaginary_unit)),X) ).

% divide_i
tff(fact_8208_complex__i__mult__minus,axiom,
    ! [X: complex] : aa_complex_complex(times_times_complex(imaginary_unit),aa_complex_complex(times_times_complex(imaginary_unit),X)) = aa_complex_complex(uminus1482373934393186551omplex,X) ).

% complex_i_mult_minus
tff(fact_8209_i__squared,axiom,
    aa_complex_complex(times_times_complex(imaginary_unit),imaginary_unit) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% i_squared
tff(fact_8210_divide__numeral__i,axiom,
    ! [Z2: complex,Nb: num] : aa_complex_complex(divide1717551699836669952omplex(Z2),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(Nb)),imaginary_unit)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(times_times_complex(imaginary_unit),Z2))),numera6690914467698888265omplex(Nb)) ).

% divide_numeral_i
tff(fact_8211_cot__npi,axiom,
    ! [Nb: nat] : aa_real_real(cot_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),pi)) = zero_zero_real ).

% cot_npi
tff(fact_8212_power2__i,axiom,
    aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,imaginary_unit),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% power2_i
tff(fact_8213_i__even__power,axiom,
    ! [Nb: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,imaginary_unit),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb) ).

% i_even_power
tff(fact_8214_exp__pi__i,axiom,
    aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,pi)),imaginary_unit)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% exp_pi_i
tff(fact_8215_exp__pi__i_H,axiom,
    aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,pi))) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% exp_pi_i'
tff(fact_8216_complex__i__not__one,axiom,
    imaginary_unit != one_one_complex ).

% complex_i_not_one
tff(fact_8217_i__times__eq__iff,axiom,
    ! [W: complex,Z2: complex] :
      ( ( aa_complex_complex(times_times_complex(imaginary_unit),W) = Z2 )
    <=> ( W = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(times_times_complex(imaginary_unit),Z2)) ) ) ).

% i_times_eq_iff
tff(fact_8218_gbinomial__Suc__Suc,axiom,
    ! [A: complex,K2: nat] : aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_nat(suc,K2)) = aa_complex_complex(plus_plus_complex(aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,A),K2)),aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,A),aa_nat_nat(suc,K2))) ).

% gbinomial_Suc_Suc
tff(fact_8219_gbinomial__Suc__Suc,axiom,
    ! [A: real,K2: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(plus_plus_real(A),one_one_real)),aa_nat_nat(suc,K2)) = aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(suc,K2))) ).

% gbinomial_Suc_Suc
tff(fact_8220_gbinomial__Suc__Suc,axiom,
    ! [A: rat,K2: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(A),one_one_rat)),aa_nat_nat(suc,K2)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(gbinomial_rat(A),K2)),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K2))) ).

% gbinomial_Suc_Suc
tff(fact_8221_gbinomial__of__nat__symmetric,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),K2) = aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K2)) ) ) ).

% gbinomial_of_nat_symmetric
tff(fact_8222_imaginary__unit_Ocode,axiom,
    imaginary_unit = aa_real_complex(complex2(zero_zero_real),one_one_real) ).

% imaginary_unit.code
tff(fact_8223_Complex__eq__i,axiom,
    ! [X: real,Y2: real] :
      ( ( aa_real_complex(complex2(X),Y2) = imaginary_unit )
    <=> ( ( X = zero_zero_real )
        & ( Y2 = one_one_real ) ) ) ).

% Complex_eq_i
tff(fact_8224_i__mult__Complex,axiom,
    ! [A: real,B: real] : aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(complex2(A),B)) = aa_real_complex(complex2(aa_real_real(uminus_uminus_real,B)),A) ).

% i_mult_Complex
tff(fact_8225_Complex__mult__i,axiom,
    ! [A: real,B: real] : aa_complex_complex(times_times_complex(aa_real_complex(complex2(A),B)),imaginary_unit) = aa_real_complex(complex2(aa_real_real(uminus_uminus_real,B)),A) ).

% Complex_mult_i
tff(fact_8226_gbinomial__addition__formula,axiom,
    ! [A: complex,K2: nat] : aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,A),aa_nat_nat(suc,K2)) = aa_complex_complex(plus_plus_complex(aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,aa_complex_complex(minus_minus_complex(A),one_one_complex)),aa_nat_nat(suc,K2))),aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,aa_complex_complex(minus_minus_complex(A),one_one_complex)),K2)) ).

% gbinomial_addition_formula
tff(fact_8227_gbinomial__addition__formula,axiom,
    ! [A: real,K2: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(suc,K2)) = aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(minus_minus_real(A),one_one_real)),aa_nat_nat(suc,K2))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(minus_minus_real(A),one_one_real)),K2)) ).

% gbinomial_addition_formula
tff(fact_8228_gbinomial__addition__formula,axiom,
    ! [A: rat,K2: nat] : aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K2)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),one_one_rat)),aa_nat_nat(suc,K2))),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),one_one_rat)),K2)) ).

% gbinomial_addition_formula
tff(fact_8229_gbinomial__absorb__comp,axiom,
    ! [A: complex,K2: nat] : aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,K2))),aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,A),K2)) = aa_complex_complex(times_times_complex(A),aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,aa_complex_complex(minus_minus_complex(A),one_one_complex)),K2)) ).

% gbinomial_absorb_comp
tff(fact_8230_gbinomial__absorb__comp,axiom,
    ! [A: rat,K2: nat] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A),aa_nat_rat(semiri681578069525770553at_rat,K2))),aa_nat_rat(gbinomial_rat(A),K2)) = aa_rat_rat(times_times_rat(A),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),one_one_rat)),K2)) ).

% gbinomial_absorb_comp
tff(fact_8231_gbinomial__absorb__comp,axiom,
    ! [A: real,K2: nat] : aa_real_real(times_times_real(aa_real_real(minus_minus_real(A),aa_nat_real(semiri5074537144036343181t_real,K2))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2)) = aa_real_real(times_times_real(A),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(minus_minus_real(A),one_one_real)),K2)) ).

% gbinomial_absorb_comp
tff(fact_8232_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [K2: nat,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,K2)),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(divide_divide_real(A),aa_nat_real(semiri5074537144036343181t_real,K2))),K2)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2)) ) ).

% gbinomial_ge_n_over_k_pow_k
tff(fact_8233_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [K2: nat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,K2)),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_rat_rat(divide_divide_rat(A),aa_nat_rat(semiri681578069525770553at_rat,K2))),K2)),aa_nat_rat(gbinomial_rat(A),K2)) ) ).

% gbinomial_ge_n_over_k_pow_k
tff(fact_8234_gbinomial__mult__1,axiom,
    ! [A: rat,K2: nat] : aa_rat_rat(times_times_rat(A),aa_nat_rat(gbinomial_rat(A),K2)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,K2)),aa_nat_rat(gbinomial_rat(A),K2))),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K2))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K2)))) ).

% gbinomial_mult_1
tff(fact_8235_gbinomial__mult__1,axiom,
    ! [A: real,K2: nat] : aa_real_real(times_times_real(A),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,K2)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2))),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K2))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(suc,K2)))) ).

% gbinomial_mult_1
tff(fact_8236_gbinomial__mult__1_H,axiom,
    ! [A: rat,K2: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(A),K2)),A) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,K2)),aa_nat_rat(gbinomial_rat(A),K2))),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K2))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K2)))) ).

% gbinomial_mult_1'
tff(fact_8237_gbinomial__mult__1_H,axiom,
    ! [A: real,K2: nat] : aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2)),A) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,K2)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2))),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K2))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(suc,K2)))) ).

% gbinomial_mult_1'
tff(fact_8238_cot__def,axiom,
    ! [X2: complex] : aa_complex_complex(cot_complex,X2) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(cos_complex,X2)),aa_complex_complex(sin_complex,X2)) ).

% cot_def
tff(fact_8239_cot__def,axiom,
    ! [X2: real] : aa_real_real(cot_real,X2) = aa_real_real(divide_divide_real(aa_real_real(cos_real,X2)),aa_real_real(sin_real,X2)) ).

% cot_def
tff(fact_8240_Suc__times__gbinomial,axiom,
    ! [K2: nat,A: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K2))),aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_nat(suc,K2))) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,A),K2)) ).

% Suc_times_gbinomial
tff(fact_8241_Suc__times__gbinomial,axiom,
    ! [K2: nat,A: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K2))),aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(A),one_one_rat)),aa_nat_nat(suc,K2))) = aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A),one_one_rat)),aa_nat_rat(gbinomial_rat(A),K2)) ).

% Suc_times_gbinomial
tff(fact_8242_Suc__times__gbinomial,axiom,
    ! [K2: nat,A: real] : aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K2))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(plus_plus_real(A),one_one_real)),aa_nat_nat(suc,K2))) = aa_real_real(times_times_real(aa_real_real(plus_plus_real(A),one_one_real)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2)) ).

% Suc_times_gbinomial
tff(fact_8243_gbinomial__absorption,axiom,
    ! [K2: nat,A: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K2))),aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,A),aa_nat_nat(suc,K2))) = aa_complex_complex(times_times_complex(A),aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,aa_complex_complex(minus_minus_complex(A),one_one_complex)),K2)) ).

% gbinomial_absorption
tff(fact_8244_gbinomial__absorption,axiom,
    ! [K2: nat,A: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K2))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K2))) = aa_rat_rat(times_times_rat(A),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),one_one_rat)),K2)) ).

% gbinomial_absorption
tff(fact_8245_gbinomial__absorption,axiom,
    ! [K2: nat,A: real] : aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K2))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(suc,K2))) = aa_real_real(times_times_real(A),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(minus_minus_real(A),one_one_real)),K2)) ).

% gbinomial_absorption
tff(fact_8246_gbinomial__trinomial__revision,axiom,
    ! [K2: nat,Ma: nat,A: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Ma)
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,A),Ma)),aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,aa_nat_complex(semiri8010041392384452111omplex,Ma)),K2)) = aa_complex_complex(times_times_complex(aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,A),K2)),aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,aa_complex_complex(minus_minus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,K2))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),K2))) ) ) ).

% gbinomial_trinomial_revision
tff(fact_8247_gbinomial__trinomial__revision,axiom,
    ! [K2: nat,Ma: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Ma)
     => ( aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(A),Ma)),aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,Ma)),K2)) = aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(A),K2)),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),aa_nat_rat(semiri681578069525770553at_rat,K2))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),K2))) ) ) ).

% gbinomial_trinomial_revision
tff(fact_8248_gbinomial__trinomial__revision,axiom,
    ! [K2: nat,Ma: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Ma)
     => ( aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),Ma)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_nat_real(semiri5074537144036343181t_real,Ma)),K2)) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(minus_minus_real(A),aa_nat_real(semiri5074537144036343181t_real,K2))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),K2))) ) ) ).

% gbinomial_trinomial_revision
tff(fact_8249_complex__of__real__i,axiom,
    ! [R: real] : aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,R)),imaginary_unit) = aa_real_complex(complex2(zero_zero_real),R) ).

% complex_of_real_i
tff(fact_8250_i__complex__of__real,axiom,
    ! [R: real] : aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,R)) = aa_real_complex(complex2(zero_zero_real),R) ).

% i_complex_of_real
tff(fact_8251_Complex__eq,axiom,
    ! [A: real,B: real] : aa_real_complex(complex2(A),B) = aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,A)),aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,B))) ).

% Complex_eq
tff(fact_8252_gbinomial__factors,axiom,
    ! [A: complex,K2: nat] : aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_nat(suc,K2)) = aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K2)))),aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,A),K2)) ).

% gbinomial_factors
tff(fact_8253_gbinomial__factors,axiom,
    ! [A: rat,K2: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(A),one_one_rat)),aa_nat_nat(suc,K2)) = aa_rat_rat(times_times_rat(aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(A),one_one_rat)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K2)))),aa_nat_rat(gbinomial_rat(A),K2)) ).

% gbinomial_factors
tff(fact_8254_gbinomial__factors,axiom,
    ! [A: real,K2: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(plus_plus_real(A),one_one_real)),aa_nat_nat(suc,K2)) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(A),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K2)))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2)) ).

% gbinomial_factors
tff(fact_8255_gbinomial__rec,axiom,
    ! [A: complex,K2: nat] : aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_nat(suc,K2)) = aa_complex_complex(times_times_complex(aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,A),K2)),aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K2)))) ).

% gbinomial_rec
tff(fact_8256_gbinomial__rec,axiom,
    ! [A: rat,K2: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(A),one_one_rat)),aa_nat_nat(suc,K2)) = aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(A),K2)),aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(A),one_one_rat)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K2)))) ).

% gbinomial_rec
tff(fact_8257_gbinomial__rec,axiom,
    ! [A: real,K2: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(plus_plus_real(A),one_one_real)),aa_nat_nat(suc,K2)) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2)),aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(A),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K2)))) ).

% gbinomial_rec
tff(fact_8258_gbinomial__negated__upper,axiom,
    ! [A: complex,K2: nat] : aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,A),K2) = aa_complex_complex(times_times_complex(aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K2)),aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,aa_complex_complex(minus_minus_complex(aa_complex_complex(minus_minus_complex(aa_nat_complex(semiri8010041392384452111omplex,K2)),A)),one_one_complex)),K2)) ).

% gbinomial_negated_upper
tff(fact_8259_gbinomial__negated__upper,axiom,
    ! [A: rat,K2: nat] : aa_nat_rat(gbinomial_rat(A),K2) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K2)),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(aa_nat_rat(semiri681578069525770553at_rat,K2)),A)),one_one_rat)),K2)) ).

% gbinomial_negated_upper
tff(fact_8260_gbinomial__negated__upper,axiom,
    ! [A: real,K2: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),K2)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(minus_minus_real(aa_real_real(minus_minus_real(aa_nat_real(semiri5074537144036343181t_real,K2)),A)),one_one_real)),K2)) ).

% gbinomial_negated_upper
tff(fact_8261_gbinomial__index__swap,axiom,
    ! [K2: nat,Nb: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K2)),aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,Nb))),one_one_complex)),K2)) = aa_complex_complex(times_times_complex(aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb)),aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,K2))),one_one_complex)),Nb)) ).

% gbinomial_index_swap
tff(fact_8262_gbinomial__index__swap,axiom,
    ! [K2: nat,Nb: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K2)),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb))),one_one_rat)),K2)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb)),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,K2))),one_one_rat)),Nb)) ).

% gbinomial_index_swap
tff(fact_8263_gbinomial__index__swap,axiom,
    ! [K2: nat,Nb: nat] : aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),K2)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Nb))),one_one_real)),K2)) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,K2))),one_one_real)),Nb)) ).

% gbinomial_index_swap
tff(fact_8264_complex__split__polar,axiom,
    ! [Z2: complex] :
    ? [R3: real,A3: real] : Z2 = aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,R3)),aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,aa_real_real(cos_real,A3))),aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sin_real,A3))))) ).

% complex_split_polar
tff(fact_8265_gbinomial__minus,axiom,
    ! [A: complex,K2: nat] : aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,aa_complex_complex(uminus1482373934393186551omplex,A)),K2) = aa_complex_complex(times_times_complex(aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K2)),aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,K2))),one_one_complex)),K2)) ).

% gbinomial_minus
tff(fact_8266_gbinomial__minus,axiom,
    ! [A: rat,K2: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(uminus_uminus_rat,A)),K2) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K2)),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),aa_nat_rat(semiri681578069525770553at_rat,K2))),one_one_rat)),K2)) ).

% gbinomial_minus
tff(fact_8267_gbinomial__minus,axiom,
    ! [A: real,K2: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(uminus_uminus_real,A)),K2) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),K2)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(A),aa_nat_real(semiri5074537144036343181t_real,K2))),one_one_real)),K2)) ).

% gbinomial_minus
tff(fact_8268_gbinomial__reduce__nat,axiom,
    ! [K2: nat,A: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
     => ( aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,A),K2) = aa_complex_complex(plus_plus_complex(aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,aa_complex_complex(minus_minus_complex(A),one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),one_one_nat))),aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,aa_complex_complex(minus_minus_complex(A),one_one_complex)),K2)) ) ) ).

% gbinomial_reduce_nat
tff(fact_8269_gbinomial__reduce__nat,axiom,
    ! [K2: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
     => ( aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2) = aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(minus_minus_real(A),one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),one_one_nat))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(minus_minus_real(A),one_one_real)),K2)) ) ) ).

% gbinomial_reduce_nat
tff(fact_8270_gbinomial__reduce__nat,axiom,
    ! [K2: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
     => ( aa_nat_rat(gbinomial_rat(A),K2) = aa_rat_rat(plus_plus_rat(aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),one_one_nat))),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),one_one_rat)),K2)) ) ) ).

% gbinomial_reduce_nat
tff(fact_8271_gbinomial__pochhammer,axiom,
    ! [A: complex,K2: nat] : aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,A),K2) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K2)),aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,aa_complex_complex(uminus1482373934393186551omplex,A)),K2))),aa_nat_complex(semiri5044797733671781792omplex,K2)) ).

% gbinomial_pochhammer
tff(fact_8272_gbinomial__pochhammer,axiom,
    ! [A: rat,K2: nat] : aa_nat_rat(gbinomial_rat(A),K2) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K2)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,A)),K2))),aa_nat_rat(semiri773545260158071498ct_rat,K2)) ).

% gbinomial_pochhammer
tff(fact_8273_gbinomial__pochhammer,axiom,
    ! [A: real,K2: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),K2)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,A)),K2))),aa_nat_real(semiri2265585572941072030t_real,K2)) ).

% gbinomial_pochhammer
tff(fact_8274_gbinomial__pochhammer_H,axiom,
    ! [A: complex,K2: nat] : aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,A),K2) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(aa_com8181001469867462503omplex(comm_s2602460028002588243omplex,aa_complex_complex(plus_plus_complex(aa_complex_complex(minus_minus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,K2))),one_one_complex)),K2)),aa_nat_complex(semiri5044797733671781792omplex,K2)) ).

% gbinomial_pochhammer'
tff(fact_8275_gbinomial__pochhammer_H,axiom,
    ! [A: rat,K2: nat] : aa_nat_rat(gbinomial_rat(A),K2) = aa_rat_rat(divide_divide_rat(aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(A),aa_nat_rat(semiri681578069525770553at_rat,K2))),one_one_rat)),K2)),aa_nat_rat(semiri773545260158071498ct_rat,K2)) ).

% gbinomial_pochhammer'
tff(fact_8276_gbinomial__pochhammer_H,axiom,
    ! [A: real,K2: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2) = aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(plus_plus_real(aa_real_real(minus_minus_real(A),aa_nat_real(semiri5074537144036343181t_real,K2))),one_one_real)),K2)),aa_nat_real(semiri2265585572941072030t_real,K2)) ).

% gbinomial_pochhammer'
tff(fact_8277_gbinomial__absorption_H,axiom,
    ! [K2: nat,A: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
     => ( aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,A),K2) = aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(A),aa_nat_complex(semiri8010041392384452111omplex,K2))),aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,aa_complex_complex(minus_minus_complex(A),one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),one_one_nat))) ) ) ).

% gbinomial_absorption'
tff(fact_8278_gbinomial__absorption_H,axiom,
    ! [K2: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
     => ( aa_nat_rat(gbinomial_rat(A),K2) = aa_rat_rat(times_times_rat(aa_rat_rat(divide_divide_rat(A),aa_nat_rat(semiri681578069525770553at_rat,K2))),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),one_one_nat))) ) ) ).

% gbinomial_absorption'
tff(fact_8279_gbinomial__absorption_H,axiom,
    ! [K2: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
     => ( aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K2) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(A),aa_nat_real(semiri5074537144036343181t_real,K2))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(minus_minus_real(A),one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),one_one_nat))) ) ) ).

% gbinomial_absorption'
tff(fact_8280_cmod__unit__one,axiom,
    ! [A: real] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,aa_real_real(cos_real,A))),aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sin_real,A))))) = one_one_real ).

% cmod_unit_one
tff(fact_8281_cmod__complex__polar,axiom,
    ! [R: real,A: real] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,R)),aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,aa_real_real(cos_real,A))),aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sin_real,A)))))) = aa_real_real(abs_abs_real,R) ).

% cmod_complex_polar
tff(fact_8282_cot__gt__zero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(cot_real,X)) ) ) ).

% cot_gt_zero
tff(fact_8283_tan__cot_H,axiom,
    ! [X: real] : aa_real_real(tan_real,aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),X)) = aa_real_real(cot_real,X) ).

% tan_cot'
tff(fact_8284_Arg__minus__ii,axiom,
    aa_complex_real(arg,aa_complex_complex(uminus1482373934393186551omplex,imaginary_unit)) = aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% Arg_minus_ii
tff(fact_8285_csqrt__ii,axiom,
    aa_complex_complex(csqrt,imaginary_unit) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(one_one_complex),imaginary_unit)),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% csqrt_ii
tff(fact_8286_Arg__ii,axiom,
    aa_complex_real(arg,imaginary_unit) = aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% Arg_ii
tff(fact_8287_cis__minus__pi__half,axiom,
    aa_real_complex(cis,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) = aa_complex_complex(uminus1482373934393186551omplex,imaginary_unit) ).

% cis_minus_pi_half
tff(fact_8288_arctan__def,axiom,
    ! [Y2: real] : aa_real_real(arctan,Y2) = the_real(aa_real_fun_real_o(aTP_Lamm_hn,Y2)) ).

% arctan_def
tff(fact_8289_csqrt__eq__1,axiom,
    ! [Z2: complex] :
      ( ( aa_complex_complex(csqrt,Z2) = one_one_complex )
    <=> ( Z2 = one_one_complex ) ) ).

% csqrt_eq_1
tff(fact_8290_csqrt__1,axiom,
    aa_complex_complex(csqrt,one_one_complex) = one_one_complex ).

% csqrt_1
tff(fact_8291_norm__cis,axiom,
    ! [A: real] : aa_complex_real(real_V1022390504157884413omplex,aa_real_complex(cis,A)) = one_one_real ).

% norm_cis
tff(fact_8292_cis__zero,axiom,
    aa_real_complex(cis,zero_zero_real) = one_one_complex ).

% cis_zero
tff(fact_8293_cis__pi,axiom,
    aa_real_complex(cis,pi) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% cis_pi
tff(fact_8294_power2__csqrt,axiom,
    ! [Z2: complex] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(csqrt,Z2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Z2 ).

% power2_csqrt
tff(fact_8295_cis__pi__half,axiom,
    aa_real_complex(cis,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = imaginary_unit ).

% cis_pi_half
tff(fact_8296_cis__2pi,axiom,
    aa_real_complex(cis,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)) = one_one_complex ).

% cis_2pi
tff(fact_8297_DeMoivre,axiom,
    ! [A: real,Nb: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_real_complex(cis,A)),Nb) = aa_real_complex(cis,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),A)) ).

% DeMoivre
tff(fact_8298_cis__mult,axiom,
    ! [A: real,B: real] : aa_complex_complex(times_times_complex(aa_real_complex(cis,A)),aa_real_complex(cis,B)) = aa_real_complex(cis,aa_real_real(plus_plus_real(A),B)) ).

% cis_mult
tff(fact_8299_ln__real__def,axiom,
    ! [X: real] : aa_real_real(ln_ln_real,X) = the_real(aa_real_fun_real_o(aTP_Lamm_ho,X)) ).

% ln_real_def
tff(fact_8300_suminf__def,axiom,
    ! [F: fun_nat_int] : suminf_int(F) = the_int(sums_int(F)) ).

% suminf_def
tff(fact_8301_suminf__def,axiom,
    ! [F: fun_nat_complex] : suminf_complex(F) = the_complex(sums_complex(F)) ).

% suminf_def
tff(fact_8302_suminf__def,axiom,
    ! [F: fun_nat_real] : suminf_real(F) = the_real(sums_real(F)) ).

% suminf_def
tff(fact_8303_ln__neg__is__const,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),zero_zero_real)
     => ( aa_real_real(ln_ln_real,X) = the_real(aTP_Lamm_hp) ) ) ).

% ln_neg_is_const
tff(fact_8304_cis__conv__exp,axiom,
    ! [B: real] : aa_real_complex(cis,B) = aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,B))) ).

% cis_conv_exp
tff(fact_8305_arccos__def,axiom,
    ! [Y2: real] : aa_real_real(arccos,Y2) = the_real(aa_real_fun_real_o(aTP_Lamm_hq,Y2)) ).

% arccos_def
tff(fact_8306_of__real__sqrt,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sqrt,X)) = aa_complex_complex(csqrt,aa_real_complex(real_V4546457046886955230omplex,X)) ) ) ).

% of_real_sqrt
tff(fact_8307_Arg__bounded,axiom,
    ! [Z2: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),aa_complex_real(arg,Z2))
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(arg,Z2)),pi) ) ).

% Arg_bounded
tff(fact_8308_pi__half,axiom,
    aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) = the_real(aTP_Lamm_hr) ).

% pi_half
tff(fact_8309_pi__def,axiom,
    pi = aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),the_real(aTP_Lamm_hr)) ).

% pi_def
tff(fact_8310_arcsin__def,axiom,
    ! [Y2: real] : aa_real_real(arcsin,Y2) = the_real(aa_real_fun_real_o(aTP_Lamm_hs,Y2)) ).

% arcsin_def
tff(fact_8311_rec__enat__def,axiom,
    ! [X2: fun_nat_real,Xa3: real,Xb2: extended_enat] : extend3259767177765611793t_real(X2,Xa3,Xb2) = the_real(extend4979364747348822211t_real(X2,Xa3,Xb2)) ).

% rec_enat_def
tff(fact_8312_rec__enat__def,axiom,
    ! [X2: fun_nat_int,Xa3: int,Xb2: extended_enat] : extend7766020554117459729at_int(X2,Xa3,Xb2) = the_int(extend5266846564383770563at_int(X2,Xa3,Xb2)) ).

% rec_enat_def
tff(fact_8313_the__sym__eq__trivial,axiom,
    ! [X: real] : the_real(aa_real_fun_real_o(fequal_real,X)) = X ).

% the_sym_eq_trivial
tff(fact_8314_the__sym__eq__trivial,axiom,
    ! [X: int] : the_int(aa_int_fun_int_o(fequal_int,X)) = X ).

% the_sym_eq_trivial
tff(fact_8315_the__eq__trivial,axiom,
    ! [A: real] : the_real(aa_real_fun_real_o(aTP_Lamm_ht,A)) = A ).

% the_eq_trivial
tff(fact_8316_the__eq__trivial,axiom,
    ! [A: int] : the_int(aa_int_fun_int_o(aTP_Lamm_hu,A)) = A ).

% the_eq_trivial
tff(fact_8317_the__equality,axiom,
    ! [P: fun_real_o,A: real] :
      ( aa_real_o(P,A)
     => ( ! [X4: real] :
            ( aa_real_o(P,X4)
           => ( X4 = A ) )
       => ( the_real(P) = A ) ) ) ).

% the_equality
tff(fact_8318_the__equality,axiom,
    ! [P: fun_int_o,A: int] :
      ( aa_int_o(P,A)
     => ( ! [X4: int] :
            ( aa_int_o(P,X4)
           => ( X4 = A ) )
       => ( the_int(P) = A ) ) ) ).

% the_equality
tff(fact_8319_modulo__int__unfold,axiom,
    ! [K2: int,Ma: nat,L: int,Nb: nat] :
      aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K2)),aa_nat_int(semiri1314217659103216013at_int,Ma))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,Nb))) = $ite(
        ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
        | ( aa_int_int(sgn_sgn_int,K2) = zero_zero_int )
        | ( Nb = zero_zero_nat ) ),
        aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K2)),aa_nat_int(semiri1314217659103216013at_int,Ma)),
        $ite(aa_int_int(sgn_sgn_int,K2) = aa_int_int(sgn_sgn_int,L),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),Nb))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),Ma))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),Nb))))) ) ).

% modulo_int_unfold
tff(fact_8320_sgn__one,axiom,
    aa_real_real(sgn_sgn_real,one_one_real) = one_one_real ).

% sgn_one
tff(fact_8321_sgn__one,axiom,
    aa_complex_complex(sgn_sgn_complex,one_one_complex) = one_one_complex ).

% sgn_one
tff(fact_8322_sgn__1,axiom,
    aa_int_int(sgn_sgn_int,one_one_int) = one_one_int ).

% sgn_1
tff(fact_8323_sgn__1,axiom,
    aa_real_real(sgn_sgn_real,one_one_real) = one_one_real ).

% sgn_1
tff(fact_8324_sgn__1,axiom,
    aa_complex_complex(sgn_sgn_complex,one_one_complex) = one_one_complex ).

% sgn_1
tff(fact_8325_sgn__1,axiom,
    aa_rat_rat(sgn_sgn_rat,one_one_rat) = one_one_rat ).

% sgn_1
tff(fact_8326_sgn__1,axiom,
    aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,one_one_Code_integer) = one_one_Code_integer ).

% sgn_1
tff(fact_8327_sgn__divide,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(sgn_sgn_complex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(sgn_sgn_complex,A)),aa_complex_complex(sgn_sgn_complex,B)) ).

% sgn_divide
tff(fact_8328_sgn__divide,axiom,
    ! [A: real,B: real] : aa_real_real(sgn_sgn_real,aa_real_real(divide_divide_real(A),B)) = aa_real_real(divide_divide_real(aa_real_real(sgn_sgn_real,A)),aa_real_real(sgn_sgn_real,B)) ).

% sgn_divide
tff(fact_8329_sgn__divide,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(sgn_sgn_rat,aa_rat_rat(divide_divide_rat(A),B)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(sgn_sgn_rat,A)),aa_rat_rat(sgn_sgn_rat,B)) ).

% sgn_divide
tff(fact_8330_power__sgn,axiom,
    ! [A: rat,Nb: nat] : aa_rat_rat(sgn_sgn_rat,aa_nat_rat(power_power_rat(A),Nb)) = aa_nat_rat(power_power_rat(aa_rat_rat(sgn_sgn_rat,A)),Nb) ).

% power_sgn
tff(fact_8331_power__sgn,axiom,
    ! [A: code_integer,Nb: nat] : aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)),Nb) ).

% power_sgn
tff(fact_8332_power__sgn,axiom,
    ! [A: real,Nb: nat] : aa_real_real(sgn_sgn_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sgn_sgn_real,A)),Nb) ).

% power_sgn
tff(fact_8333_power__sgn,axiom,
    ! [A: int,Nb: nat] : aa_int_int(sgn_sgn_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb)) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(sgn_sgn_int,A)),Nb) ).

% power_sgn
tff(fact_8334_The__split__eq,axiom,
    ! [X: code_integer,Y2: $o] : the_Pr406357557219058217eger_o(produc7828578312038201481er_o_o(aa_o_f2459543391104474489un_o_o(aTP_Lamm_hv(X),(Y2)))) = aa_o_P4192039665577170892eger_o(aa_Cod3473573201699832512eger_o(produc6677183202524767010eger_o,X),(Y2)) ).

% The_split_eq
tff(fact_8335_The__split__eq,axiom,
    ! [X: num,Y2: num] : the_Pr6395592806576110876um_num(produc5703948589228662326_num_o(aa_num5226579579167915582_num_o(aTP_Lamm_hw(X),Y2))) = aa_num3156681643174609981um_num(aa_num8681986212680877308um_num(product_Pair_num_num,X),Y2) ).

% The_split_eq
tff(fact_8336_The__split__eq,axiom,
    ! [X: nat,Y2: num] : the_Pr8265262403268641490at_num(produc4927758841916487424_num_o(aa_num3446304307616776968_num_o(aTP_Lamm_hx(X),Y2))) = aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,X),Y2) ).

% The_split_eq
tff(fact_8337_The__split__eq,axiom,
    ! [X: nat,Y2: nat] : the_Pr7557018466319803784at_nat(produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_hy(X),Y2))) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X),Y2) ).

% The_split_eq
tff(fact_8338_The__split__eq,axiom,
    ! [X: int,Y2: int] : the_Pr4378521158711661632nt_int(produc4947309494688390418_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_hz(X),Y2))) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Y2) ).

% The_split_eq
tff(fact_8339_sgn__less,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)),zero_z3403309356797280102nteger)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ).

% sgn_less
tff(fact_8340_sgn__less,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sgn_sgn_real,A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% sgn_less
tff(fact_8341_sgn__less,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(sgn_sgn_rat,A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% sgn_less
tff(fact_8342_sgn__less,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(sgn_sgn_int,A)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% sgn_less
tff(fact_8343_sgn__greater,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A) ) ).

% sgn_greater
tff(fact_8344_sgn__greater,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sgn_sgn_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% sgn_greater
tff(fact_8345_sgn__greater,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(sgn_sgn_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% sgn_greater
tff(fact_8346_sgn__greater,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(sgn_sgn_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ).

% sgn_greater
tff(fact_8347_divide__sgn,axiom,
    ! [A: real,B: real] : aa_real_real(divide_divide_real(A),aa_real_real(sgn_sgn_real,B)) = aa_real_real(times_times_real(A),aa_real_real(sgn_sgn_real,B)) ).

% divide_sgn
tff(fact_8348_divide__sgn,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(divide_divide_rat(A),aa_rat_rat(sgn_sgn_rat,B)) = aa_rat_rat(times_times_rat(A),aa_rat_rat(sgn_sgn_rat,B)) ).

% divide_sgn
tff(fact_8349_sgn__pos,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) = one_one_Code_integer ) ) ).

% sgn_pos
tff(fact_8350_sgn__pos,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_real(sgn_sgn_real,A) = one_one_real ) ) ).

% sgn_pos
tff(fact_8351_sgn__pos,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_rat(sgn_sgn_rat,A) = one_one_rat ) ) ).

% sgn_pos
tff(fact_8352_sgn__pos,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_int(sgn_sgn_int,A) = one_one_int ) ) ).

% sgn_pos
tff(fact_8353_abs__sgn__eq__1,axiom,
    ! [A: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( abs_abs_Code_integer(aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)) = one_one_Code_integer ) ) ).

% abs_sgn_eq_1
tff(fact_8354_abs__sgn__eq__1,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(abs_abs_real,aa_real_real(sgn_sgn_real,A)) = one_one_real ) ) ).

% abs_sgn_eq_1
tff(fact_8355_abs__sgn__eq__1,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(abs_abs_rat,aa_rat_rat(sgn_sgn_rat,A)) = one_one_rat ) ) ).

% abs_sgn_eq_1
tff(fact_8356_abs__sgn__eq__1,axiom,
    ! [A: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(abs_abs_int,aa_int_int(sgn_sgn_int,A)) = one_one_int ) ) ).

% abs_sgn_eq_1
tff(fact_8357_sgn__mult__self__eq,axiom,
    ! [A: real] : aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,A)),aa_real_real(sgn_sgn_real,A)) = zero_n3304061248610475627l_real(A != zero_zero_real) ).

% sgn_mult_self_eq
tff(fact_8358_sgn__mult__self__eq,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(sgn_sgn_rat,A)),aa_rat_rat(sgn_sgn_rat,A)) = zero_n2052037380579107095ol_rat(A != zero_zero_rat) ).

% sgn_mult_self_eq
tff(fact_8359_sgn__mult__self__eq,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,A)),aa_int_int(sgn_sgn_int,A)) = aa_o_int(zero_n2684676970156552555ol_int,A != zero_zero_int) ).

% sgn_mult_self_eq
tff(fact_8360_sgn__mult__self__eq,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)),aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)) = zero_n356916108424825756nteger(A != zero_z3403309356797280102nteger) ).

% sgn_mult_self_eq
tff(fact_8361_sgn__mult__dvd__iff,axiom,
    ! [R: int,L: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,R)),L)),K2)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K2)
        & ( ( R = zero_zero_int )
         => ( K2 = zero_zero_int ) ) ) ) ).

% sgn_mult_dvd_iff
tff(fact_8362_mult__sgn__dvd__iff,axiom,
    ! [L: int,R: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,L),aa_int_int(sgn_sgn_int,R))),K2)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K2)
        & ( ( R = zero_zero_int )
         => ( K2 = zero_zero_int ) ) ) ) ).

% mult_sgn_dvd_iff
tff(fact_8363_dvd__sgn__mult__iff,axiom,
    ! [L: int,R: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,R)),K2))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K2)
        | ( R = zero_zero_int ) ) ) ).

% dvd_sgn_mult_iff
tff(fact_8364_dvd__mult__sgn__iff,axiom,
    ! [L: int,K2: int,R: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),aa_int_int(aa_int_fun_int_int(times_times_int,K2),aa_int_int(sgn_sgn_int,R)))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K2)
        | ( R = zero_zero_int ) ) ) ).

% dvd_mult_sgn_iff
tff(fact_8365_sgn__neg,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_real(sgn_sgn_real,A) = aa_real_real(uminus_uminus_real,one_one_real) ) ) ).

% sgn_neg
tff(fact_8366_sgn__neg,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_int(sgn_sgn_int,A) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ).

% sgn_neg
tff(fact_8367_sgn__neg,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_rat(sgn_sgn_rat,A) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ).

% sgn_neg
tff(fact_8368_sgn__neg,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)
     => ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ).

% sgn_neg
tff(fact_8369_sgn__of__nat,axiom,
    ! [Nb: nat] : aa_rat_rat(sgn_sgn_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb)) = zero_n2052037380579107095ol_rat(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).

% sgn_of_nat
tff(fact_8370_sgn__of__nat,axiom,
    ! [Nb: nat] : aa_real_real(sgn_sgn_real,aa_nat_real(semiri5074537144036343181t_real,Nb)) = zero_n3304061248610475627l_real(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).

% sgn_of_nat
tff(fact_8371_sgn__of__nat,axiom,
    ! [Nb: nat] : aa_int_int(sgn_sgn_int,aa_nat_int(semiri1314217659103216013at_int,Nb)) = aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).

% sgn_of_nat
tff(fact_8372_sgn__of__nat,axiom,
    ! [Nb: nat] : aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)) = zero_n356916108424825756nteger(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).

% sgn_of_nat
tff(fact_8373_Real__Vector__Spaces_Osgn__mult,axiom,
    ! [X: complex,Y2: complex] : aa_complex_complex(sgn_sgn_complex,aa_complex_complex(times_times_complex(X),Y2)) = aa_complex_complex(times_times_complex(aa_complex_complex(sgn_sgn_complex,X)),aa_complex_complex(sgn_sgn_complex,Y2)) ).

% Real_Vector_Spaces.sgn_mult
tff(fact_8374_Real__Vector__Spaces_Osgn__mult,axiom,
    ! [X: real,Y2: real] : aa_real_real(sgn_sgn_real,aa_real_real(times_times_real(X),Y2)) = aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,X)),aa_real_real(sgn_sgn_real,Y2)) ).

% Real_Vector_Spaces.sgn_mult
tff(fact_8375_sgn__mult,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(sgn_sgn_complex,aa_complex_complex(times_times_complex(A),B)) = aa_complex_complex(times_times_complex(aa_complex_complex(sgn_sgn_complex,A)),aa_complex_complex(sgn_sgn_complex,B)) ).

% sgn_mult
tff(fact_8376_sgn__mult,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)),aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,B)) ).

% sgn_mult
tff(fact_8377_sgn__mult,axiom,
    ! [A: real,B: real] : aa_real_real(sgn_sgn_real,aa_real_real(times_times_real(A),B)) = aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,A)),aa_real_real(sgn_sgn_real,B)) ).

% sgn_mult
tff(fact_8378_sgn__mult,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(sgn_sgn_rat,aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(times_times_rat(aa_rat_rat(sgn_sgn_rat,A)),aa_rat_rat(sgn_sgn_rat,B)) ).

% sgn_mult
tff(fact_8379_sgn__mult,axiom,
    ! [A: int,B: int] : aa_int_int(sgn_sgn_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,A)),aa_int_int(sgn_sgn_int,B)) ).

% sgn_mult
tff(fact_8380_same__sgn__sgn__add,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,B) = aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) )
     => ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,plus_p5714425477246183910nteger(A,B)) = aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) ) ) ).

% same_sgn_sgn_add
tff(fact_8381_same__sgn__sgn__add,axiom,
    ! [B: real,A: real] :
      ( ( aa_real_real(sgn_sgn_real,B) = aa_real_real(sgn_sgn_real,A) )
     => ( aa_real_real(sgn_sgn_real,aa_real_real(plus_plus_real(A),B)) = aa_real_real(sgn_sgn_real,A) ) ) ).

% same_sgn_sgn_add
tff(fact_8382_same__sgn__sgn__add,axiom,
    ! [B: rat,A: rat] :
      ( ( aa_rat_rat(sgn_sgn_rat,B) = aa_rat_rat(sgn_sgn_rat,A) )
     => ( aa_rat_rat(sgn_sgn_rat,aa_rat_rat(plus_plus_rat(A),B)) = aa_rat_rat(sgn_sgn_rat,A) ) ) ).

% same_sgn_sgn_add
tff(fact_8383_same__sgn__sgn__add,axiom,
    ! [B: int,A: int] :
      ( ( aa_int_int(sgn_sgn_int,B) = aa_int_int(sgn_sgn_int,A) )
     => ( aa_int_int(sgn_sgn_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = aa_int_int(sgn_sgn_int,A) ) ) ).

% same_sgn_sgn_add
tff(fact_8384_sgn__minus__1,axiom,
    aa_real_real(sgn_sgn_real,aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,one_one_real) ).

% sgn_minus_1
tff(fact_8385_sgn__minus__1,axiom,
    aa_int_int(sgn_sgn_int,aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% sgn_minus_1
tff(fact_8386_sgn__minus__1,axiom,
    aa_complex_complex(sgn_sgn_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% sgn_minus_1
tff(fact_8387_sgn__minus__1,axiom,
    aa_rat_rat(sgn_sgn_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% sgn_minus_1
tff(fact_8388_sgn__minus__1,axiom,
    aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% sgn_minus_1
tff(fact_8389_mult__sgn__abs,axiom,
    ! [X: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,X)),abs_abs_Code_integer(X)) = X ).

% mult_sgn_abs
tff(fact_8390_mult__sgn__abs,axiom,
    ! [X: real] : aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,X)),aa_real_real(abs_abs_real,X)) = X ).

% mult_sgn_abs
tff(fact_8391_mult__sgn__abs,axiom,
    ! [X: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(sgn_sgn_rat,X)),aa_rat_rat(abs_abs_rat,X)) = X ).

% mult_sgn_abs
tff(fact_8392_mult__sgn__abs,axiom,
    ! [X: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,X)),aa_int_int(abs_abs_int,X)) = X ).

% mult_sgn_abs
tff(fact_8393_sgn__mult__abs,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(sgn_sgn_complex,A)),aa_complex_complex(abs_abs_complex,A)) = A ).

% sgn_mult_abs
tff(fact_8394_sgn__mult__abs,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)),abs_abs_Code_integer(A)) = A ).

% sgn_mult_abs
tff(fact_8395_sgn__mult__abs,axiom,
    ! [A: real] : aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,A)),aa_real_real(abs_abs_real,A)) = A ).

% sgn_mult_abs
tff(fact_8396_sgn__mult__abs,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(sgn_sgn_rat,A)),aa_rat_rat(abs_abs_rat,A)) = A ).

% sgn_mult_abs
tff(fact_8397_sgn__mult__abs,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,A)),aa_int_int(abs_abs_int,A)) = A ).

% sgn_mult_abs
tff(fact_8398_abs__mult__sgn,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(abs_abs_complex,A)),aa_complex_complex(sgn_sgn_complex,A)) = A ).

% abs_mult_sgn
tff(fact_8399_abs__mult__sgn,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(A)),aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)) = A ).

% abs_mult_sgn
tff(fact_8400_abs__mult__sgn,axiom,
    ! [A: real] : aa_real_real(times_times_real(aa_real_real(abs_abs_real,A)),aa_real_real(sgn_sgn_real,A)) = A ).

% abs_mult_sgn
tff(fact_8401_abs__mult__sgn,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(sgn_sgn_rat,A)) = A ).

% abs_mult_sgn
tff(fact_8402_abs__mult__sgn,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(sgn_sgn_int,A)) = A ).

% abs_mult_sgn
tff(fact_8403_linordered__idom__class_Oabs__sgn,axiom,
    ! [K2: code_integer] : abs_abs_Code_integer(K2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K2),aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K2)) ).

% linordered_idom_class.abs_sgn
tff(fact_8404_linordered__idom__class_Oabs__sgn,axiom,
    ! [K2: real] : aa_real_real(abs_abs_real,K2) = aa_real_real(times_times_real(K2),aa_real_real(sgn_sgn_real,K2)) ).

% linordered_idom_class.abs_sgn
tff(fact_8405_linordered__idom__class_Oabs__sgn,axiom,
    ! [K2: rat] : aa_rat_rat(abs_abs_rat,K2) = aa_rat_rat(times_times_rat(K2),aa_rat_rat(sgn_sgn_rat,K2)) ).

% linordered_idom_class.abs_sgn
tff(fact_8406_linordered__idom__class_Oabs__sgn,axiom,
    ! [K2: int] : aa_int_int(abs_abs_int,K2) = aa_int_int(aa_int_fun_int_int(times_times_int,K2),aa_int_int(sgn_sgn_int,K2)) ).

% linordered_idom_class.abs_sgn
tff(fact_8407_int__sgnE,axiom,
    ! [K2: int] :
      ~ ! [N: nat,L2: int] : K2 != aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L2)),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% int_sgnE
tff(fact_8408_same__sgn__abs__add,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,B) = aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) )
     => ( abs_abs_Code_integer(plus_p5714425477246183910nteger(A,B)) = plus_p5714425477246183910nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B)) ) ) ).

% same_sgn_abs_add
tff(fact_8409_same__sgn__abs__add,axiom,
    ! [B: real,A: real] :
      ( ( aa_real_real(sgn_sgn_real,B) = aa_real_real(sgn_sgn_real,A) )
     => ( aa_real_real(abs_abs_real,aa_real_real(plus_plus_real(A),B)) = aa_real_real(plus_plus_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ) ) ).

% same_sgn_abs_add
tff(fact_8410_same__sgn__abs__add,axiom,
    ! [B: rat,A: rat] :
      ( ( aa_rat_rat(sgn_sgn_rat,B) = aa_rat_rat(sgn_sgn_rat,A) )
     => ( aa_rat_rat(abs_abs_rat,aa_rat_rat(plus_plus_rat(A),B)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ) ) ).

% same_sgn_abs_add
tff(fact_8411_same__sgn__abs__add,axiom,
    ! [B: int,A: int] :
      ( ( aa_int_int(sgn_sgn_int,B) = aa_int_int(sgn_sgn_int,A) )
     => ( aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)) ) ) ).

% same_sgn_abs_add
tff(fact_8412_sgn__1__pos,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) = one_one_Code_integer )
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A) ) ).

% sgn_1_pos
tff(fact_8413_sgn__1__pos,axiom,
    ! [A: real] :
      ( ( aa_real_real(sgn_sgn_real,A) = one_one_real )
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% sgn_1_pos
tff(fact_8414_sgn__1__pos,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(sgn_sgn_rat,A) = one_one_rat )
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% sgn_1_pos
tff(fact_8415_sgn__1__pos,axiom,
    ! [A: int] :
      ( ( aa_int_int(sgn_sgn_int,A) = one_one_int )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ).

% sgn_1_pos
tff(fact_8416_abs__sgn__eq,axiom,
    ! [A: code_integer] :
      abs_abs_Code_integer(aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)) = $ite(A = zero_z3403309356797280102nteger,zero_z3403309356797280102nteger,one_one_Code_integer) ).

% abs_sgn_eq
tff(fact_8417_abs__sgn__eq,axiom,
    ! [A: real] :
      aa_real_real(abs_abs_real,aa_real_real(sgn_sgn_real,A)) = $ite(A = zero_zero_real,zero_zero_real,one_one_real) ).

% abs_sgn_eq
tff(fact_8418_abs__sgn__eq,axiom,
    ! [A: rat] :
      aa_rat_rat(abs_abs_rat,aa_rat_rat(sgn_sgn_rat,A)) = $ite(A = zero_zero_rat,zero_zero_rat,one_one_rat) ).

% abs_sgn_eq
tff(fact_8419_abs__sgn__eq,axiom,
    ! [A: int] :
      aa_int_int(abs_abs_int,aa_int_int(sgn_sgn_int,A)) = $ite(A = zero_zero_int,zero_zero_int,one_one_int) ).

% abs_sgn_eq
tff(fact_8420_sgn__mod,axiom,
    ! [L: int,K2: int] :
      ( ( L != zero_zero_int )
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K2)
       => ( aa_int_int(sgn_sgn_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),L)) = aa_int_int(sgn_sgn_int,L) ) ) ) ).

% sgn_mod
tff(fact_8421_sgn__1__neg,axiom,
    ! [A: real] :
      ( ( aa_real_real(sgn_sgn_real,A) = aa_real_real(uminus_uminus_real,one_one_real) )
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% sgn_1_neg
tff(fact_8422_sgn__1__neg,axiom,
    ! [A: int] :
      ( ( aa_int_int(sgn_sgn_int,A) = aa_int_int(uminus_uminus_int,one_one_int) )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% sgn_1_neg
tff(fact_8423_sgn__1__neg,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(sgn_sgn_rat,A) = aa_rat_rat(uminus_uminus_rat,one_one_rat) )
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% sgn_1_neg
tff(fact_8424_sgn__1__neg,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ).

% sgn_1_neg
tff(fact_8425_sgn__if,axiom,
    ! [X: real] :
      aa_real_real(sgn_sgn_real,X) = $ite(
        X = zero_zero_real,
        zero_zero_real,
        $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X),one_one_real,aa_real_real(uminus_uminus_real,one_one_real)) ) ).

% sgn_if
tff(fact_8426_sgn__if,axiom,
    ! [X: int] :
      aa_int_int(sgn_sgn_int,X) = $ite(
        X = zero_zero_int,
        zero_zero_int,
        $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),X),one_one_int,aa_int_int(uminus_uminus_int,one_one_int)) ) ).

% sgn_if
tff(fact_8427_sgn__if,axiom,
    ! [X: rat] :
      aa_rat_rat(sgn_sgn_rat,X) = $ite(
        X = zero_zero_rat,
        zero_zero_rat,
        $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),X),one_one_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) ) ).

% sgn_if
tff(fact_8428_sgn__if,axiom,
    ! [X: code_integer] :
      aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,X) = $ite(
        X = zero_z3403309356797280102nteger,
        zero_z3403309356797280102nteger,
        $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),X),one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ) ).

% sgn_if
tff(fact_8429_zsgn__def,axiom,
    ! [I2: int] :
      aa_int_int(sgn_sgn_int,I2) = $ite(
        I2 = zero_zero_int,
        zero_zero_int,
        $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),I2),one_one_int,aa_int_int(uminus_uminus_int,one_one_int)) ) ).

% zsgn_def
tff(fact_8430_norm__sgn,axiom,
    ! [X: real] :
      aa_real_real(real_V7735802525324610683m_real,aa_real_real(sgn_sgn_real,X)) = $ite(X = zero_zero_real,zero_zero_real,one_one_real) ).

% norm_sgn
tff(fact_8431_norm__sgn,axiom,
    ! [X: complex] :
      aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(sgn_sgn_complex,X)) = $ite(X = zero_zero_complex,zero_zero_real,one_one_real) ).

% norm_sgn
tff(fact_8432_div__sgn__abs__cancel,axiom,
    ! [V: int,K2: int,L: int] :
      ( ( V != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,V)),aa_int_int(abs_abs_int,K2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,V)),aa_int_int(abs_abs_int,L))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K2)),aa_int_int(abs_abs_int,L)) ) ) ).

% div_sgn_abs_cancel
tff(fact_8433_div__dvd__sgn__abs,axiom,
    ! [L: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K2)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),L) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K2)),aa_int_int(sgn_sgn_int,L))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K2)),aa_int_int(abs_abs_int,L))) ) ) ).

% div_dvd_sgn_abs
tff(fact_8434_theI,axiom,
    ! [P: fun_real_o,A: real] :
      ( aa_real_o(P,A)
     => ( ! [X4: real] :
            ( aa_real_o(P,X4)
           => ( X4 = A ) )
       => aa_real_o(P,the_real(P)) ) ) ).

% theI
tff(fact_8435_theI,axiom,
    ! [P: fun_int_o,A: int] :
      ( aa_int_o(P,A)
     => ( ! [X4: int] :
            ( aa_int_o(P,X4)
           => ( X4 = A ) )
       => aa_int_o(P,the_int(P)) ) ) ).

% theI
tff(fact_8436_theI_H,axiom,
    ! [P: fun_real_o] :
      ( ? [X2: real] :
          ( aa_real_o(P,X2)
          & ! [Y4: real] :
              ( aa_real_o(P,Y4)
             => ( Y4 = X2 ) ) )
     => aa_real_o(P,the_real(P)) ) ).

% theI'
tff(fact_8437_theI_H,axiom,
    ! [P: fun_int_o] :
      ( ? [X2: int] :
          ( aa_int_o(P,X2)
          & ! [Y4: int] :
              ( aa_int_o(P,Y4)
             => ( Y4 = X2 ) ) )
     => aa_int_o(P,the_int(P)) ) ).

% theI'
tff(fact_8438_theI2,axiom,
    ! [P: fun_real_o,A: real,Q2: fun_real_o] :
      ( aa_real_o(P,A)
     => ( ! [X4: real] :
            ( aa_real_o(P,X4)
           => ( X4 = A ) )
       => ( ! [X4: real] :
              ( aa_real_o(P,X4)
             => aa_real_o(Q2,X4) )
         => aa_real_o(Q2,the_real(P)) ) ) ) ).

% theI2
tff(fact_8439_theI2,axiom,
    ! [P: fun_int_o,A: int,Q2: fun_int_o] :
      ( aa_int_o(P,A)
     => ( ! [X4: int] :
            ( aa_int_o(P,X4)
           => ( X4 = A ) )
       => ( ! [X4: int] :
              ( aa_int_o(P,X4)
             => aa_int_o(Q2,X4) )
         => aa_int_o(Q2,the_int(P)) ) ) ) ).

% theI2
tff(fact_8440_If__def,axiom,
    ! [P: $o,X: real,Y2: real] :
      $ite((P),X,Y2) = the_real(aa_real_fun_real_o(aa_rea1652550732481586312real_o(aTP_Lamm_ia((P)),X),Y2)) ).

% If_def
tff(fact_8441_If__def,axiom,
    ! [P: $o,X: int,Y2: int] :
      $ite((P),X,Y2) = the_int(aa_int_fun_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_ib((P)),X),Y2)) ).

% If_def
tff(fact_8442_the1I2,axiom,
    ! [P: fun_real_o,Q2: fun_real_o] :
      ( ? [X2: real] :
          ( aa_real_o(P,X2)
          & ! [Y4: real] :
              ( aa_real_o(P,Y4)
             => ( Y4 = X2 ) ) )
     => ( ! [X4: real] :
            ( aa_real_o(P,X4)
           => aa_real_o(Q2,X4) )
       => aa_real_o(Q2,the_real(P)) ) ) ).

% the1I2
tff(fact_8443_the1I2,axiom,
    ! [P: fun_int_o,Q2: fun_int_o] :
      ( ? [X2: int] :
          ( aa_int_o(P,X2)
          & ! [Y4: int] :
              ( aa_int_o(P,Y4)
             => ( Y4 = X2 ) ) )
     => ( ! [X4: int] :
            ( aa_int_o(P,X4)
           => aa_int_o(Q2,X4) )
       => aa_int_o(Q2,the_int(P)) ) ) ).

% the1I2
tff(fact_8444_the1__equality,axiom,
    ! [P: fun_real_o,A: real] :
      ( ? [X2: real] :
          ( aa_real_o(P,X2)
          & ! [Y4: real] :
              ( aa_real_o(P,Y4)
             => ( Y4 = X2 ) ) )
     => ( aa_real_o(P,A)
       => ( the_real(P) = A ) ) ) ).

% the1_equality
tff(fact_8445_the1__equality,axiom,
    ! [P: fun_int_o,A: int] :
      ( ? [X2: int] :
          ( aa_int_o(P,X2)
          & ! [Y4: int] :
              ( aa_int_o(P,Y4)
             => ( Y4 = X2 ) ) )
     => ( aa_int_o(P,A)
       => ( the_int(P) = A ) ) ) ).

% the1_equality
tff(fact_8446_eucl__rel__int__remainderI,axiom,
    ! [R: int,L: int,K2: int,Q: int] :
      ( ( aa_int_int(sgn_sgn_int,R) = aa_int_int(sgn_sgn_int,L) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R)),aa_int_int(abs_abs_int,L))
       => ( ( K2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q),L)),R) )
         => aa_Pro4592068669816069595_int_o(eucl_rel_int(K2,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q),R)) ) ) ) ).

% eucl_rel_int_remainderI
tff(fact_8447_eucl__rel__int_Osimps,axiom,
    ! [A1: int,A22: int,A32: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A1,A22),A32)
    <=> ( ? [K3: int] :
            ( ( A1 = K3 )
            & ( A22 = zero_zero_int )
            & ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),K3) ) )
        | ? [L3: int,K3: int,Q5: int] :
            ( ( A1 = K3 )
            & ( A22 = L3 )
            & ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q5),zero_zero_int) )
            & ( L3 != zero_zero_int )
            & ( K3 = aa_int_int(aa_int_fun_int_int(times_times_int,Q5),L3) ) )
        | ? [R5: int,L3: int,K3: int,Q5: int] :
            ( ( A1 = K3 )
            & ( A22 = L3 )
            & ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q5),R5) )
            & ( aa_int_int(sgn_sgn_int,R5) = aa_int_int(sgn_sgn_int,L3) )
            & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R5)),aa_int_int(abs_abs_int,L3))
            & ( K3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q5),L3)),R5) ) ) ) ) ).

% eucl_rel_int.simps
tff(fact_8448_eucl__rel__int_Ocases,axiom,
    ! [A1: int,A22: int,A32: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A1,A22),A32)
     => ( ( ( A22 = zero_zero_int )
         => ( A32 != aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),A1) ) )
       => ( ! [Q3: int] :
              ( ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),zero_zero_int) )
             => ( ( A22 != zero_zero_int )
               => ( A1 != aa_int_int(aa_int_fun_int_int(times_times_int,Q3),A22) ) ) )
         => ~ ! [R3: int,Q3: int] :
                ( ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),R3) )
               => ( ( aa_int_int(sgn_sgn_int,R3) = aa_int_int(sgn_sgn_int,A22) )
                 => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R3)),aa_int_int(abs_abs_int,A22))
                   => ( A1 != aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q3),A22)),R3) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
tff(fact_8449_floor__real__def,axiom,
    ! [X: real] : aa_real_int(archim6058952711729229775r_real,X) = the_int(aTP_Lamm_ic(X)) ).

% floor_real_def
tff(fact_8450_divide__int__unfold,axiom,
    ! [K2: int,Ma: nat,L: int,Nb: nat] :
      aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K2)),aa_nat_int(semiri1314217659103216013at_int,Ma))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,Nb))) = $ite(
        ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
        | ( aa_int_int(sgn_sgn_int,K2) = zero_zero_int )
        | ( Nb = zero_zero_nat ) ),
        zero_zero_int,
        $ite(aa_int_int(sgn_sgn_int,K2) = aa_int_int(sgn_sgn_int,L),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),Nb)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),Nb)),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),Ma)))))) ) ).

% divide_int_unfold
tff(fact_8451_modulo__int__def,axiom,
    ! [K2: int,L: int] :
      aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),L) = $ite(
        L = zero_zero_int,
        K2,
        $ite(aa_int_int(sgn_sgn_int,K2) = aa_int_int(sgn_sgn_int,L),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K2))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,L)),aa_o_int(zero_n2684676970156552555ol_int,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K2)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K2))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L))))))) ) ).

% modulo_int_def
tff(fact_8452_divide__int__def,axiom,
    ! [K2: int,L: int] :
      aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),L) = $ite(
        L = zero_zero_int,
        zero_zero_int,
        $ite(aa_int_int(sgn_sgn_int,K2) = aa_int_int(sgn_sgn_int,L),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K2))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K2))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K2)))))) ) ).

% divide_int_def
tff(fact_8453_powr__int,axiom,
    ! [X: real,I2: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_real(powr_real(X),aa_int_real(ring_1_of_int_real,I2)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),I2),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_int_nat(nat2,I2)),aa_real_real(divide_divide_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,I2))))) ) ) ).

% powr_int
tff(fact_8454_arctan__inverse,axiom,
    ! [X: real] :
      ( ( X != zero_zero_real )
     => ( aa_real_real(arctan,aa_real_real(divide_divide_real(one_one_real),X)) = aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,X)),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_real_real(arctan,X)) ) ) ).

% arctan_inverse
tff(fact_8455_cis__multiple__2pi,axiom,
    ! [Nb: real] :
      ( member_real(Nb,ring_1_Ints_real)
     => ( aa_real_complex(cis,aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),Nb)) = one_one_complex ) ) ).

% cis_multiple_2pi
tff(fact_8456_nat__numeral,axiom,
    ! [K2: num] : aa_int_nat(nat2,aa_num_int(numeral_numeral_int,K2)) = aa_num_nat(numeral_numeral_nat,K2) ).

% nat_numeral
tff(fact_8457_sgn__le__0__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sgn_sgn_real,X)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),zero_zero_real) ) ).

% sgn_le_0_iff
tff(fact_8458_zero__le__sgn__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sgn_sgn_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X) ) ).

% zero_le_sgn_iff
tff(fact_8459_nat__1,axiom,
    aa_int_nat(nat2,one_one_int) = aa_nat_nat(suc,zero_zero_nat) ).

% nat_1
tff(fact_8460_nat__0__iff,axiom,
    ! [I2: int] :
      ( ( aa_int_nat(nat2,I2) = zero_zero_nat )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),zero_zero_int) ) ).

% nat_0_iff
tff(fact_8461_nat__le__0,axiom,
    ! [Z2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z2),zero_zero_int)
     => ( aa_int_nat(nat2,Z2) = zero_zero_nat ) ) ).

% nat_le_0
tff(fact_8462_zless__nat__conj,axiom,
    ! [W: int,Z2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z2))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z2)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z2) ) ) ).

% zless_nat_conj
tff(fact_8463_int__nat__eq,axiom,
    ! [Z2: int] :
      aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z2)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z2),Z2,zero_zero_int) ).

% int_nat_eq
tff(fact_8464_floor__add2,axiom,
    ! [X: real,Y2: real] :
      ( ( member_real(X,ring_1_Ints_real)
        | member_real(Y2,ring_1_Ints_real) )
     => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(plus_plus_real(X),Y2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,X)),aa_real_int(archim6058952711729229775r_real,Y2)) ) ) ).

% floor_add2
tff(fact_8465_floor__add2,axiom,
    ! [X: rat,Y2: rat] :
      ( ( member_rat(X,ring_1_Ints_rat)
        | member_rat(Y2,ring_1_Ints_rat) )
     => ( aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(plus_plus_rat(X),Y2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,X)),aa_rat_int(archim3151403230148437115or_rat,Y2)) ) ) ).

% floor_add2
tff(fact_8466_of__nat__nat__take__bit__eq,axiom,
    ! [Nb: nat,K2: int] : aa_nat_rat(semiri681578069525770553at_rat,aa_int_nat(nat2,aa_int_int(bit_se2923211474154528505it_int(Nb),K2))) = aa_int_rat(ring_1_of_int_rat,aa_int_int(bit_se2923211474154528505it_int(Nb),K2)) ).

% of_nat_nat_take_bit_eq
tff(fact_8467_of__nat__nat__take__bit__eq,axiom,
    ! [Nb: nat,K2: int] : aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,aa_int_int(bit_se2923211474154528505it_int(Nb),K2))) = aa_int_real(ring_1_of_int_real,aa_int_int(bit_se2923211474154528505it_int(Nb),K2)) ).

% of_nat_nat_take_bit_eq
tff(fact_8468_of__nat__nat__take__bit__eq,axiom,
    ! [Nb: nat,K2: int] : aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,aa_int_int(bit_se2923211474154528505it_int(Nb),K2))) = aa_int_int(ring_1_of_int_int,aa_int_int(bit_se2923211474154528505it_int(Nb),K2)) ).

% of_nat_nat_take_bit_eq
tff(fact_8469_frac__gt__0__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(archim2898591450579166408c_real,X))
    <=> ~ member_real(X,ring_1_Ints_real) ) ).

% frac_gt_0_iff
tff(fact_8470_frac__gt__0__iff,axiom,
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(archimedean_frac_rat,X))
    <=> ~ member_rat(X,ring_1_Ints_rat) ) ).

% frac_gt_0_iff
tff(fact_8471_zero__less__nat__eq,axiom,
    ! [Z2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_int_nat(nat2,Z2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z2) ) ).

% zero_less_nat_eq
tff(fact_8472_of__nat__nat,axiom,
    ! [Z2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z2)
     => ( aa_nat_rat(semiri681578069525770553at_rat,aa_int_nat(nat2,Z2)) = aa_int_rat(ring_1_of_int_rat,Z2) ) ) ).

% of_nat_nat
tff(fact_8473_of__nat__nat,axiom,
    ! [Z2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z2)
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,Z2)) = aa_int_real(ring_1_of_int_real,Z2) ) ) ).

% of_nat_nat
tff(fact_8474_of__nat__nat,axiom,
    ! [Z2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z2)
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z2)) = aa_int_int(ring_1_of_int_int,Z2) ) ) ).

% of_nat_nat
tff(fact_8475_diff__nat__numeral,axiom,
    ! [V: num,V3: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,V)),aa_num_nat(numeral_numeral_nat,V3)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,V3))) ).

% diff_nat_numeral
tff(fact_8476_nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y2: int,X: num,Nb: nat] :
      ( ( aa_int_nat(nat2,Y2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),Nb) )
    <=> ( Y2 = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X)),Nb) ) ) ).

% nat_eq_numeral_power_cancel_iff
tff(fact_8477_numeral__power__eq__nat__cancel__iff,axiom,
    ! [X: num,Nb: nat,Y2: int] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),Nb) = aa_int_nat(nat2,Y2) )
    <=> ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X)),Nb) = Y2 ) ) ).

% numeral_power_eq_nat_cancel_iff
tff(fact_8478_nat__ceiling__le__eq,axiom,
    ! [X: real,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,X))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_nat_real(semiri5074537144036343181t_real,A)) ) ).

% nat_ceiling_le_eq
tff(fact_8479_one__less__nat__eq,axiom,
    ! [Z2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),aa_int_nat(nat2,Z2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Z2) ) ).

% one_less_nat_eq
tff(fact_8480_nat__numeral__diff__1,axiom,
    ! [V: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,V)),one_one_nat) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,V)),one_one_int)) ).

% nat_numeral_diff_1
tff(fact_8481_nat__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X)),Nb)) ) ).

% nat_less_numeral_power_cancel_iff
tff(fact_8482_numeral__power__less__nat__cancel__iff,axiom,
    ! [X: num,Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),Nb)),aa_int_nat(nat2,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X)),Nb)),A) ) ).

% numeral_power_less_nat_cancel_iff
tff(fact_8483_numeral__power__le__nat__cancel__iff,axiom,
    ! [X: num,Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),Nb)),aa_int_nat(nat2,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X)),Nb)),A) ) ).

% numeral_power_le_nat_cancel_iff
tff(fact_8484_nat__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,X)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,X)),Nb)) ) ).

% nat_le_numeral_power_cancel_iff
tff(fact_8485_Ints__numeral,axiom,
    ! [Nb: num] : member_complex(numera6690914467698888265omplex(Nb),ring_1_Ints_complex) ).

% Ints_numeral
tff(fact_8486_Ints__numeral,axiom,
    ! [Nb: num] : member_real(aa_num_real(numeral_numeral_real,Nb),ring_1_Ints_real) ).

% Ints_numeral
tff(fact_8487_Ints__numeral,axiom,
    ! [Nb: num] : member_rat(aa_num_rat(numeral_numeral_rat,Nb),ring_1_Ints_rat) ).

% Ints_numeral
tff(fact_8488_Ints__numeral,axiom,
    ! [Nb: num] : member_int(aa_num_int(numeral_numeral_int,Nb),ring_1_Ints_int) ).

% Ints_numeral
tff(fact_8489_Ints__add,axiom,
    ! [A: complex,B: complex] :
      ( member_complex(A,ring_1_Ints_complex)
     => ( member_complex(B,ring_1_Ints_complex)
       => member_complex(aa_complex_complex(plus_plus_complex(A),B),ring_1_Ints_complex) ) ) ).

% Ints_add
tff(fact_8490_Ints__add,axiom,
    ! [A: real,B: real] :
      ( member_real(A,ring_1_Ints_real)
     => ( member_real(B,ring_1_Ints_real)
       => member_real(aa_real_real(plus_plus_real(A),B),ring_1_Ints_real) ) ) ).

% Ints_add
tff(fact_8491_Ints__add,axiom,
    ! [A: rat,B: rat] :
      ( member_rat(A,ring_1_Ints_rat)
     => ( member_rat(B,ring_1_Ints_rat)
       => member_rat(aa_rat_rat(plus_plus_rat(A),B),ring_1_Ints_rat) ) ) ).

% Ints_add
tff(fact_8492_Ints__add,axiom,
    ! [A: int,B: int] :
      ( member_int(A,ring_1_Ints_int)
     => ( member_int(B,ring_1_Ints_int)
       => member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B),ring_1_Ints_int) ) ) ).

% Ints_add
tff(fact_8493_Ints__1,axiom,
    member_complex(one_one_complex,ring_1_Ints_complex) ).

% Ints_1
tff(fact_8494_Ints__1,axiom,
    member_rat(one_one_rat,ring_1_Ints_rat) ).

% Ints_1
tff(fact_8495_Ints__1,axiom,
    member_int(one_one_int,ring_1_Ints_int) ).

% Ints_1
tff(fact_8496_Ints__1,axiom,
    member_real(one_one_real,ring_1_Ints_real) ).

% Ints_1
tff(fact_8497_Ints__mult,axiom,
    ! [A: complex,B: complex] :
      ( member_complex(A,ring_1_Ints_complex)
     => ( member_complex(B,ring_1_Ints_complex)
       => member_complex(aa_complex_complex(times_times_complex(A),B),ring_1_Ints_complex) ) ) ).

% Ints_mult
tff(fact_8498_Ints__mult,axiom,
    ! [A: real,B: real] :
      ( member_real(A,ring_1_Ints_real)
     => ( member_real(B,ring_1_Ints_real)
       => member_real(aa_real_real(times_times_real(A),B),ring_1_Ints_real) ) ) ).

% Ints_mult
tff(fact_8499_Ints__mult,axiom,
    ! [A: rat,B: rat] :
      ( member_rat(A,ring_1_Ints_rat)
     => ( member_rat(B,ring_1_Ints_rat)
       => member_rat(aa_rat_rat(times_times_rat(A),B),ring_1_Ints_rat) ) ) ).

% Ints_mult
tff(fact_8500_Ints__mult,axiom,
    ! [A: int,B: int] :
      ( member_int(A,ring_1_Ints_int)
     => ( member_int(B,ring_1_Ints_int)
       => member_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B),ring_1_Ints_int) ) ) ).

% Ints_mult
tff(fact_8501_Ints__power,axiom,
    ! [A: real,Nb: nat] :
      ( member_real(A,ring_1_Ints_real)
     => member_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb),ring_1_Ints_real) ) ).

% Ints_power
tff(fact_8502_Ints__power,axiom,
    ! [A: int,Nb: nat] :
      ( member_int(A,ring_1_Ints_int)
     => member_int(aa_nat_int(aa_int_fun_nat_int(power_power_int,A),Nb),ring_1_Ints_int) ) ).

% Ints_power
tff(fact_8503_Ints__power,axiom,
    ! [A: complex,Nb: nat] :
      ( member_complex(A,ring_1_Ints_complex)
     => member_complex(aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),Nb),ring_1_Ints_complex) ) ).

% Ints_power
tff(fact_8504_nat__zero__as__int,axiom,
    zero_zero_nat = aa_int_nat(nat2,zero_zero_int) ).

% nat_zero_as_int
tff(fact_8505_nat__numeral__as__int,axiom,
    ! [X2: num] : aa_num_nat(numeral_numeral_nat,X2) = aa_int_nat(nat2,aa_num_int(numeral_numeral_int,X2)) ).

% nat_numeral_as_int
tff(fact_8506_nat__mono,axiom,
    ! [X: int,Y2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,X)),aa_int_nat(nat2,Y2)) ) ).

% nat_mono
tff(fact_8507_eq__nat__nat__iff,axiom,
    ! [Z2: int,Z4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z4)
       => ( ( aa_int_nat(nat2,Z2) = aa_int_nat(nat2,Z4) )
        <=> ( Z2 = Z4 ) ) ) ) ).

% eq_nat_nat_iff
tff(fact_8508_all__nat,axiom,
    ! [P: fun_nat_o] :
      ( ! [X_12: nat] : aa_nat_o(P,X_12)
    <=> ! [X3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X3)
         => aa_nat_o(P,aa_int_nat(nat2,X3)) ) ) ).

% all_nat
tff(fact_8509_ex__nat,axiom,
    ! [P: fun_nat_o] :
      ( ? [X_12: nat] : aa_nat_o(P,X_12)
    <=> ? [X3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X3)
          & aa_nat_o(P,aa_int_nat(nat2,X3)) ) ) ).

% ex_nat
tff(fact_8510_nat__one__as__int,axiom,
    one_one_nat = aa_int_nat(nat2,one_one_int) ).

% nat_one_as_int
tff(fact_8511_Ints__double__eq__0__iff,axiom,
    ! [A: complex] :
      ( member_complex(A,ring_1_Ints_complex)
     => ( ( aa_complex_complex(plus_plus_complex(A),A) = zero_zero_complex )
      <=> ( A = zero_zero_complex ) ) ) ).

% Ints_double_eq_0_iff
tff(fact_8512_Ints__double__eq__0__iff,axiom,
    ! [A: real] :
      ( member_real(A,ring_1_Ints_real)
     => ( ( aa_real_real(plus_plus_real(A),A) = zero_zero_real )
      <=> ( A = zero_zero_real ) ) ) ).

% Ints_double_eq_0_iff
tff(fact_8513_Ints__double__eq__0__iff,axiom,
    ! [A: rat] :
      ( member_rat(A,ring_1_Ints_rat)
     => ( ( aa_rat_rat(plus_plus_rat(A),A) = zero_zero_rat )
      <=> ( A = zero_zero_rat ) ) ) ).

% Ints_double_eq_0_iff
tff(fact_8514_Ints__double__eq__0__iff,axiom,
    ! [A: int] :
      ( member_int(A,ring_1_Ints_int)
     => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A) = zero_zero_int )
      <=> ( A = zero_zero_int ) ) ) ).

% Ints_double_eq_0_iff
tff(fact_8515_unset__bit__nat__def,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,Ma),Nb) = aa_int_nat(nat2,aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Ma),aa_nat_int(semiri1314217659103216013at_int,Nb))) ).

% unset_bit_nat_def
tff(fact_8516_nat__mask__eq,axiom,
    ! [Nb: nat] : aa_int_nat(nat2,aa_nat_int(bit_se2000444600071755411sk_int,Nb)) = aa_nat_nat(bit_se2002935070580805687sk_nat,Nb) ).

% nat_mask_eq
tff(fact_8517_nat__mono__iff,axiom,
    ! [Z2: int,W: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z2))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z2) ) ) ).

% nat_mono_iff
tff(fact_8518_of__nat__ceiling,axiom,
    ! [R: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,R),aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,R)))) ).

% of_nat_ceiling
tff(fact_8519_of__nat__ceiling,axiom,
    ! [R: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,R),aa_nat_rat(semiri681578069525770553at_rat,aa_int_nat(nat2,aa_rat_int(archim2889992004027027881ng_rat,R)))) ).

% of_nat_ceiling
tff(fact_8520_zless__nat__eq__int__zless,axiom,
    ! [Ma: nat,Z2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_int_nat(nat2,Z2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Ma)),Z2) ) ).

% zless_nat_eq_int_zless
tff(fact_8521_nat__le__iff,axiom,
    ! [X: int,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,X)),Nb)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_nat_int(semiri1314217659103216013at_int,Nb)) ) ).

% nat_le_iff
tff(fact_8522_int__eq__iff,axiom,
    ! [Ma: nat,Z2: int] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Ma) = Z2 )
    <=> ( ( Ma = aa_int_nat(nat2,Z2) )
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z2) ) ) ).

% int_eq_iff
tff(fact_8523_nat__0__le,axiom,
    ! [Z2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z2)
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,Z2)) = Z2 ) ) ).

% nat_0_le
tff(fact_8524_nat__int__add,axiom,
    ! [A: nat,B: nat] : aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) ).

% nat_int_add
tff(fact_8525_int__minus,axiom,
    ! [Nb: nat,Ma: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Ma)) = aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_nat_int(semiri1314217659103216013at_int,Ma)))) ).

% int_minus
tff(fact_8526_nat__abs__mult__distrib,axiom,
    ! [W: int,Z2: int] : aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z2))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,W))),aa_int_nat(nat2,aa_int_int(abs_abs_int,Z2))) ).

% nat_abs_mult_distrib
tff(fact_8527_Ints__odd__nonzero,axiom,
    ! [A: complex] :
      ( member_complex(A,ring_1_Ints_complex)
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(one_one_complex),A)),A) != zero_zero_complex ) ) ).

% Ints_odd_nonzero
tff(fact_8528_Ints__odd__nonzero,axiom,
    ! [A: real] :
      ( member_real(A,ring_1_Ints_real)
     => ( aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(one_one_real),A)),A) != zero_zero_real ) ) ).

% Ints_odd_nonzero
tff(fact_8529_Ints__odd__nonzero,axiom,
    ! [A: rat] :
      ( member_rat(A,ring_1_Ints_rat)
     => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(one_one_rat),A)),A) != zero_zero_rat ) ) ).

% Ints_odd_nonzero
tff(fact_8530_Ints__odd__nonzero,axiom,
    ! [A: int] :
      ( member_int(A,ring_1_Ints_int)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),A) != zero_zero_int ) ) ).

% Ints_odd_nonzero
tff(fact_8531_of__int__divide__in__Ints,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => member_complex(aa_complex_complex(divide1717551699836669952omplex(aa_int_complex(ring_17405671764205052669omplex,A)),aa_int_complex(ring_17405671764205052669omplex,B)),ring_1_Ints_complex) ) ).

% of_int_divide_in_Ints
tff(fact_8532_of__int__divide__in__Ints,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => member_real(aa_real_real(divide_divide_real(aa_int_real(ring_1_of_int_real,A)),aa_int_real(ring_1_of_int_real,B)),ring_1_Ints_real) ) ).

% of_int_divide_in_Ints
tff(fact_8533_of__int__divide__in__Ints,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => member_rat(aa_rat_rat(divide_divide_rat(aa_int_rat(ring_1_of_int_rat,A)),aa_int_rat(ring_1_of_int_rat,B)),ring_1_Ints_rat) ) ).

% of_int_divide_in_Ints
tff(fact_8534_of__int__divide__in__Ints,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => member_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(ring_1_of_int_int,A)),aa_int_int(ring_1_of_int_int,B)),ring_1_Ints_int) ) ).

% of_int_divide_in_Ints
tff(fact_8535_and__nat__def,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Ma),Nb) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_nat_int(semiri1314217659103216013at_int,Ma)),aa_nat_int(semiri1314217659103216013at_int,Nb))) ).

% and_nat_def
tff(fact_8536_nat__plus__as__int,axiom,
    ! [X2: nat,Xa3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),Xa3) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,X2)),aa_nat_int(semiri1314217659103216013at_int,Xa3))) ).

% nat_plus_as_int
tff(fact_8537_nat__times__as__int,axiom,
    ! [X2: nat,Xa3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X2),Xa3) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,X2)),aa_nat_int(semiri1314217659103216013at_int,Xa3))) ).

% nat_times_as_int
tff(fact_8538_real__nat__ceiling__ge,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,X)))) ).

% real_nat_ceiling_ge
tff(fact_8539_nat__minus__as__int,axiom,
    ! [X2: nat,Xa3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X2),Xa3) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,X2)),aa_nat_int(semiri1314217659103216013at_int,Xa3))) ).

% nat_minus_as_int
tff(fact_8540_nat__div__as__int,axiom,
    ! [X2: nat,Xa3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X2),Xa3) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(semiri1314217659103216013at_int,X2)),aa_nat_int(semiri1314217659103216013at_int,Xa3))) ).

% nat_div_as_int
tff(fact_8541_nat__mod__as__int,axiom,
    ! [X2: nat,Xa3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X2),Xa3) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(semiri1314217659103216013at_int,X2)),aa_nat_int(semiri1314217659103216013at_int,Xa3))) ).

% nat_mod_as_int
tff(fact_8542_sgn__real__def,axiom,
    ! [A: real] :
      aa_real_real(sgn_sgn_real,A) = $ite(
        A = zero_zero_real,
        zero_zero_real,
        $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A),one_one_real,aa_real_real(uminus_uminus_real,one_one_real)) ) ).

% sgn_real_def
tff(fact_8543_of__nat__floor,axiom,
    ! [R: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),R)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,R)))),R) ) ).

% of_nat_floor
tff(fact_8544_of__nat__floor,axiom,
    ! [R: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),R)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_int_nat(nat2,aa_rat_int(archim3151403230148437115or_rat,R)))),R) ) ).

% of_nat_floor
tff(fact_8545_nat__less__eq__zless,axiom,
    ! [W: int,Z2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z2))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z2) ) ) ).

% nat_less_eq_zless
tff(fact_8546_nat__le__eq__zle,axiom,
    ! [W: int,Z2: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),W)
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z2))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Z2) ) ) ).

% nat_le_eq_zle
tff(fact_8547_nat__eq__iff2,axiom,
    ! [Ma: nat,W: int] :
      ( ( Ma = aa_int_nat(nat2,W) )
    <=> $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W),W = aa_nat_int(semiri1314217659103216013at_int,Ma),Ma = zero_zero_nat) ) ).

% nat_eq_iff2
tff(fact_8548_nat__eq__iff,axiom,
    ! [W: int,Ma: nat] :
      ( ( aa_int_nat(nat2,W) = Ma )
    <=> $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W),W = aa_nat_int(semiri1314217659103216013at_int,Ma),Ma = zero_zero_nat) ) ).

% nat_eq_iff
tff(fact_8549_le__mult__nat__floor,axiom,
    ! [A: real,B: real] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,A))),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,B)))),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,aa_real_real(times_times_real(A),B)))) ).

% le_mult_nat_floor
tff(fact_8550_le__mult__nat__floor,axiom,
    ! [A: rat,B: rat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,aa_rat_int(archim3151403230148437115or_rat,A))),aa_int_nat(nat2,aa_rat_int(archim3151403230148437115or_rat,B)))),aa_int_nat(nat2,aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(times_times_rat(A),B)))) ).

% le_mult_nat_floor
tff(fact_8551_le__nat__iff,axiom,
    ! [K2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_int_nat(nat2,K2))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),K2) ) ) ).

% le_nat_iff
tff(fact_8552_nat__add__distrib,axiom,
    ! [Z2: int,Z4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z4)
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z2),Z4)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(nat2,Z2)),aa_int_nat(nat2,Z4)) ) ) ) ).

% nat_add_distrib
tff(fact_8553_nat__mult__distrib,axiom,
    ! [Z2: int,Z4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z2)
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(times_times_int,Z2),Z4)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,Z2)),aa_int_nat(nat2,Z4)) ) ) ).

% nat_mult_distrib
tff(fact_8554_Suc__as__int,axiom,
    ! [X2: nat] : aa_nat_nat(suc,X2) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,X2)),one_one_int)) ).

% Suc_as_int
tff(fact_8555_nat__diff__distrib,axiom,
    ! [Z4: int,Z2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z4)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z4),Z2)
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z2),Z4)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z2)),aa_int_nat(nat2,Z4)) ) ) ) ).

% nat_diff_distrib
tff(fact_8556_nat__diff__distrib_H,axiom,
    ! [X: int,Y2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y2)
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,X)),aa_int_nat(nat2,Y2)) ) ) ) ).

% nat_diff_distrib'
tff(fact_8557_nat__abs__triangle__ineq,axiom,
    ! [K2: int,L: int] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K2),L)))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K2))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))) ).

% nat_abs_triangle_ineq
tff(fact_8558_nat__div__distrib,axiom,
    ! [X: int,Y2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,X)),aa_int_nat(nat2,Y2)) ) ) ).

% nat_div_distrib
tff(fact_8559_nat__div__distrib_H,axiom,
    ! [Y2: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y2)
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,X)),aa_int_nat(nat2,Y2)) ) ) ).

% nat_div_distrib'
tff(fact_8560_nat__floor__neg,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),zero_zero_real)
     => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,X)) = zero_zero_nat ) ) ).

% nat_floor_neg
tff(fact_8561_nat__power__eq,axiom,
    ! [Z2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z2)
     => ( aa_int_nat(nat2,aa_nat_int(aa_int_fun_nat_int(power_power_int,Z2),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_int_nat(nat2,Z2)),Nb) ) ) ).

% nat_power_eq
tff(fact_8562_nat__mod__distrib,axiom,
    ! [X: int,Y2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y2)
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X),Y2)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,X)),aa_int_nat(nat2,Y2)) ) ) ) ).

% nat_mod_distrib
tff(fact_8563_div__abs__eq__div__nat,axiom,
    ! [K2: int,L: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K2)),aa_int_int(abs_abs_int,L)) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K2))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))) ).

% div_abs_eq_div_nat
tff(fact_8564_Ints__odd__less__0,axiom,
    ! [A: real] :
      ( member_real(A,ring_1_Ints_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(one_one_real),A)),A)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ) ).

% Ints_odd_less_0
tff(fact_8565_Ints__odd__less__0,axiom,
    ! [A: rat] :
      ( member_rat(A,ring_1_Ints_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(one_one_rat),A)),A)),zero_zero_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ) ).

% Ints_odd_less_0
tff(fact_8566_Ints__odd__less__0,axiom,
    ! [A: int] :
      ( member_int(A,ring_1_Ints_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),A)),zero_zero_int)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ) ).

% Ints_odd_less_0
tff(fact_8567_floor__eq3,axiom,
    ! [Nb: nat,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb)))
       => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,X)) = Nb ) ) ) ).

% floor_eq3
tff(fact_8568_le__nat__floor,axiom,
    ! [X: nat,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,X)),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,A))) ) ).

% le_nat_floor
tff(fact_8569_Ints__nonzero__abs__ge1,axiom,
    ! [X: code_integer] :
      ( member_Code_integer(X,ring_11222124179247155820nteger)
     => ( ( X != zero_z3403309356797280102nteger )
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),abs_abs_Code_integer(X)) ) ) ).

% Ints_nonzero_abs_ge1
tff(fact_8570_Ints__nonzero__abs__ge1,axiom,
    ! [X: real] :
      ( member_real(X,ring_1_Ints_real)
     => ( ( X != zero_zero_real )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(abs_abs_real,X)) ) ) ).

% Ints_nonzero_abs_ge1
tff(fact_8571_Ints__nonzero__abs__ge1,axiom,
    ! [X: rat] :
      ( member_rat(X,ring_1_Ints_rat)
     => ( ( X != zero_zero_rat )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(abs_abs_rat,X)) ) ) ).

% Ints_nonzero_abs_ge1
tff(fact_8572_Ints__nonzero__abs__ge1,axiom,
    ! [X: int] :
      ( member_int(X,ring_1_Ints_int)
     => ( ( X != zero_zero_int )
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_int_int(abs_abs_int,X)) ) ) ).

% Ints_nonzero_abs_ge1
tff(fact_8573_mod__abs__eq__div__nat,axiom,
    ! [K2: int,L: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(abs_abs_int,K2)),aa_int_int(abs_abs_int,L)) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K2))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))) ).

% mod_abs_eq_div_nat
tff(fact_8574_Ints__nonzero__abs__less1,axiom,
    ! [X: code_integer] :
      ( member_Code_integer(X,ring_11222124179247155820nteger)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(X)),one_one_Code_integer)
       => ( X = zero_z3403309356797280102nteger ) ) ) ).

% Ints_nonzero_abs_less1
tff(fact_8575_Ints__nonzero__abs__less1,axiom,
    ! [X: real] :
      ( member_real(X,ring_1_Ints_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,X)),one_one_real)
       => ( X = zero_zero_real ) ) ) ).

% Ints_nonzero_abs_less1
tff(fact_8576_Ints__nonzero__abs__less1,axiom,
    ! [X: rat] :
      ( member_rat(X,ring_1_Ints_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,X)),one_one_rat)
       => ( X = zero_zero_rat ) ) ) ).

% Ints_nonzero_abs_less1
tff(fact_8577_Ints__nonzero__abs__less1,axiom,
    ! [X: int] :
      ( member_int(X,ring_1_Ints_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,X)),one_one_int)
       => ( X = zero_zero_int ) ) ) ).

% Ints_nonzero_abs_less1
tff(fact_8578_Ints__eq__abs__less1,axiom,
    ! [X: code_integer,Y2: code_integer] :
      ( member_Code_integer(X,ring_11222124179247155820nteger)
     => ( member_Code_integer(Y2,ring_11222124179247155820nteger)
       => ( ( X = Y2 )
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(X,Y2))),one_one_Code_integer) ) ) ) ).

% Ints_eq_abs_less1
tff(fact_8579_Ints__eq__abs__less1,axiom,
    ! [X: real,Y2: real] :
      ( member_real(X,ring_1_Ints_real)
     => ( member_real(Y2,ring_1_Ints_real)
       => ( ( X = Y2 )
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(X),Y2))),one_one_real) ) ) ) ).

% Ints_eq_abs_less1
tff(fact_8580_Ints__eq__abs__less1,axiom,
    ! [X: rat,Y2: rat] :
      ( member_rat(X,ring_1_Ints_rat)
     => ( member_rat(Y2,ring_1_Ints_rat)
       => ( ( X = Y2 )
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(X),Y2))),one_one_rat) ) ) ) ).

% Ints_eq_abs_less1
tff(fact_8581_Ints__eq__abs__less1,axiom,
    ! [X: int,Y2: int] :
      ( member_int(X,ring_1_Ints_int)
     => ( member_int(Y2,ring_1_Ints_int)
       => ( ( X = Y2 )
        <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y2))),one_one_int) ) ) ) ).

% Ints_eq_abs_less1
tff(fact_8582_nat__take__bit__eq,axiom,
    ! [K2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
     => ( aa_int_nat(nat2,aa_int_int(bit_se2923211474154528505it_int(Nb),K2)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_int_nat(nat2,K2)) ) ) ).

% nat_take_bit_eq
tff(fact_8583_take__bit__nat__eq,axiom,
    ! [K2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_int_nat(nat2,K2)) = aa_int_nat(nat2,aa_int_int(bit_se2923211474154528505it_int(Nb),K2)) ) ) ).

% take_bit_nat_eq
tff(fact_8584_sin__times__pi__eq__0,axiom,
    ! [X: real] :
      ( ( aa_real_real(sin_real,aa_real_real(times_times_real(X),pi)) = zero_zero_real )
    <=> member_real(X,ring_1_Ints_real) ) ).

% sin_times_pi_eq_0
tff(fact_8585_bit__nat__iff,axiom,
    ! [K2: int,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_int_nat(nat2,K2)),Nb)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
        & aa_nat_o(bit_se1146084159140164899it_int(K2),Nb) ) ) ).

% bit_nat_iff
tff(fact_8586_nat__2,axiom,
    aa_int_nat(nat2,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat)) ).

% nat_2
tff(fact_8587_sgn__power__injE,axiom,
    ! [A: real,Nb: nat,X: real,B: real] :
      ( ( aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),Nb)) = X )
     => ( ( X = aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,B)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,B)),Nb)) )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => ( A = B ) ) ) ) ).

% sgn_power_injE
tff(fact_8588_Suc__nat__eq__nat__zadd1,axiom,
    ! [Z2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z2)
     => ( aa_nat_nat(suc,aa_int_nat(nat2,Z2)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Z2)) ) ) ).

% Suc_nat_eq_nat_zadd1
tff(fact_8589_nat__less__iff,axiom,
    ! [W: int,Ma: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W)),Ma)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),aa_nat_int(semiri1314217659103216013at_int,Ma)) ) ) ).

% nat_less_iff
tff(fact_8590_nat__mult__distrib__neg,axiom,
    ! [Z2: int,Z4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z2),zero_zero_int)
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(times_times_int,Z2),Z4)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Z2))),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Z4))) ) ) ).

% nat_mult_distrib_neg
tff(fact_8591_nat__abs__int__diff,axiom,
    ! [A: nat,B: nat] :
      aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)) ).

% nat_abs_int_diff
tff(fact_8592_floor__eq4,axiom,
    ! [Nb: nat,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb)))
       => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,X)) = Nb ) ) ) ).

% floor_eq4
tff(fact_8593_diff__nat__eq__if,axiom,
    ! [Z2: int,Z4: int] :
      aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z2)),aa_int_nat(nat2,Z4)) = $ite(
        aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),zero_zero_int),
        aa_int_nat(nat2,Z2),
        $let(
          d: int,
          d:= aa_int_int(aa_int_fun_int_int(minus_minus_int,Z2),Z4),
          $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,d),zero_zero_int),zero_zero_nat,aa_int_nat(nat2,d)) ) ) ).

% diff_nat_eq_if
tff(fact_8594_frac__neg,axiom,
    ! [X: real] :
      aa_real_real(archim2898591450579166408c_real,aa_real_real(uminus_uminus_real,X)) = $ite(member_real(X,ring_1_Ints_real),zero_zero_real,aa_real_real(minus_minus_real(one_one_real),aa_real_real(archim2898591450579166408c_real,X))) ).

% frac_neg
tff(fact_8595_frac__neg,axiom,
    ! [X: rat] :
      aa_rat_rat(archimedean_frac_rat,aa_rat_rat(uminus_uminus_rat,X)) = $ite(member_rat(X,ring_1_Ints_rat),zero_zero_rat,aa_rat_rat(minus_minus_rat(one_one_rat),aa_rat_rat(archimedean_frac_rat,X))) ).

% frac_neg
tff(fact_8596_nat__dvd__iff,axiom,
    ! [Z2: int,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_int_nat(nat2,Z2)),Ma)
    <=> $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z2),aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Z2),aa_nat_int(semiri1314217659103216013at_int,Ma)),Ma = zero_zero_nat) ) ).

% nat_dvd_iff
tff(fact_8597_Arg__correct,axiom,
    ! [Z2: complex] :
      ( ( Z2 != zero_zero_complex )
     => ( ( aa_complex_complex(sgn_sgn_complex,Z2) = aa_real_complex(cis,aa_complex_real(arg,Z2)) )
        & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),aa_complex_real(arg,Z2))
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(arg,Z2)),pi) ) ) ).

% Arg_correct
tff(fact_8598_cis__Arg__unique,axiom,
    ! [Z2: complex,X: real] :
      ( ( aa_complex_complex(sgn_sgn_complex,Z2) = aa_real_complex(cis,X) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),X)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),pi)
         => ( aa_complex_real(arg,Z2) = X ) ) ) ) ).

% cis_Arg_unique
tff(fact_8599_le__mult__floor__Ints,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( member_real(A,ring_1_Ints_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(archim6058952711729229775r_real,A)),aa_real_int(archim6058952711729229775r_real,B)))),aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,aa_real_real(times_times_real(A),B)))) ) ) ).

% le_mult_floor_Ints
tff(fact_8600_le__mult__floor__Ints,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( member_real(A,ring_1_Ints_real)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(archim6058952711729229775r_real,A)),aa_real_int(archim6058952711729229775r_real,B)))),aa_int_rat(ring_1_of_int_rat,aa_real_int(archim6058952711729229775r_real,aa_real_real(times_times_real(A),B)))) ) ) ).

% le_mult_floor_Ints
tff(fact_8601_le__mult__floor__Ints,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( member_real(A,ring_1_Ints_real)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(archim6058952711729229775r_real,A)),aa_real_int(archim6058952711729229775r_real,B)))),aa_int_int(ring_1_of_int_int,aa_real_int(archim6058952711729229775r_real,aa_real_real(times_times_real(A),B)))) ) ) ).

% le_mult_floor_Ints
tff(fact_8602_le__mult__floor__Ints,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( member_rat(A,ring_1_Ints_rat)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(times_times_int,aa_rat_int(archim3151403230148437115or_rat,A)),aa_rat_int(archim3151403230148437115or_rat,B)))),aa_int_real(ring_1_of_int_real,aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(times_times_rat(A),B)))) ) ) ).

% le_mult_floor_Ints
tff(fact_8603_le__mult__floor__Ints,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( member_rat(A,ring_1_Ints_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(times_times_int,aa_rat_int(archim3151403230148437115or_rat,A)),aa_rat_int(archim3151403230148437115or_rat,B)))),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(times_times_rat(A),B)))) ) ) ).

% le_mult_floor_Ints
tff(fact_8604_le__mult__floor__Ints,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( member_rat(A,ring_1_Ints_rat)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_rat_int(archim3151403230148437115or_rat,A)),aa_rat_int(archim3151403230148437115or_rat,B)))),aa_int_int(ring_1_of_int_int,aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(times_times_rat(A),B)))) ) ) ).

% le_mult_floor_Ints
tff(fact_8605_frac__unique__iff,axiom,
    ! [X: real,A: real] :
      ( ( aa_real_real(archim2898591450579166408c_real,X) = A )
    <=> ( member_real(aa_real_real(minus_minus_real(X),A),ring_1_Ints_real)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real) ) ) ).

% frac_unique_iff
tff(fact_8606_frac__unique__iff,axiom,
    ! [X: rat,A: rat] :
      ( ( aa_rat_rat(archimedean_frac_rat,X) = A )
    <=> ( member_rat(aa_rat_rat(minus_minus_rat(X),A),ring_1_Ints_rat)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat) ) ) ).

% frac_unique_iff
tff(fact_8607_mult__ceiling__le__Ints,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( member_real(A,ring_1_Ints_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,aa_real_real(times_times_real(A),B)))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(archim7802044766580827645g_real,A)),aa_real_int(archim7802044766580827645g_real,B)))) ) ) ).

% mult_ceiling_le_Ints
tff(fact_8608_mult__ceiling__le__Ints,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( member_real(A,ring_1_Ints_real)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_real_int(archim7802044766580827645g_real,aa_real_real(times_times_real(A),B)))),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(archim7802044766580827645g_real,A)),aa_real_int(archim7802044766580827645g_real,B)))) ) ) ).

% mult_ceiling_le_Ints
tff(fact_8609_mult__ceiling__le__Ints,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( member_real(A,ring_1_Ints_real)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(times_times_real(A),B)))),aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(archim7802044766580827645g_real,A)),aa_real_int(archim7802044766580827645g_real,B)))) ) ) ).

% mult_ceiling_le_Ints
tff(fact_8610_mult__ceiling__le__Ints,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( member_rat(A,ring_1_Ints_rat)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(times_times_rat(A),B)))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(times_times_int,aa_rat_int(archim2889992004027027881ng_rat,A)),aa_rat_int(archim2889992004027027881ng_rat,B)))) ) ) ).

% mult_ceiling_le_Ints
tff(fact_8611_mult__ceiling__le__Ints,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( member_rat(A,ring_1_Ints_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(times_times_rat(A),B)))),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(times_times_int,aa_rat_int(archim2889992004027027881ng_rat,A)),aa_rat_int(archim2889992004027027881ng_rat,B)))) ) ) ).

% mult_ceiling_le_Ints
tff(fact_8612_mult__ceiling__le__Ints,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( member_rat(A,ring_1_Ints_rat)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(times_times_rat(A),B)))),aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_rat_int(archim2889992004027027881ng_rat,A)),aa_rat_int(archim2889992004027027881ng_rat,B)))) ) ) ).

% mult_ceiling_le_Ints
tff(fact_8613_sin__integer__2pi,axiom,
    ! [Nb: real] :
      ( member_real(Nb,ring_1_Ints_real)
     => ( aa_real_real(sin_real,aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),Nb)) = zero_zero_real ) ) ).

% sin_integer_2pi
tff(fact_8614_cos__integer__2pi,axiom,
    ! [Nb: real] :
      ( member_real(Nb,ring_1_Ints_real)
     => ( aa_real_real(cos_real,aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),Nb)) = one_one_real ) ) ).

% cos_integer_2pi
tff(fact_8615_even__nat__iff,axiom,
    ! [K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_int_nat(nat2,K2))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K2) ) ) ).

% even_nat_iff
tff(fact_8616_floor__rat__def,axiom,
    ! [X: rat] : aa_rat_int(archim3151403230148437115or_rat,X) = the_int(aTP_Lamm_id(X)) ).

% floor_rat_def
tff(fact_8617_powr__real__of__int,axiom,
    ! [X: real,Nb: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_real(powr_real(X),aa_int_real(ring_1_of_int_real,Nb)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Nb),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_int_nat(nat2,Nb)),aa_real_real(inverse_inverse_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Nb))))) ) ) ).

% powr_real_of_int
tff(fact_8618_case__prod__app,axiom,
    ! [F: fun_na1668100129766719005at_nat,X: product_prod_nat_nat,Y2: product_prod_nat_nat] : aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(F),X),Y2) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aa_Pro6895555690143131534at_nat(aTP_Lamm_ie(F),Y2)),X) ).

% case_prod_app
tff(fact_8619_case__prod__app,axiom,
    ! [F: fun_na5914473681949881652_nat_o,X: product_prod_nat_nat,Y2: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(F),X),Y2)
    <=> aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(aa_Pro1957341097241355813_nat_o(aTP_Lamm_if(F),Y2)),X) ) ).

% case_prod_app
tff(fact_8620_Suc__0__xor__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% Suc_0_xor_eq
tff(fact_8621_inverse__eq__iff__eq,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(inverse_inverse_real,A) = aa_real_real(inverse_inverse_real,B) )
    <=> ( A = B ) ) ).

% inverse_eq_iff_eq
tff(fact_8622_inverse__eq__iff__eq,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(invers8013647133539491842omplex,A) = aa_complex_complex(invers8013647133539491842omplex,B) )
    <=> ( A = B ) ) ).

% inverse_eq_iff_eq
tff(fact_8623_inverse__eq__iff__eq,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(inverse_inverse_rat,A) = aa_rat_rat(inverse_inverse_rat,B) )
    <=> ( A = B ) ) ).

% inverse_eq_iff_eq
tff(fact_8624_inverse__inverse__eq,axiom,
    ! [A: real] : aa_real_real(inverse_inverse_real,aa_real_real(inverse_inverse_real,A)) = A ).

% inverse_inverse_eq
tff(fact_8625_inverse__inverse__eq,axiom,
    ! [A: complex] : aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(invers8013647133539491842omplex,A)) = A ).

% inverse_inverse_eq
tff(fact_8626_inverse__inverse__eq,axiom,
    ! [A: rat] : aa_rat_rat(inverse_inverse_rat,aa_rat_rat(inverse_inverse_rat,A)) = A ).

% inverse_inverse_eq
tff(fact_8627_bit_Oxor__left__self,axiom,
    ! [X: int,Y2: int] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,X),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,X),Y2)) = Y2 ).

% bit.xor_left_self
tff(fact_8628_inverse__nonzero__iff__nonzero,axiom,
    ! [A: real] :
      ( ( aa_real_real(inverse_inverse_real,A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% inverse_nonzero_iff_nonzero
tff(fact_8629_inverse__nonzero__iff__nonzero,axiom,
    ! [A: complex] :
      ( ( aa_complex_complex(invers8013647133539491842omplex,A) = zero_zero_complex )
    <=> ( A = zero_zero_complex ) ) ).

% inverse_nonzero_iff_nonzero
tff(fact_8630_inverse__nonzero__iff__nonzero,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(inverse_inverse_rat,A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% inverse_nonzero_iff_nonzero
tff(fact_8631_inverse__zero,axiom,
    aa_real_real(inverse_inverse_real,zero_zero_real) = zero_zero_real ).

% inverse_zero
tff(fact_8632_inverse__zero,axiom,
    aa_complex_complex(invers8013647133539491842omplex,zero_zero_complex) = zero_zero_complex ).

% inverse_zero
tff(fact_8633_inverse__zero,axiom,
    aa_rat_rat(inverse_inverse_rat,zero_zero_rat) = zero_zero_rat ).

% inverse_zero
tff(fact_8634_inverse__mult__distrib,axiom,
    ! [A: real,B: real] : aa_real_real(inverse_inverse_real,aa_real_real(times_times_real(A),B)) = aa_real_real(times_times_real(aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)) ).

% inverse_mult_distrib
tff(fact_8635_inverse__mult__distrib,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(times_times_complex(A),B)) = aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(invers8013647133539491842omplex,B)) ).

% inverse_mult_distrib
tff(fact_8636_inverse__mult__distrib,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(inverse_inverse_rat,aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(times_times_rat(aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B)) ).

% inverse_mult_distrib
tff(fact_8637_inverse__eq__1__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(inverse_inverse_real,X) = one_one_real )
    <=> ( X = one_one_real ) ) ).

% inverse_eq_1_iff
tff(fact_8638_inverse__eq__1__iff,axiom,
    ! [X: complex] :
      ( ( aa_complex_complex(invers8013647133539491842omplex,X) = one_one_complex )
    <=> ( X = one_one_complex ) ) ).

% inverse_eq_1_iff
tff(fact_8639_inverse__eq__1__iff,axiom,
    ! [X: rat] :
      ( ( aa_rat_rat(inverse_inverse_rat,X) = one_one_rat )
    <=> ( X = one_one_rat ) ) ).

% inverse_eq_1_iff
tff(fact_8640_inverse__1,axiom,
    aa_real_real(inverse_inverse_real,one_one_real) = one_one_real ).

% inverse_1
tff(fact_8641_inverse__1,axiom,
    aa_complex_complex(invers8013647133539491842omplex,one_one_complex) = one_one_complex ).

% inverse_1
tff(fact_8642_inverse__1,axiom,
    aa_rat_rat(inverse_inverse_rat,one_one_rat) = one_one_rat ).

% inverse_1
tff(fact_8643_inverse__divide,axiom,
    ! [A: real,B: real] : aa_real_real(inverse_inverse_real,aa_real_real(divide_divide_real(A),B)) = aa_real_real(divide_divide_real(B),A) ).

% inverse_divide
tff(fact_8644_inverse__divide,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = aa_complex_complex(divide1717551699836669952omplex(B),A) ).

% inverse_divide
tff(fact_8645_inverse__divide,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(inverse_inverse_rat,aa_rat_rat(divide_divide_rat(A),B)) = aa_rat_rat(divide_divide_rat(B),A) ).

% inverse_divide
tff(fact_8646_inverse__minus__eq,axiom,
    ! [A: real] : aa_real_real(inverse_inverse_real,aa_real_real(uminus_uminus_real,A)) = aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,A)) ).

% inverse_minus_eq
tff(fact_8647_inverse__minus__eq,axiom,
    ! [A: complex] : aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(uminus1482373934393186551omplex,A)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(invers8013647133539491842omplex,A)) ).

% inverse_minus_eq
tff(fact_8648_inverse__minus__eq,axiom,
    ! [A: rat] : aa_rat_rat(inverse_inverse_rat,aa_rat_rat(uminus_uminus_rat,A)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(inverse_inverse_rat,A)) ).

% inverse_minus_eq
tff(fact_8649_bit_Oxor__self,axiom,
    ! [X: int] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,X),X) = zero_zero_int ).

% bit.xor_self
tff(fact_8650_xor__self__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,A),A) = zero_zero_nat ).

% xor_self_eq
tff(fact_8651_xor__self__eq,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,A),A) = zero_zero_int ).

% xor_self_eq
tff(fact_8652_xor_Oleft__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,zero_zero_nat),A) = A ).

% xor.left_neutral
tff(fact_8653_xor_Oleft__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,zero_zero_int),A) = A ).

% xor.left_neutral
tff(fact_8654_xor_Oright__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,A),zero_zero_nat) = A ).

% xor.right_neutral
tff(fact_8655_xor_Oright__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,A),zero_zero_int) = A ).

% xor.right_neutral
tff(fact_8656_abs__inverse,axiom,
    ! [A: real] : aa_real_real(abs_abs_real,aa_real_real(inverse_inverse_real,A)) = aa_real_real(inverse_inverse_real,aa_real_real(abs_abs_real,A)) ).

% abs_inverse
tff(fact_8657_abs__inverse,axiom,
    ! [A: complex] : aa_complex_complex(abs_abs_complex,aa_complex_complex(invers8013647133539491842omplex,A)) = aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(abs_abs_complex,A)) ).

% abs_inverse
tff(fact_8658_abs__inverse,axiom,
    ! [A: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(inverse_inverse_rat,A)) = aa_rat_rat(inverse_inverse_rat,aa_rat_rat(abs_abs_rat,A)) ).

% abs_inverse
tff(fact_8659_sgn__inverse,axiom,
    ! [A: real] : aa_real_real(sgn_sgn_real,aa_real_real(inverse_inverse_real,A)) = aa_real_real(inverse_inverse_real,aa_real_real(sgn_sgn_real,A)) ).

% sgn_inverse
tff(fact_8660_sgn__inverse,axiom,
    ! [A: complex] : aa_complex_complex(sgn_sgn_complex,aa_complex_complex(invers8013647133539491842omplex,A)) = aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(sgn_sgn_complex,A)) ).

% sgn_inverse
tff(fact_8661_sgn__inverse,axiom,
    ! [A: rat] : aa_rat_rat(sgn_sgn_rat,aa_rat_rat(inverse_inverse_rat,A)) = aa_rat_rat(inverse_inverse_rat,aa_rat_rat(sgn_sgn_rat,A)) ).

% sgn_inverse
tff(fact_8662_inverse__sgn,axiom,
    ! [A: real] : aa_real_real(inverse_inverse_real,aa_real_real(sgn_sgn_real,A)) = aa_real_real(sgn_sgn_real,A) ).

% inverse_sgn
tff(fact_8663_inverse__sgn,axiom,
    ! [A: rat] : aa_rat_rat(inverse_inverse_rat,aa_rat_rat(sgn_sgn_rat,A)) = aa_rat_rat(sgn_sgn_rat,A) ).

% inverse_sgn
tff(fact_8664_take__bit__xor,axiom,
    ! [Nb: nat,A: int,B: int] : aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,A),B)) = aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_int_int(bit_se2923211474154528505it_int(Nb),A)),aa_int_int(bit_se2923211474154528505it_int(Nb),B)) ).

% take_bit_xor
tff(fact_8665_take__bit__xor,axiom,
    ! [Nb: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),B)) ).

% take_bit_xor
tff(fact_8666_inverse__nonpositive__iff__nonpositive,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(inverse_inverse_real,A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% inverse_nonpositive_iff_nonpositive
tff(fact_8667_inverse__nonpositive__iff__nonpositive,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% inverse_nonpositive_iff_nonpositive
tff(fact_8668_inverse__nonnegative__iff__nonnegative,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(inverse_inverse_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% inverse_nonnegative_iff_nonnegative
tff(fact_8669_inverse__nonnegative__iff__nonnegative,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(inverse_inverse_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% inverse_nonnegative_iff_nonnegative
tff(fact_8670_inverse__less__iff__less,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).

% inverse_less_iff_less
tff(fact_8671_inverse__less__iff__less,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% inverse_less_iff_less
tff(fact_8672_inverse__less__iff__less__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ) ).

% inverse_less_iff_less_neg
tff(fact_8673_inverse__less__iff__less__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ) ).

% inverse_less_iff_less_neg
tff(fact_8674_inverse__negative__iff__negative,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% inverse_negative_iff_negative
tff(fact_8675_inverse__negative__iff__negative,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% inverse_negative_iff_negative
tff(fact_8676_inverse__positive__iff__positive,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(inverse_inverse_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% inverse_positive_iff_positive
tff(fact_8677_inverse__positive__iff__positive,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(inverse_inverse_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% inverse_positive_iff_positive
tff(fact_8678_inverse__le__iff__le__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).

% inverse_le_iff_le_neg
tff(fact_8679_inverse__le__iff__le__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).

% inverse_le_iff_le_neg
tff(fact_8680_inverse__le__iff__le,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).

% inverse_le_iff_le
tff(fact_8681_inverse__le__iff__le,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ) ).

% inverse_le_iff_le
tff(fact_8682_left__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(times_times_real(aa_real_real(inverse_inverse_real,A)),A) = one_one_real ) ) ).

% left_inverse
tff(fact_8683_left__inverse,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,A)),A) = one_one_complex ) ) ).

% left_inverse
tff(fact_8684_left__inverse,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(times_times_rat(aa_rat_rat(inverse_inverse_rat,A)),A) = one_one_rat ) ) ).

% left_inverse
tff(fact_8685_right__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(times_times_real(A),aa_real_real(inverse_inverse_real,A)) = one_one_real ) ) ).

% right_inverse
tff(fact_8686_right__inverse,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(times_times_complex(A),aa_complex_complex(invers8013647133539491842omplex,A)) = one_one_complex ) ) ).

% right_inverse
tff(fact_8687_right__inverse,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(times_times_rat(A),aa_rat_rat(inverse_inverse_rat,A)) = one_one_rat ) ) ).

% right_inverse
tff(fact_8688_inverse__eq__divide__numeral,axiom,
    ! [W: num] : aa_real_real(inverse_inverse_real,aa_num_real(numeral_numeral_real,W)) = aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,W)) ).

% inverse_eq_divide_numeral
tff(fact_8689_inverse__eq__divide__numeral,axiom,
    ! [W: num] : aa_complex_complex(invers8013647133539491842omplex,numera6690914467698888265omplex(W)) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),numera6690914467698888265omplex(W)) ).

% inverse_eq_divide_numeral
tff(fact_8690_inverse__eq__divide__numeral,axiom,
    ! [W: num] : aa_rat_rat(inverse_inverse_rat,aa_num_rat(numeral_numeral_rat,W)) = aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,W)) ).

% inverse_eq_divide_numeral
tff(fact_8691_inverse__eq__divide__neg__numeral,axiom,
    ! [W: num] : aa_real_real(inverse_inverse_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) = aa_real_real(divide_divide_real(one_one_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) ).

% inverse_eq_divide_neg_numeral
tff(fact_8692_inverse__eq__divide__neg__numeral,axiom,
    ! [W: num] : aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) ).

% inverse_eq_divide_neg_numeral
tff(fact_8693_inverse__eq__divide__neg__numeral,axiom,
    ! [W: num] : aa_rat_rat(inverse_inverse_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))) = aa_rat_rat(divide_divide_rat(one_one_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))) ).

% inverse_eq_divide_neg_numeral
tff(fact_8694_xor__numerals_I3_J,axiom,
    ! [X: num,Y2: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y2))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,X)),aa_num_nat(numeral_numeral_nat,Y2))) ).

% xor_numerals(3)
tff(fact_8695_xor__numerals_I3_J,axiom,
    ! [X: num,Y2: num] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,X))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y2))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_num_int(numeral_numeral_int,X)),aa_num_int(numeral_numeral_int,Y2))) ).

% xor_numerals(3)
tff(fact_8696_xor__numerals_I1_J,axiom,
    ! [Y2: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y2))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y2)) ).

% xor_numerals(1)
tff(fact_8697_xor__numerals_I1_J,axiom,
    ! [Y2: num] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y2))) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y2)) ).

% xor_numerals(1)
tff(fact_8698_xor__numerals_I2_J,axiom,
    ! [Y2: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y2))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y2)) ).

% xor_numerals(2)
tff(fact_8699_xor__numerals_I2_J,axiom,
    ! [Y2: num] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y2))) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y2)) ).

% xor_numerals(2)
tff(fact_8700_xor__numerals_I5_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X))),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X)) ).

% xor_numerals(5)
tff(fact_8701_xor__numerals_I5_J,axiom,
    ! [X: num] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,X))),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,X)) ).

% xor_numerals(5)
tff(fact_8702_xor__numerals_I8_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X)) ).

% xor_numerals(8)
tff(fact_8703_xor__numerals_I8_J,axiom,
    ! [X: num] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,X))),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,X)) ).

% xor_numerals(8)
tff(fact_8704_xor__numerals_I7_J,axiom,
    ! [X: num,Y2: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y2))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,X)),aa_num_nat(numeral_numeral_nat,Y2))) ).

% xor_numerals(7)
tff(fact_8705_xor__numerals_I7_J,axiom,
    ! [X: num,Y2: num] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,X))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y2))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_num_int(numeral_numeral_int,X)),aa_num_int(numeral_numeral_int,Y2))) ).

% xor_numerals(7)
tff(fact_8706_xor__nat__numerals_I4_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X)) ).

% xor_nat_numerals(4)
tff(fact_8707_xor__nat__numerals_I3_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X)) ).

% xor_nat_numerals(3)
tff(fact_8708_xor__nat__numerals_I2_J,axiom,
    ! [Y2: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y2))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y2)) ).

% xor_nat_numerals(2)
tff(fact_8709_xor__nat__numerals_I1_J,axiom,
    ! [Y2: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y2))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y2)) ).

% xor_nat_numerals(1)
tff(fact_8710_xor__numerals_I4_J,axiom,
    ! [X: num,Y2: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,X)),aa_num_nat(numeral_numeral_nat,Y2)))) ).

% xor_numerals(4)
tff(fact_8711_xor__numerals_I4_J,axiom,
    ! [X: num,Y2: num] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,X))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_num_int(numeral_numeral_int,X)),aa_num_int(numeral_numeral_int,Y2)))) ).

% xor_numerals(4)
tff(fact_8712_xor__numerals_I6_J,axiom,
    ! [X: num,Y2: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,X)),aa_num_nat(numeral_numeral_nat,Y2)))) ).

% xor_numerals(6)
tff(fact_8713_xor__numerals_I6_J,axiom,
    ! [X: num,Y2: num] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,X))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_num_int(numeral_numeral_int,X)),aa_num_int(numeral_numeral_int,Y2)))) ).

% xor_numerals(6)
tff(fact_8714_sgn__rat__def,axiom,
    ! [A: rat] :
      aa_rat_rat(sgn_sgn_rat,A) = $ite(
        A = zero_zero_rat,
        zero_zero_rat,
        $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A),one_one_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) ) ).

% sgn_rat_def
tff(fact_8715_power__inverse,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(inverse_inverse_real,A)),Nb) = aa_real_real(inverse_inverse_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ).

% power_inverse
tff(fact_8716_power__inverse,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(invers8013647133539491842omplex,A)),Nb) = aa_complex_complex(invers8013647133539491842omplex,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,A),Nb)) ).

% power_inverse
tff(fact_8717_power__inverse,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(inverse_inverse_rat,A)),Nb) = aa_rat_rat(inverse_inverse_rat,aa_nat_rat(power_power_rat(A),Nb)) ).

% power_inverse
tff(fact_8718_mult__commute__imp__mult__inverse__commute,axiom,
    ! [Y2: real,X: real] :
      ( ( aa_real_real(times_times_real(Y2),X) = aa_real_real(times_times_real(X),Y2) )
     => ( aa_real_real(times_times_real(aa_real_real(inverse_inverse_real,Y2)),X) = aa_real_real(times_times_real(X),aa_real_real(inverse_inverse_real,Y2)) ) ) ).

% mult_commute_imp_mult_inverse_commute
tff(fact_8719_mult__commute__imp__mult__inverse__commute,axiom,
    ! [Y2: complex,X: complex] :
      ( ( aa_complex_complex(times_times_complex(Y2),X) = aa_complex_complex(times_times_complex(X),Y2) )
     => ( aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,Y2)),X) = aa_complex_complex(times_times_complex(X),aa_complex_complex(invers8013647133539491842omplex,Y2)) ) ) ).

% mult_commute_imp_mult_inverse_commute
tff(fact_8720_mult__commute__imp__mult__inverse__commute,axiom,
    ! [Y2: rat,X: rat] :
      ( ( aa_rat_rat(times_times_rat(Y2),X) = aa_rat_rat(times_times_rat(X),Y2) )
     => ( aa_rat_rat(times_times_rat(aa_rat_rat(inverse_inverse_rat,Y2)),X) = aa_rat_rat(times_times_rat(X),aa_rat_rat(inverse_inverse_rat,Y2)) ) ) ).

% mult_commute_imp_mult_inverse_commute
tff(fact_8721_real__sqrt__inverse,axiom,
    ! [X: real] : aa_real_real(sqrt,aa_real_real(inverse_inverse_real,X)) = aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X)) ).

% real_sqrt_inverse
tff(fact_8722_nonzero__imp__inverse__nonzero,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(inverse_inverse_real,A) != zero_zero_real ) ) ).

% nonzero_imp_inverse_nonzero
tff(fact_8723_nonzero__imp__inverse__nonzero,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(invers8013647133539491842omplex,A) != zero_zero_complex ) ) ).

% nonzero_imp_inverse_nonzero
tff(fact_8724_nonzero__imp__inverse__nonzero,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(inverse_inverse_rat,A) != zero_zero_rat ) ) ).

% nonzero_imp_inverse_nonzero
tff(fact_8725_nonzero__inverse__inverse__eq,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(inverse_inverse_real,aa_real_real(inverse_inverse_real,A)) = A ) ) ).

% nonzero_inverse_inverse_eq
tff(fact_8726_nonzero__inverse__inverse__eq,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(invers8013647133539491842omplex,A)) = A ) ) ).

% nonzero_inverse_inverse_eq
tff(fact_8727_nonzero__inverse__inverse__eq,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(inverse_inverse_rat,aa_rat_rat(inverse_inverse_rat,A)) = A ) ) ).

% nonzero_inverse_inverse_eq
tff(fact_8728_nonzero__inverse__eq__imp__eq,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(inverse_inverse_real,A) = aa_real_real(inverse_inverse_real,B) )
     => ( ( A != zero_zero_real )
       => ( ( B != zero_zero_real )
         => ( A = B ) ) ) ) ).

% nonzero_inverse_eq_imp_eq
tff(fact_8729_nonzero__inverse__eq__imp__eq,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(invers8013647133539491842omplex,A) = aa_complex_complex(invers8013647133539491842omplex,B) )
     => ( ( A != zero_zero_complex )
       => ( ( B != zero_zero_complex )
         => ( A = B ) ) ) ) ).

% nonzero_inverse_eq_imp_eq
tff(fact_8730_nonzero__inverse__eq__imp__eq,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(inverse_inverse_rat,A) = aa_rat_rat(inverse_inverse_rat,B) )
     => ( ( A != zero_zero_rat )
       => ( ( B != zero_zero_rat )
         => ( A = B ) ) ) ) ).

% nonzero_inverse_eq_imp_eq
tff(fact_8731_inverse__zero__imp__zero,axiom,
    ! [A: real] :
      ( ( aa_real_real(inverse_inverse_real,A) = zero_zero_real )
     => ( A = zero_zero_real ) ) ).

% inverse_zero_imp_zero
tff(fact_8732_inverse__zero__imp__zero,axiom,
    ! [A: complex] :
      ( ( aa_complex_complex(invers8013647133539491842omplex,A) = zero_zero_complex )
     => ( A = zero_zero_complex ) ) ).

% inverse_zero_imp_zero
tff(fact_8733_inverse__zero__imp__zero,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(inverse_inverse_rat,A) = zero_zero_rat )
     => ( A = zero_zero_rat ) ) ).

% inverse_zero_imp_zero
tff(fact_8734_field__class_Ofield__inverse__zero,axiom,
    aa_real_real(inverse_inverse_real,zero_zero_real) = zero_zero_real ).

% field_class.field_inverse_zero
tff(fact_8735_field__class_Ofield__inverse__zero,axiom,
    aa_complex_complex(invers8013647133539491842omplex,zero_zero_complex) = zero_zero_complex ).

% field_class.field_inverse_zero
tff(fact_8736_field__class_Ofield__inverse__zero,axiom,
    aa_rat_rat(inverse_inverse_rat,zero_zero_rat) = zero_zero_rat ).

% field_class.field_inverse_zero
tff(fact_8737_less__eq__rat__def,axiom,
    ! [X: rat,Y2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y2)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y2)
        | ( X = Y2 ) ) ) ).

% less_eq_rat_def
tff(fact_8738_inverse__eq__imp__eq,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(inverse_inverse_real,A) = aa_real_real(inverse_inverse_real,B) )
     => ( A = B ) ) ).

% inverse_eq_imp_eq
tff(fact_8739_inverse__eq__imp__eq,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(invers8013647133539491842omplex,A) = aa_complex_complex(invers8013647133539491842omplex,B) )
     => ( A = B ) ) ).

% inverse_eq_imp_eq
tff(fact_8740_inverse__eq__imp__eq,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(inverse_inverse_rat,A) = aa_rat_rat(inverse_inverse_rat,B) )
     => ( A = B ) ) ).

% inverse_eq_imp_eq
tff(fact_8741_xor_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,B),C)) ).

% xor.assoc
tff(fact_8742_xor_Oassoc,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,A),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,B),C)) ).

% xor.assoc
tff(fact_8743_xor_Ocommute,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,B),A) ).

% xor.commute
tff(fact_8744_xor_Ocommute,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,A),B) = aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,B),A) ).

% xor.commute
tff(fact_8745_xor_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,A),C)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,B),C)) ).

% xor.left_commute
tff(fact_8746_xor_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,B),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,A),C)) = aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,A),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,B),C)) ).

% xor.left_commute
tff(fact_8747_of__int__xor__eq,axiom,
    ! [K2: int,L: int] : aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K2),L)) = aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_int_int(ring_1_of_int_int,K2)),aa_int_int(ring_1_of_int_int,L)) ).

% of_int_xor_eq
tff(fact_8748_obtain__pos__sum,axiom,
    ! [R: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R)
     => ~ ! [S2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),S2)
           => ! [T3: rat] :
                ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),T3)
               => ( R != aa_rat_rat(plus_plus_rat(S2),T3) ) ) ) ) ).

% obtain_pos_sum
tff(fact_8749_of__nat__xor__eq,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Ma),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(semiri1316708129612266289at_nat,Ma)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% of_nat_xor_eq
tff(fact_8750_of__nat__xor__eq,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Ma),Nb)) = aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_nat_int(semiri1314217659103216013at_int,Ma)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% of_nat_xor_eq
tff(fact_8751_bit__xor__iff,axiom,
    ! [A: nat,B: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,A),B)),Nb)
    <=> ~ ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Nb)
        <=> aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,B),Nb) ) ) ).

% bit_xor_iff
tff(fact_8752_bit__xor__iff,axiom,
    ! [A: int,B: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,A),B)),Nb)
    <=> ~ ( aa_nat_o(bit_se1146084159140164899it_int(A),Nb)
        <=> aa_nat_o(bit_se1146084159140164899it_int(B),Nb) ) ) ).

% bit_xor_iff
tff(fact_8753_bit_Oconj__xor__distrib2,axiom,
    ! [Y2: int,Z2: int,X: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,Y2),Z2)),X) = aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Y2),X)),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Z2),X)) ).

% bit.conj_xor_distrib2
tff(fact_8754_bit_Oconj__xor__distrib,axiom,
    ! [X: int,Y2: int,Z2: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,Y2),Z2)) = aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),Y2)),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),Z2)) ).

% bit.conj_xor_distrib
tff(fact_8755_norm__inverse__le__norm,axiom,
    ! [R: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,R),aa_real_real(real_V7735802525324610683m_real,X))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(inverse_inverse_real,X))),aa_real_real(inverse_inverse_real,R)) ) ) ).

% norm_inverse_le_norm
tff(fact_8756_norm__inverse__le__norm,axiom,
    ! [R: real,X: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,R),aa_complex_real(real_V1022390504157884413omplex,X))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(invers8013647133539491842omplex,X))),aa_real_real(inverse_inverse_real,R)) ) ) ).

% norm_inverse_le_norm
tff(fact_8757_inverse__less__imp__less,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ).

% inverse_less_imp_less
tff(fact_8758_inverse__less__imp__less,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ).

% inverse_less_imp_less
tff(fact_8759_less__imp__inverse__less,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,B)),aa_real_real(inverse_inverse_real,A)) ) ) ).

% less_imp_inverse_less
tff(fact_8760_less__imp__inverse__less,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,B)),aa_rat_rat(inverse_inverse_rat,A)) ) ) ).

% less_imp_inverse_less
tff(fact_8761_inverse__less__imp__less__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) ) ) ).

% inverse_less_imp_less_neg
tff(fact_8762_inverse__less__imp__less__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) ) ) ).

% inverse_less_imp_less_neg
tff(fact_8763_less__imp__inverse__less__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,B)),aa_real_real(inverse_inverse_real,A)) ) ) ).

% less_imp_inverse_less_neg
tff(fact_8764_less__imp__inverse__less__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,B)),aa_rat_rat(inverse_inverse_rat,A)) ) ) ).

% less_imp_inverse_less_neg
tff(fact_8765_inverse__negative__imp__negative,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,A)),zero_zero_real)
     => ( ( A != zero_zero_real )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ) ).

% inverse_negative_imp_negative
tff(fact_8766_inverse__negative__imp__negative,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),zero_zero_rat)
     => ( ( A != zero_zero_rat )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ) ).

% inverse_negative_imp_negative
tff(fact_8767_inverse__positive__imp__positive,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(inverse_inverse_real,A))
     => ( ( A != zero_zero_real )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ) ).

% inverse_positive_imp_positive
tff(fact_8768_inverse__positive__imp__positive,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(inverse_inverse_rat,A))
     => ( ( A != zero_zero_rat )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ) ).

% inverse_positive_imp_positive
tff(fact_8769_negative__imp__inverse__negative,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,A)),zero_zero_real) ) ).

% negative_imp_inverse_negative
tff(fact_8770_negative__imp__inverse__negative,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),zero_zero_rat) ) ).

% negative_imp_inverse_negative
tff(fact_8771_positive__imp__inverse__positive,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(inverse_inverse_real,A)) ) ).

% positive_imp_inverse_positive
tff(fact_8772_positive__imp__inverse__positive,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(inverse_inverse_rat,A)) ) ).

% positive_imp_inverse_positive
tff(fact_8773_nonzero__inverse__mult__distrib,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( aa_real_real(inverse_inverse_real,aa_real_real(times_times_real(A),B)) = aa_real_real(times_times_real(aa_real_real(inverse_inverse_real,B)),aa_real_real(inverse_inverse_real,A)) ) ) ) ).

% nonzero_inverse_mult_distrib
tff(fact_8774_nonzero__inverse__mult__distrib,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B != zero_zero_complex )
       => ( aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(times_times_complex(A),B)) = aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,B)),aa_complex_complex(invers8013647133539491842omplex,A)) ) ) ) ).

% nonzero_inverse_mult_distrib
tff(fact_8775_nonzero__inverse__mult__distrib,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( aa_rat_rat(inverse_inverse_rat,aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(times_times_rat(aa_rat_rat(inverse_inverse_rat,B)),aa_rat_rat(inverse_inverse_rat,A)) ) ) ) ).

% nonzero_inverse_mult_distrib
tff(fact_8776_nonzero__inverse__minus__eq,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(inverse_inverse_real,aa_real_real(uminus_uminus_real,A)) = aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,A)) ) ) ).

% nonzero_inverse_minus_eq
tff(fact_8777_nonzero__inverse__minus__eq,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(uminus1482373934393186551omplex,A)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(invers8013647133539491842omplex,A)) ) ) ).

% nonzero_inverse_minus_eq
tff(fact_8778_nonzero__inverse__minus__eq,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(inverse_inverse_rat,aa_rat_rat(uminus_uminus_rat,A)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(inverse_inverse_rat,A)) ) ) ).

% nonzero_inverse_minus_eq
tff(fact_8779_inverse__numeral__1,axiom,
    aa_real_real(inverse_inverse_real,aa_num_real(numeral_numeral_real,one)) = aa_num_real(numeral_numeral_real,one) ).

% inverse_numeral_1
tff(fact_8780_inverse__numeral__1,axiom,
    aa_complex_complex(invers8013647133539491842omplex,numera6690914467698888265omplex(one)) = numera6690914467698888265omplex(one) ).

% inverse_numeral_1
tff(fact_8781_inverse__numeral__1,axiom,
    aa_rat_rat(inverse_inverse_rat,aa_num_rat(numeral_numeral_rat,one)) = aa_num_rat(numeral_numeral_rat,one) ).

% inverse_numeral_1
tff(fact_8782_inverse__unique,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(times_times_real(A),B) = one_one_real )
     => ( aa_real_real(inverse_inverse_real,A) = B ) ) ).

% inverse_unique
tff(fact_8783_inverse__unique,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_complex(times_times_complex(A),B) = one_one_complex )
     => ( aa_complex_complex(invers8013647133539491842omplex,A) = B ) ) ).

% inverse_unique
tff(fact_8784_inverse__unique,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(times_times_rat(A),B) = one_one_rat )
     => ( aa_rat_rat(inverse_inverse_rat,A) = B ) ) ).

% inverse_unique
tff(fact_8785_divide__inverse__commute,axiom,
    ! [A: real,B: real] : aa_real_real(divide_divide_real(A),B) = aa_real_real(times_times_real(aa_real_real(inverse_inverse_real,B)),A) ).

% divide_inverse_commute
tff(fact_8786_divide__inverse__commute,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(divide1717551699836669952omplex(A),B) = aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,B)),A) ).

% divide_inverse_commute
tff(fact_8787_divide__inverse__commute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(divide_divide_rat(A),B) = aa_rat_rat(times_times_rat(aa_rat_rat(inverse_inverse_rat,B)),A) ).

% divide_inverse_commute
tff(fact_8788_divide__inverse,axiom,
    ! [A: real,B: real] : aa_real_real(divide_divide_real(A),B) = aa_real_real(times_times_real(A),aa_real_real(inverse_inverse_real,B)) ).

% divide_inverse
tff(fact_8789_divide__inverse,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(divide1717551699836669952omplex(A),B) = aa_complex_complex(times_times_complex(A),aa_complex_complex(invers8013647133539491842omplex,B)) ).

% divide_inverse
tff(fact_8790_divide__inverse,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(divide_divide_rat(A),B) = aa_rat_rat(times_times_rat(A),aa_rat_rat(inverse_inverse_rat,B)) ).

% divide_inverse
tff(fact_8791_field__class_Ofield__divide__inverse,axiom,
    ! [A: real,B: real] : aa_real_real(divide_divide_real(A),B) = aa_real_real(times_times_real(A),aa_real_real(inverse_inverse_real,B)) ).

% field_class.field_divide_inverse
tff(fact_8792_field__class_Ofield__divide__inverse,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(divide1717551699836669952omplex(A),B) = aa_complex_complex(times_times_complex(A),aa_complex_complex(invers8013647133539491842omplex,B)) ).

% field_class.field_divide_inverse
tff(fact_8793_field__class_Ofield__divide__inverse,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(divide_divide_rat(A),B) = aa_rat_rat(times_times_rat(A),aa_rat_rat(inverse_inverse_rat,B)) ).

% field_class.field_divide_inverse
tff(fact_8794_inverse__eq__divide,axiom,
    ! [A: real] : aa_real_real(inverse_inverse_real,A) = aa_real_real(divide_divide_real(one_one_real),A) ).

% inverse_eq_divide
tff(fact_8795_inverse__eq__divide,axiom,
    ! [A: complex] : aa_complex_complex(invers8013647133539491842omplex,A) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),A) ).

% inverse_eq_divide
tff(fact_8796_inverse__eq__divide,axiom,
    ! [A: rat] : aa_rat_rat(inverse_inverse_rat,A) = aa_rat_rat(divide_divide_rat(one_one_rat),A) ).

% inverse_eq_divide
tff(fact_8797_power__mult__power__inverse__commute,axiom,
    ! [X: real,Ma: nat,Nb: nat] : aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Ma)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(inverse_inverse_real,X)),Nb)) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(inverse_inverse_real,X)),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Ma)) ).

% power_mult_power_inverse_commute
tff(fact_8798_power__mult__power__inverse__commute,axiom,
    ! [X: complex,Ma: nat,Nb: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,X),Ma)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(invers8013647133539491842omplex,X)),Nb)) = aa_complex_complex(times_times_complex(aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(invers8013647133539491842omplex,X)),Nb)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,X),Ma)) ).

% power_mult_power_inverse_commute
tff(fact_8799_power__mult__power__inverse__commute,axiom,
    ! [X: rat,Ma: nat,Nb: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(X),Ma)),aa_nat_rat(power_power_rat(aa_rat_rat(inverse_inverse_rat,X)),Nb)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(inverse_inverse_rat,X)),Nb)),aa_nat_rat(power_power_rat(X),Ma)) ).

% power_mult_power_inverse_commute
tff(fact_8800_power__mult__inverse__distrib,axiom,
    ! [X: real,Ma: nat] : aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Ma)),aa_real_real(inverse_inverse_real,X)) = aa_real_real(times_times_real(aa_real_real(inverse_inverse_real,X)),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Ma)) ).

% power_mult_inverse_distrib
tff(fact_8801_power__mult__inverse__distrib,axiom,
    ! [X: complex,Ma: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,X),Ma)),aa_complex_complex(invers8013647133539491842omplex,X)) = aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,X)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,X),Ma)) ).

% power_mult_inverse_distrib
tff(fact_8802_power__mult__inverse__distrib,axiom,
    ! [X: rat,Ma: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(X),Ma)),aa_rat_rat(inverse_inverse_rat,X)) = aa_rat_rat(times_times_rat(aa_rat_rat(inverse_inverse_rat,X)),aa_nat_rat(power_power_rat(X),Ma)) ).

% power_mult_inverse_distrib
tff(fact_8803_mult__inverse__of__nat__commute,axiom,
    ! [Xa: nat,X: real] : aa_real_real(times_times_real(aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,Xa))),X) = aa_real_real(times_times_real(X),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,Xa))) ).

% mult_inverse_of_nat_commute
tff(fact_8804_mult__inverse__of__nat__commute,axiom,
    ! [Xa: nat,X: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,aa_nat_complex(semiri8010041392384452111omplex,Xa))),X) = aa_complex_complex(times_times_complex(X),aa_complex_complex(invers8013647133539491842omplex,aa_nat_complex(semiri8010041392384452111omplex,Xa))) ).

% mult_inverse_of_nat_commute
tff(fact_8805_mult__inverse__of__nat__commute,axiom,
    ! [Xa: nat,X: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(inverse_inverse_rat,aa_nat_rat(semiri681578069525770553at_rat,Xa))),X) = aa_rat_rat(times_times_rat(X),aa_rat_rat(inverse_inverse_rat,aa_nat_rat(semiri681578069525770553at_rat,Xa))) ).

% mult_inverse_of_nat_commute
tff(fact_8806_nonzero__abs__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(abs_abs_real,aa_real_real(inverse_inverse_real,A)) = aa_real_real(inverse_inverse_real,aa_real_real(abs_abs_real,A)) ) ) ).

% nonzero_abs_inverse
tff(fact_8807_nonzero__abs__inverse,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(abs_abs_rat,aa_rat_rat(inverse_inverse_rat,A)) = aa_rat_rat(inverse_inverse_rat,aa_rat_rat(abs_abs_rat,A)) ) ) ).

% nonzero_abs_inverse
tff(fact_8808_mult__inverse__of__int__commute,axiom,
    ! [Xa: int,X: real] : aa_real_real(times_times_real(aa_real_real(inverse_inverse_real,aa_int_real(ring_1_of_int_real,Xa))),X) = aa_real_real(times_times_real(X),aa_real_real(inverse_inverse_real,aa_int_real(ring_1_of_int_real,Xa))) ).

% mult_inverse_of_int_commute
tff(fact_8809_mult__inverse__of__int__commute,axiom,
    ! [Xa: int,X: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,aa_int_complex(ring_17405671764205052669omplex,Xa))),X) = aa_complex_complex(times_times_complex(X),aa_complex_complex(invers8013647133539491842omplex,aa_int_complex(ring_17405671764205052669omplex,Xa))) ).

% mult_inverse_of_int_commute
tff(fact_8810_mult__inverse__of__int__commute,axiom,
    ! [Xa: int,X: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(inverse_inverse_rat,aa_int_rat(ring_1_of_int_rat,Xa))),X) = aa_rat_rat(times_times_rat(X),aa_rat_rat(inverse_inverse_rat,aa_int_rat(ring_1_of_int_rat,Xa))) ).

% mult_inverse_of_int_commute
tff(fact_8811_divide__real__def,axiom,
    ! [X: real,Y2: real] : aa_real_real(divide_divide_real(X),Y2) = aa_real_real(times_times_real(X),aa_real_real(inverse_inverse_real,Y2)) ).

% divide_real_def
tff(fact_8812_exp__fdiffs,axiom,
    ! [X2: nat] : aa_nat_complex(diffs_complex(aTP_Lamm_ig),X2) = aa_complex_complex(invers8013647133539491842omplex,aa_nat_complex(semiri5044797733671781792omplex,X2)) ).

% exp_fdiffs
tff(fact_8813_exp__fdiffs,axiom,
    ! [X2: nat] : aa_nat_real(diffs_real(aTP_Lamm_ih),X2) = aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,X2)) ).

% exp_fdiffs
tff(fact_8814_le__imp__inverse__le__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(inverse_inverse_real,B)),aa_real_real(inverse_inverse_real,A)) ) ) ).

% le_imp_inverse_le_neg
tff(fact_8815_le__imp__inverse__le__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,B)),aa_rat_rat(inverse_inverse_rat,A)) ) ) ).

% le_imp_inverse_le_neg
tff(fact_8816_inverse__le__imp__le__neg,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ).

% inverse_le_imp_le_neg
tff(fact_8817_inverse__le__imp__le__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).

% inverse_le_imp_le_neg
tff(fact_8818_le__imp__inverse__le,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(inverse_inverse_real,B)),aa_real_real(inverse_inverse_real,A)) ) ) ).

% le_imp_inverse_le
tff(fact_8819_le__imp__inverse__le,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,B)),aa_rat_rat(inverse_inverse_rat,A)) ) ) ).

% le_imp_inverse_le
tff(fact_8820_inverse__le__imp__le,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ).

% inverse_le_imp_le
tff(fact_8821_inverse__le__imp__le,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).

% inverse_le_imp_le
tff(fact_8822_inverse__le__1__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(inverse_inverse_real,X)),one_one_real)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),zero_zero_real)
        | aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),X) ) ) ).

% inverse_le_1_iff
tff(fact_8823_inverse__le__1__iff,axiom,
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,X)),one_one_rat)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),zero_zero_rat)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),X) ) ) ).

% inverse_le_1_iff
tff(fact_8824_one__less__inverse,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(inverse_inverse_real,A)) ) ) ).

% one_less_inverse
tff(fact_8825_one__less__inverse,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(inverse_inverse_rat,A)) ) ) ).

% one_less_inverse
tff(fact_8826_one__less__inverse__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(inverse_inverse_real,X))
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,X),one_one_real) ) ) ).

% one_less_inverse_iff
tff(fact_8827_one__less__inverse__iff,axiom,
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(inverse_inverse_rat,X))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),X)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),one_one_rat) ) ) ).

% one_less_inverse_iff
tff(fact_8828_field__class_Ofield__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(times_times_real(aa_real_real(inverse_inverse_real,A)),A) = one_one_real ) ) ).

% field_class.field_inverse
tff(fact_8829_field__class_Ofield__inverse,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,A)),A) = one_one_complex ) ) ).

% field_class.field_inverse
tff(fact_8830_field__class_Ofield__inverse,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(times_times_rat(aa_rat_rat(inverse_inverse_rat,A)),A) = one_one_rat ) ) ).

% field_class.field_inverse
tff(fact_8831_division__ring__inverse__add,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( aa_real_real(plus_plus_real(aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(inverse_inverse_real,A)),aa_real_real(plus_plus_real(A),B))),aa_real_real(inverse_inverse_real,B)) ) ) ) ).

% division_ring_inverse_add
tff(fact_8832_division__ring__inverse__add,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(invers8013647133539491842omplex,B)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(plus_plus_complex(A),B))),aa_complex_complex(invers8013647133539491842omplex,B)) ) ) ) ).

% division_ring_inverse_add
tff(fact_8833_division__ring__inverse__add,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B)) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(plus_plus_rat(A),B))),aa_rat_rat(inverse_inverse_rat,B)) ) ) ) ).

% division_ring_inverse_add
tff(fact_8834_inverse__add,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( aa_real_real(plus_plus_real(aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(plus_plus_real(A),B)),aa_real_real(inverse_inverse_real,A))),aa_real_real(inverse_inverse_real,B)) ) ) ) ).

% inverse_add
tff(fact_8835_inverse__add,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(invers8013647133539491842omplex,B)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(A),B)),aa_complex_complex(invers8013647133539491842omplex,A))),aa_complex_complex(invers8013647133539491842omplex,B)) ) ) ) ).

% inverse_add
tff(fact_8836_inverse__add,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B)) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A),B)),aa_rat_rat(inverse_inverse_rat,A))),aa_rat_rat(inverse_inverse_rat,B)) ) ) ) ).

% inverse_add
tff(fact_8837_division__ring__inverse__diff,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( aa_real_real(minus_minus_real(aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(inverse_inverse_real,A)),aa_real_real(minus_minus_real(B),A))),aa_real_real(inverse_inverse_real,B)) ) ) ) ).

% division_ring_inverse_diff
tff(fact_8838_division__ring__inverse__diff,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(invers8013647133539491842omplex,B)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(minus_minus_complex(B),A))),aa_complex_complex(invers8013647133539491842omplex,B)) ) ) ) ).

% division_ring_inverse_diff
tff(fact_8839_division__ring__inverse__diff,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B)) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(minus_minus_rat(B),A))),aa_rat_rat(inverse_inverse_rat,B)) ) ) ) ).

% division_ring_inverse_diff
tff(fact_8840_nonzero__inverse__eq__divide,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(inverse_inverse_real,A) = aa_real_real(divide_divide_real(one_one_real),A) ) ) ).

% nonzero_inverse_eq_divide
tff(fact_8841_nonzero__inverse__eq__divide,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(invers8013647133539491842omplex,A) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),A) ) ) ).

% nonzero_inverse_eq_divide
tff(fact_8842_nonzero__inverse__eq__divide,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(inverse_inverse_rat,A) = aa_rat_rat(divide_divide_rat(one_one_rat),A) ) ) ).

% nonzero_inverse_eq_divide
tff(fact_8843_inverse__powr,axiom,
    ! [Y2: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y2)
     => ( aa_real_real(powr_real(aa_real_real(inverse_inverse_real,Y2)),A) = aa_real_real(inverse_inverse_real,aa_real_real(powr_real(Y2),A)) ) ) ).

% inverse_powr
tff(fact_8844_inverse__less__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(times_times_real(A),B))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B),A) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),B)),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ) ).

% inverse_less_iff
tff(fact_8845_inverse__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ) ).

% inverse_less_iff
tff(fact_8846_inverse__le__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(times_times_real(A),B))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),B)),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ) ).

% inverse_le_iff
tff(fact_8847_inverse__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A),B))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),B)),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ) ).

% inverse_le_iff
tff(fact_8848_one__le__inverse,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(inverse_inverse_real,A)) ) ) ).

% one_le_inverse
tff(fact_8849_one__le__inverse,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(inverse_inverse_rat,A)) ) ) ).

% one_le_inverse
tff(fact_8850_inverse__less__1__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,X)),one_one_real)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),zero_zero_real)
        | aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X) ) ) ).

% inverse_less_1_iff
tff(fact_8851_inverse__less__1__iff,axiom,
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,X)),one_one_rat)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),zero_zero_rat)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),X) ) ) ).

% inverse_less_1_iff
tff(fact_8852_one__le__inverse__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(inverse_inverse_real,X))
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),one_one_real) ) ) ).

% one_le_inverse_iff
tff(fact_8853_one__le__inverse__iff,axiom,
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(inverse_inverse_rat,X))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),X)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),one_one_rat) ) ) ).

% one_le_inverse_iff
tff(fact_8854_inverse__diff__inverse,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( aa_real_real(minus_minus_real(aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)) = aa_real_real(uminus_uminus_real,aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(inverse_inverse_real,A)),aa_real_real(minus_minus_real(A),B))),aa_real_real(inverse_inverse_real,B))) ) ) ) ).

% inverse_diff_inverse
tff(fact_8855_inverse__diff__inverse,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(invers8013647133539491842omplex,B)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(minus_minus_complex(A),B))),aa_complex_complex(invers8013647133539491842omplex,B))) ) ) ) ).

% inverse_diff_inverse
tff(fact_8856_inverse__diff__inverse,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(minus_minus_rat(A),B))),aa_rat_rat(inverse_inverse_rat,B))) ) ) ) ).

% inverse_diff_inverse
tff(fact_8857_reals__Archimedean,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N)))),X) ) ).

% reals_Archimedean
tff(fact_8858_reals__Archimedean,axiom,
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),X)
     => ? [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N)))),X) ) ).

% reals_Archimedean
tff(fact_8859_forall__pos__mono__1,axiom,
    ! [P: fun_real_o,E: real] :
      ( ! [D2: real,E2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,D2),E2)
         => ( aa_real_o(P,D2)
           => aa_real_o(P,E2) ) )
     => ( ! [N: nat] : aa_real_o(P,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
         => aa_real_o(P,E) ) ) ) ).

% forall_pos_mono_1
tff(fact_8860_forall__pos__mono,axiom,
    ! [P: fun_real_o,E: real] :
      ( ! [D2: real,E2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,D2),E2)
         => ( aa_real_o(P,D2)
           => aa_real_o(P,E2) ) )
     => ( ! [N: nat] :
            ( ( N != zero_zero_nat )
           => aa_real_o(P,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N))) )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
         => aa_real_o(P,E) ) ) ) ).

% forall_pos_mono
tff(fact_8861_real__arch__inverse,axiom,
    ! [E: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
    <=> ? [N4: nat] :
          ( ( N4 != zero_zero_nat )
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N4)))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N4))),E) ) ) ).

% real_arch_inverse
tff(fact_8862_even__xor__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se3222712562003087583nteger(A,B))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),B) ) ) ).

% even_xor_iff
tff(fact_8863_even__xor__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,A),B))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B) ) ) ).

% even_xor_iff
tff(fact_8864_even__xor__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,A),B))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B) ) ) ).

% even_xor_iff
tff(fact_8865_sqrt__divide__self__eq,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_real(divide_divide_real(aa_real_real(sqrt,X)),X) = aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X)) ) ) ).

% sqrt_divide_self_eq
tff(fact_8866_summable__exp,axiom,
    ! [X: complex] : summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_ii,X)) ).

% summable_exp
tff(fact_8867_summable__exp,axiom,
    ! [X: real] : summable_real(aa_real_fun_nat_real(aTP_Lamm_ij,X)) ).

% summable_exp
tff(fact_8868_ex__inverse__of__nat__less,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ? [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N))),X) ) ) ).

% ex_inverse_of_nat_less
tff(fact_8869_ex__inverse__of__nat__less,axiom,
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),X)
     => ? [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,aa_nat_rat(semiri681578069525770553at_rat,N))),X) ) ) ).

% ex_inverse_of_nat_less
tff(fact_8870_power__diff__conv__inverse,axiom,
    ! [X: real,Ma: nat,Nb: nat] :
      ( ( X != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Ma)) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(inverse_inverse_real,X)),Ma)) ) ) ) ).

% power_diff_conv_inverse
tff(fact_8871_power__diff__conv__inverse,axiom,
    ! [X: complex,Ma: nat,Nb: nat] :
      ( ( X != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
       => ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,X),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Ma)) = aa_complex_complex(times_times_complex(aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,X),Nb)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(invers8013647133539491842omplex,X)),Ma)) ) ) ) ).

% power_diff_conv_inverse
tff(fact_8872_power__diff__conv__inverse,axiom,
    ! [X: rat,Ma: nat,Nb: nat] :
      ( ( X != zero_zero_rat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
       => ( aa_nat_rat(power_power_rat(X),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Ma)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(X),Nb)),aa_nat_rat(power_power_rat(aa_rat_rat(inverse_inverse_rat,X)),Ma)) ) ) ) ).

% power_diff_conv_inverse
tff(fact_8873_log__inverse,axiom,
    ! [A: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
         => ( aa_real_real(log(A),aa_real_real(inverse_inverse_real,X)) = aa_real_real(uminus_uminus_real,aa_real_real(log(A),X)) ) ) ) ) ).

% log_inverse
tff(fact_8874_exp__plus__inverse__exp,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(plus_plus_real(aa_real_real(exp_real,X)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,X)))) ).

% exp_plus_inverse_exp
tff(fact_8875_plus__inverse__ge__2,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(plus_plus_real(X),aa_real_real(inverse_inverse_real,X))) ) ).

% plus_inverse_ge_2
tff(fact_8876_real__inv__sqrt__pow2,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(inverse_inverse_real,X) ) ) ).

% real_inv_sqrt_pow2
tff(fact_8877_tan__cot,axiom,
    ! [X: real] : aa_real_real(tan_real,aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),X)) = aa_real_real(inverse_inverse_real,aa_real_real(tan_real,X)) ).

% tan_cot
tff(fact_8878_xor__nat__unfold,axiom,
    ! [Ma: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Ma),Nb) = $ite(
        Ma = zero_zero_nat,
        Nb,
        $ite(Nb = zero_zero_nat,Ma,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) ).

% xor_nat_unfold
tff(fact_8879_real__le__x__sinh,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(exp_real,X)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,X)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ).

% real_le_x_sinh
tff(fact_8880_xor__nat__rec,axiom,
    ! [Ma: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Ma),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
          aa_o_nat(zero_n2687167440665602831ol_nat,
            ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma) != ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% xor_nat_rec
tff(fact_8881_xor__one__eq,axiom,
    ! [A: code_integer] : bit_se3222712562003087583nteger(A,one_one_Code_integer) = minus_8373710615458151222nteger(plus_p5714425477246183910nteger(A,zero_n356916108424825756nteger(aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))),zero_n356916108424825756nteger(~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))) ).

% xor_one_eq
tff(fact_8882_xor__one__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,A),one_one_nat) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)))),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))) ).

% xor_one_eq
tff(fact_8883_xor__one__eq,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,A),one_one_int) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_o_int(zero_n2684676970156552555ol_int,aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)))),aa_o_int(zero_n2684676970156552555ol_int,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))) ).

% xor_one_eq
tff(fact_8884_one__xor__eq,axiom,
    ! [A: code_integer] : bit_se3222712562003087583nteger(one_one_Code_integer,A) = minus_8373710615458151222nteger(plus_p5714425477246183910nteger(A,zero_n356916108424825756nteger(aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))),zero_n356916108424825756nteger(~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))) ).

% one_xor_eq
tff(fact_8885_one__xor__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,one_one_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)))),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))) ).

% one_xor_eq
tff(fact_8886_one__xor__eq,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,one_one_int),A) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_o_int(zero_n2684676970156552555ol_int,aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)))),aa_o_int(zero_n2684676970156552555ol_int,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))) ).

% one_xor_eq
tff(fact_8887_real__le__abs__sinh,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X)),aa_real_real(abs_abs_real,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(exp_real,X)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,X)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% real_le_abs_sinh
tff(fact_8888_tan__sec,axiom,
    ! [X: real] :
      ( ( aa_real_real(cos_real,X) != zero_zero_real )
     => ( aa_real_real(plus_plus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(tan_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(inverse_inverse_real,aa_real_real(cos_real,X))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% tan_sec
tff(fact_8889_tan__sec,axiom,
    ! [X: complex] :
      ( ( aa_complex_complex(cos_complex,X) != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(one_one_complex),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(tan_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(cos_complex,X))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% tan_sec
tff(fact_8890_case__prod__Pair__iden,axiom,
    ! [P2: produc6271795597528267376eger_o] : produc8281089916252573759eger_o(produc6677183202524767010eger_o,P2) = P2 ).

% case_prod_Pair_iden
tff(fact_8891_case__prod__Pair__iden,axiom,
    ! [P2: product_prod_num_num] : produc64540874165560627um_num(product_Pair_num_num,P2) = P2 ).

% case_prod_Pair_iden
tff(fact_8892_case__prod__Pair__iden,axiom,
    ! [P2: product_prod_nat_num] : produc49306077274653107at_num(product_Pair_nat_num,P2) = P2 ).

% case_prod_Pair_iden
tff(fact_8893_case__prod__Pair__iden,axiom,
    ! [P2: product_prod_nat_nat] : aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(product_Pair_nat_nat),P2) = P2 ).

% case_prod_Pair_iden
tff(fact_8894_case__prod__Pair__iden,axiom,
    ! [P2: product_prod_int_int] : aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(product_Pair_int_int),P2) = P2 ).

% case_prod_Pair_iden
tff(fact_8895_xor__Suc__0__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% xor_Suc_0_eq
tff(fact_8896_exp__first__two__terms,axiom,
    ! [X: real] : aa_real_real(exp_real,X) = aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(one_one_real),X)),suminf_real(aa_real_fun_nat_real(aTP_Lamm_ik,X))) ).

% exp_first_two_terms
tff(fact_8897_exp__first__two__terms,axiom,
    ! [X: complex] : aa_complex_complex(exp_complex,X) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(one_one_complex),X)),suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_il,X))) ).

% exp_first_two_terms
tff(fact_8898_Arg__def,axiom,
    ! [Z2: complex] :
      aa_complex_real(arg,Z2) = $ite(Z2 = zero_zero_complex,zero_zero_real,fChoice_real(aTP_Lamm_im(Z2))) ).

% Arg_def
tff(fact_8899_sinh__ln__real,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_real(sinh_real,aa_real_real(ln_ln_real,X)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(X),aa_real_real(inverse_inverse_real,X))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% sinh_ln_real
tff(fact_8900_cosh__ln__real,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_real_real(cosh_real,aa_real_real(ln_ln_real,X)) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(X),aa_real_real(inverse_inverse_real,X))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% cosh_ln_real
tff(fact_8901_mult__scaleR__left,axiom,
    ! [A: real,X: real,Y2: real] : aa_real_real(times_times_real(aa_real_real(real_V1485227260804924795R_real(A),X)),Y2) = aa_real_real(real_V1485227260804924795R_real(A),aa_real_real(times_times_real(X),Y2)) ).

% mult_scaleR_left
tff(fact_8902_mult__scaleR__left,axiom,
    ! [A: real,X: complex,Y2: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(real_V2046097035970521341omplex(A),X)),Y2) = aa_complex_complex(real_V2046097035970521341omplex(A),aa_complex_complex(times_times_complex(X),Y2)) ).

% mult_scaleR_left
tff(fact_8903_mult__scaleR__right,axiom,
    ! [X: real,A: real,Y2: real] : aa_real_real(times_times_real(X),aa_real_real(real_V1485227260804924795R_real(A),Y2)) = aa_real_real(real_V1485227260804924795R_real(A),aa_real_real(times_times_real(X),Y2)) ).

% mult_scaleR_right
tff(fact_8904_mult__scaleR__right,axiom,
    ! [X: complex,A: real,Y2: complex] : aa_complex_complex(times_times_complex(X),aa_complex_complex(real_V2046097035970521341omplex(A),Y2)) = aa_complex_complex(real_V2046097035970521341omplex(A),aa_complex_complex(times_times_complex(X),Y2)) ).

% mult_scaleR_right
tff(fact_8905_scaleR__one,axiom,
    ! [X: real] : aa_real_real(real_V1485227260804924795R_real(one_one_real),X) = X ).

% scaleR_one
tff(fact_8906_scaleR__one,axiom,
    ! [X: complex] : aa_complex_complex(real_V2046097035970521341omplex(one_one_real),X) = X ).

% scaleR_one
tff(fact_8907_scaleR__scaleR,axiom,
    ! [A: real,B: real,X: real] : aa_real_real(real_V1485227260804924795R_real(A),aa_real_real(real_V1485227260804924795R_real(B),X)) = aa_real_real(real_V1485227260804924795R_real(aa_real_real(times_times_real(A),B)),X) ).

% scaleR_scaleR
tff(fact_8908_scaleR__scaleR,axiom,
    ! [A: real,B: real,X: complex] : aa_complex_complex(real_V2046097035970521341omplex(A),aa_complex_complex(real_V2046097035970521341omplex(B),X)) = aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(times_times_real(A),B)),X) ).

% scaleR_scaleR
tff(fact_8909_sinh__real__le__iff,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sinh_real,X)),aa_real_real(sinh_real,Y2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y2) ) ).

% sinh_real_le_iff
tff(fact_8910_scaleR__eq__iff,axiom,
    ! [B: real,U: real,A: real] :
      ( ( aa_real_real(plus_plus_real(B),aa_real_real(real_V1485227260804924795R_real(U),A)) = aa_real_real(plus_plus_real(A),aa_real_real(real_V1485227260804924795R_real(U),B)) )
    <=> ( ( A = B )
        | ( U = one_one_real ) ) ) ).

% scaleR_eq_iff
tff(fact_8911_scaleR__eq__iff,axiom,
    ! [B: complex,U: real,A: complex] :
      ( ( aa_complex_complex(plus_plus_complex(B),aa_complex_complex(real_V2046097035970521341omplex(U),A)) = aa_complex_complex(plus_plus_complex(A),aa_complex_complex(real_V2046097035970521341omplex(U),B)) )
    <=> ( ( A = B )
        | ( U = one_one_real ) ) ) ).

% scaleR_eq_iff
tff(fact_8912_cosh__0,axiom,
    aa_complex_complex(cosh_complex,zero_zero_complex) = one_one_complex ).

% cosh_0
tff(fact_8913_cosh__0,axiom,
    aa_real_real(cosh_real,zero_zero_real) = one_one_real ).

% cosh_0
tff(fact_8914_scaleR__power,axiom,
    ! [X: real,Y2: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(real_V1485227260804924795R_real(X),Y2)),Nb) = aa_real_real(real_V1485227260804924795R_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),Nb)) ).

% scaleR_power
tff(fact_8915_scaleR__power,axiom,
    ! [X: real,Y2: complex,Nb: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(real_V2046097035970521341omplex(X),Y2)),Nb) = aa_complex_complex(real_V2046097035970521341omplex(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Nb)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Y2),Nb)) ).

% scaleR_power
tff(fact_8916_sinh__real__nonneg__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sinh_real,X))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X) ) ).

% sinh_real_nonneg_iff
tff(fact_8917_sinh__real__nonpos__iff,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sinh_real,X)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),zero_zero_real) ) ).

% sinh_real_nonpos_iff
tff(fact_8918_xor__nonnegative__int__iff,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K2),L))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) ) ) ).

% xor_nonnegative_int_iff
tff(fact_8919_xor__negative__int__iff,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K2),L)),zero_zero_int)
    <=> ~ ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int)
        <=> aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ) ).

% xor_negative_int_iff
tff(fact_8920_scaleR__minus1__left,axiom,
    ! [X: real] : aa_real_real(real_V1485227260804924795R_real(aa_real_real(uminus_uminus_real,one_one_real)),X) = aa_real_real(uminus_uminus_real,X) ).

% scaleR_minus1_left
tff(fact_8921_scaleR__minus1__left,axiom,
    ! [X: complex] : aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(uminus_uminus_real,one_one_real)),X) = aa_complex_complex(uminus1482373934393186551omplex,X) ).

% scaleR_minus1_left
tff(fact_8922_scaleR__collapse,axiom,
    ! [U: real,A: real] : aa_real_real(plus_plus_real(aa_real_real(real_V1485227260804924795R_real(aa_real_real(minus_minus_real(one_one_real),U)),A)),aa_real_real(real_V1485227260804924795R_real(U),A)) = A ).

% scaleR_collapse
tff(fact_8923_scaleR__collapse,axiom,
    ! [U: real,A: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(minus_minus_real(one_one_real),U)),A)),aa_complex_complex(real_V2046097035970521341omplex(U),A)) = A ).

% scaleR_collapse
tff(fact_8924_norm__scaleR,axiom,
    ! [A: real,X: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(real_V1485227260804924795R_real(A),X)) = aa_real_real(times_times_real(aa_real_real(abs_abs_real,A)),aa_real_real(real_V7735802525324610683m_real,X)) ).

% norm_scaleR
tff(fact_8925_norm__scaleR,axiom,
    ! [A: real,X: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(real_V2046097035970521341omplex(A),X)) = aa_real_real(times_times_real(aa_real_real(abs_abs_real,A)),aa_complex_real(real_V1022390504157884413omplex,X)) ).

% norm_scaleR
tff(fact_8926_scaleR__times,axiom,
    ! [U: num,W: num,A: real] : aa_real_real(real_V1485227260804924795R_real(aa_num_real(numeral_numeral_real,U)),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,W)),A)) = aa_real_real(real_V1485227260804924795R_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,U)),aa_num_real(numeral_numeral_real,W))),A) ).

% scaleR_times
tff(fact_8927_scaleR__times,axiom,
    ! [U: num,W: num,A: complex] : aa_complex_complex(real_V2046097035970521341omplex(aa_num_real(numeral_numeral_real,U)),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),A)) = aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,U)),aa_num_real(numeral_numeral_real,W))),A) ).

% scaleR_times
tff(fact_8928_inverse__scaleR__times,axiom,
    ! [V: num,W: num,A: real] : aa_real_real(real_V1485227260804924795R_real(aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,V))),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,W)),A)) = aa_real_real(real_V1485227260804924795R_real(aa_real_real(divide_divide_real(aa_num_real(numeral_numeral_real,W)),aa_num_real(numeral_numeral_real,V))),A) ).

% inverse_scaleR_times
tff(fact_8929_inverse__scaleR__times,axiom,
    ! [V: num,W: num,A: complex] : aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,V))),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),A)) = aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(divide_divide_real(aa_num_real(numeral_numeral_real,W)),aa_num_real(numeral_numeral_real,V))),A) ).

% inverse_scaleR_times
tff(fact_8930_fraction__scaleR__times,axiom,
    ! [U: num,V: num,W: num,A: real] : aa_real_real(real_V1485227260804924795R_real(aa_real_real(divide_divide_real(aa_num_real(numeral_numeral_real,U)),aa_num_real(numeral_numeral_real,V))),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,W)),A)) = aa_real_real(real_V1485227260804924795R_real(aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,U)),aa_num_real(numeral_numeral_real,W))),aa_num_real(numeral_numeral_real,V))),A) ).

% fraction_scaleR_times
tff(fact_8931_fraction__scaleR__times,axiom,
    ! [U: num,V: num,W: num,A: complex] : aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(divide_divide_real(aa_num_real(numeral_numeral_real,U)),aa_num_real(numeral_numeral_real,V))),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),A)) = aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,U)),aa_num_real(numeral_numeral_real,W))),aa_num_real(numeral_numeral_real,V))),A) ).

% fraction_scaleR_times
tff(fact_8932_scaleR__half__double,axiom,
    ! [A: real] : aa_real_real(real_V1485227260804924795R_real(aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_real_real(plus_plus_real(A),A)) = A ).

% scaleR_half_double
tff(fact_8933_scaleR__half__double,axiom,
    ! [A: complex] : aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_complex_complex(plus_plus_complex(A),A)) = A ).

% scaleR_half_double
tff(fact_8934_sinh__le__cosh__real,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sinh_real,X)),aa_real_real(cosh_real,X)) ).

% sinh_le_cosh_real
tff(fact_8935_bit__xor__int__iff,axiom,
    ! [K2: int,L: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K2),L)),Nb)
    <=> ~ ( aa_nat_o(bit_se1146084159140164899it_int(K2),Nb)
        <=> aa_nat_o(bit_se1146084159140164899it_int(L),Nb) ) ) ).

% bit_xor_int_iff
tff(fact_8936_cosh__plus__sinh,axiom,
    ! [X: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(cosh_complex,X)),aa_complex_complex(sinh_complex,X)) = aa_complex_complex(exp_complex,X) ).

% cosh_plus_sinh
tff(fact_8937_cosh__plus__sinh,axiom,
    ! [X: real] : aa_real_real(plus_plus_real(aa_real_real(cosh_real,X)),aa_real_real(sinh_real,X)) = aa_real_real(exp_real,X) ).

% cosh_plus_sinh
tff(fact_8938_sinh__plus__cosh,axiom,
    ! [X: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(sinh_complex,X)),aa_complex_complex(cosh_complex,X)) = aa_complex_complex(exp_complex,X) ).

% sinh_plus_cosh
tff(fact_8939_sinh__plus__cosh,axiom,
    ! [X: real] : aa_real_real(plus_plus_real(aa_real_real(sinh_real,X)),aa_real_real(cosh_real,X)) = aa_real_real(exp_real,X) ).

% sinh_plus_cosh
tff(fact_8940_verit__sko__ex_H,axiom,
    ! [P: fun_real_o,A2: $o] :
      ( ( aa_real_o(P,fChoice_real(P))
      <=> (A2) )
     => ( ? [X_12: real] : aa_real_o(P,X_12)
      <=> (A2) ) ) ).

% verit_sko_ex'
tff(fact_8941_verit__sko__forall,axiom,
    ! [P: fun_real_o] :
      ( ! [X_12: real] : aa_real_o(P,X_12)
    <=> aa_real_o(P,fChoice_real(aTP_Lamm_in(P))) ) ).

% verit_sko_forall
tff(fact_8942_verit__sko__forall_H,axiom,
    ! [P: fun_real_o,A2: $o] :
      ( ( aa_real_o(P,fChoice_real(aTP_Lamm_in(P)))
      <=> (A2) )
     => ( ! [X_12: real] : aa_real_o(P,X_12)
      <=> (A2) ) ) ).

% verit_sko_forall'
tff(fact_8943_verit__sko__forall_H_H,axiom,
    ! [B3: real,A2: real,P: fun_real_o] :
      ( ( B3 = A2 )
     => ( ( fChoice_real(P) = A2 )
      <=> ( fChoice_real(P) = B3 ) ) ) ).

% verit_sko_forall''
tff(fact_8944_verit__sko__ex__indirect,axiom,
    ! [X: real,P: fun_real_o] :
      ( ( X = fChoice_real(P) )
     => ( ? [X_12: real] : aa_real_o(P,X_12)
      <=> aa_real_o(P,X) ) ) ).

% verit_sko_ex_indirect
tff(fact_8945_verit__sko__ex__indirect2,axiom,
    ! [X: real,P: fun_real_o,P3: fun_real_o] :
      ( ( X = fChoice_real(P) )
     => ( ! [X4: real] :
            ( aa_real_o(P,X4)
          <=> aa_real_o(P3,X4) )
       => ( ? [X_12: real] : aa_real_o(P3,X_12)
        <=> aa_real_o(P,X) ) ) ) ).

% verit_sko_ex_indirect2
tff(fact_8946_verit__sko__forall__indirect,axiom,
    ! [X: real,P: fun_real_o] :
      ( ( X = fChoice_real(aTP_Lamm_in(P)) )
     => ( ! [X_12: real] : aa_real_o(P,X_12)
      <=> aa_real_o(P,X) ) ) ).

% verit_sko_forall_indirect
tff(fact_8947_verit__sko__forall__indirect2,axiom,
    ! [X: real,P: fun_real_o,P3: fun_real_o] :
      ( ( X = fChoice_real(aTP_Lamm_in(P)) )
     => ( ! [X4: real] :
            ( aa_real_o(P,X4)
          <=> aa_real_o(P3,X4) )
       => ( ! [X_12: real] : aa_real_o(P3,X_12)
        <=> aa_real_o(P,X) ) ) ) ).

% verit_sko_forall_indirect2
tff(fact_8948_scaleR__right__distrib,axiom,
    ! [A: real,X: real,Y2: real] : aa_real_real(real_V1485227260804924795R_real(A),aa_real_real(plus_plus_real(X),Y2)) = aa_real_real(plus_plus_real(aa_real_real(real_V1485227260804924795R_real(A),X)),aa_real_real(real_V1485227260804924795R_real(A),Y2)) ).

% scaleR_right_distrib
tff(fact_8949_scaleR__right__distrib,axiom,
    ! [A: real,X: complex,Y2: complex] : aa_complex_complex(real_V2046097035970521341omplex(A),aa_complex_complex(plus_plus_complex(X),Y2)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(real_V2046097035970521341omplex(A),X)),aa_complex_complex(real_V2046097035970521341omplex(A),Y2)) ).

% scaleR_right_distrib
tff(fact_8950_sinh__diff,axiom,
    ! [X: complex,Y2: complex] : aa_complex_complex(sinh_complex,aa_complex_complex(minus_minus_complex(X),Y2)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(aa_complex_complex(sinh_complex,X)),aa_complex_complex(cosh_complex,Y2))),aa_complex_complex(times_times_complex(aa_complex_complex(cosh_complex,X)),aa_complex_complex(sinh_complex,Y2))) ).

% sinh_diff
tff(fact_8951_sinh__diff,axiom,
    ! [X: real,Y2: real] : aa_real_real(sinh_real,aa_real_real(minus_minus_real(X),Y2)) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(aa_real_real(sinh_real,X)),aa_real_real(cosh_real,Y2))),aa_real_real(times_times_real(aa_real_real(cosh_real,X)),aa_real_real(sinh_real,Y2))) ).

% sinh_diff
tff(fact_8952_cosh__diff,axiom,
    ! [X: complex,Y2: complex] : aa_complex_complex(cosh_complex,aa_complex_complex(minus_minus_complex(X),Y2)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(aa_complex_complex(cosh_complex,X)),aa_complex_complex(cosh_complex,Y2))),aa_complex_complex(times_times_complex(aa_complex_complex(sinh_complex,X)),aa_complex_complex(sinh_complex,Y2))) ).

% cosh_diff
tff(fact_8953_cosh__diff,axiom,
    ! [X: real,Y2: real] : aa_real_real(cosh_real,aa_real_real(minus_minus_real(X),Y2)) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(aa_real_real(cosh_real,X)),aa_real_real(cosh_real,Y2))),aa_real_real(times_times_real(aa_real_real(sinh_real,X)),aa_real_real(sinh_real,Y2))) ).

% cosh_diff
tff(fact_8954_sinh__add,axiom,
    ! [X: real,Y2: real] : aa_real_real(sinh_real,aa_real_real(plus_plus_real(X),Y2)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(sinh_real,X)),aa_real_real(cosh_real,Y2))),aa_real_real(times_times_real(aa_real_real(cosh_real,X)),aa_real_real(sinh_real,Y2))) ).

% sinh_add
tff(fact_8955_cosh__add,axiom,
    ! [X: real,Y2: real] : aa_real_real(cosh_real,aa_real_real(plus_plus_real(X),Y2)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(cosh_real,X)),aa_real_real(cosh_real,Y2))),aa_real_real(times_times_real(aa_real_real(sinh_real,X)),aa_real_real(sinh_real,Y2))) ).

% cosh_add
tff(fact_8956_real__scaleR__def,axiom,
    ! [A: real,X: real] : aa_real_real(real_V1485227260804924795R_real(A),X) = aa_real_real(times_times_real(A),X) ).

% real_scaleR_def
tff(fact_8957_tanh__def,axiom,
    ! [X: complex] : aa_complex_complex(tanh_complex,X) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(sinh_complex,X)),aa_complex_complex(cosh_complex,X)) ).

% tanh_def
tff(fact_8958_tanh__def,axiom,
    ! [X: real] : aa_real_real(tanh_real,X) = aa_real_real(divide_divide_real(aa_real_real(sinh_real,X)),aa_real_real(cosh_real,X)) ).

% tanh_def
tff(fact_8959_summable__scaleR__right,axiom,
    ! [X7: fun_nat_real,R: real] :
      ( summable_real(X7)
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_io(X7),R)) ) ).

% summable_scaleR_right
tff(fact_8960_summable__scaleR__right,axiom,
    ! [X7: fun_nat_complex,R: real] :
      ( summable_complex(X7)
     => summable_complex(aa_rea1531280491672885477omplex(aTP_Lamm_ip(X7),R)) ) ).

% summable_scaleR_right
tff(fact_8961_sums__scaleR__right,axiom,
    ! [X7: fun_nat_real,A: real,R: real] :
      ( aa_real_o(sums_real(X7),A)
     => aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_io(X7),R)),aa_real_real(real_V1485227260804924795R_real(R),A)) ) ).

% sums_scaleR_right
tff(fact_8962_sums__scaleR__right,axiom,
    ! [X7: fun_nat_complex,A: complex,R: real] :
      ( aa_complex_o(sums_complex(X7),A)
     => aa_complex_o(sums_complex(aa_rea1531280491672885477omplex(aTP_Lamm_ip(X7),R)),aa_complex_complex(real_V2046097035970521341omplex(R),A)) ) ).

% sums_scaleR_right
tff(fact_8963_XOR__lower,axiom,
    ! [X: int,Y2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,X),Y2)) ) ) ).

% XOR_lower
tff(fact_8964_scaleR__left_Oadd,axiom,
    ! [X: real,Y2: real,Xa: real] : aa_real_real(real_V1485227260804924795R_real(aa_real_real(plus_plus_real(X),Y2)),Xa) = aa_real_real(plus_plus_real(aa_real_real(real_V1485227260804924795R_real(X),Xa)),aa_real_real(real_V1485227260804924795R_real(Y2),Xa)) ).

% scaleR_left.add
tff(fact_8965_scaleR__left_Oadd,axiom,
    ! [X: real,Y2: real,Xa: complex] : aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(plus_plus_real(X),Y2)),Xa) = aa_complex_complex(plus_plus_complex(aa_complex_complex(real_V2046097035970521341omplex(X),Xa)),aa_complex_complex(real_V2046097035970521341omplex(Y2),Xa)) ).

% scaleR_left.add
tff(fact_8966_scaleR__left__distrib,axiom,
    ! [A: real,B: real,X: real] : aa_real_real(real_V1485227260804924795R_real(aa_real_real(plus_plus_real(A),B)),X) = aa_real_real(plus_plus_real(aa_real_real(real_V1485227260804924795R_real(A),X)),aa_real_real(real_V1485227260804924795R_real(B),X)) ).

% scaleR_left_distrib
tff(fact_8967_scaleR__left__distrib,axiom,
    ! [A: real,B: real,X: complex] : aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(plus_plus_real(A),B)),X) = aa_complex_complex(plus_plus_complex(aa_complex_complex(real_V2046097035970521341omplex(A),X)),aa_complex_complex(real_V2046097035970521341omplex(B),X)) ).

% scaleR_left_distrib
tff(fact_8968_cosh__real__nonneg,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(cosh_real,X)) ).

% cosh_real_nonneg
tff(fact_8969_cosh__real__nonneg__le__iff,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cosh_real,X)),aa_real_real(cosh_real,Y2))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y2) ) ) ) ).

% cosh_real_nonneg_le_iff
tff(fact_8970_cosh__real__nonpos__le__iff,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cosh_real,X)),aa_real_real(cosh_real,Y2))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),X) ) ) ) ).

% cosh_real_nonpos_le_iff
tff(fact_8971_scaleR__conv__of__real,axiom,
    ! [R: real,X: real] : aa_real_real(real_V1485227260804924795R_real(R),X) = aa_real_real(times_times_real(aa_real_real(real_V1803761363581548252l_real,R)),X) ).

% scaleR_conv_of_real
tff(fact_8972_scaleR__conv__of__real,axiom,
    ! [R: real,X: complex] : aa_complex_complex(real_V2046097035970521341omplex(R),X) = aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,R)),X) ).

% scaleR_conv_of_real
tff(fact_8973_of__real__def,axiom,
    ! [R: real] : aa_real_real(real_V1803761363581548252l_real,R) = aa_real_real(real_V1485227260804924795R_real(R),one_one_real) ).

% of_real_def
tff(fact_8974_of__real__def,axiom,
    ! [R: real] : aa_real_complex(real_V4546457046886955230omplex,R) = aa_complex_complex(real_V2046097035970521341omplex(R),one_one_complex) ).

% of_real_def
tff(fact_8975_cosh__real__ge__1,axiom,
    ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(cosh_real,X)) ).

% cosh_real_ge_1
tff(fact_8976_sinh__double,axiom,
    ! [X: complex] : aa_complex_complex(sinh_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),X)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(sinh_complex,X))),aa_complex_complex(cosh_complex,X)) ).

% sinh_double
tff(fact_8977_sinh__double,axiom,
    ! [X: real] : aa_real_real(sinh_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X)) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(sinh_real,X))),aa_real_real(cosh_real,X)) ).

% sinh_double
tff(fact_8978_complex__scaleR,axiom,
    ! [R: real,A: real,B: real] : aa_complex_complex(real_V2046097035970521341omplex(R),aa_real_complex(complex2(A),B)) = aa_real_complex(complex2(aa_real_real(times_times_real(R),A)),aa_real_real(times_times_real(R),B)) ).

% complex_scaleR
tff(fact_8979_divide__complex__def,axiom,
    ! [X: complex,Y2: complex] : aa_complex_complex(divide1717551699836669952omplex(X),Y2) = aa_complex_complex(times_times_complex(X),aa_complex_complex(invers8013647133539491842omplex,Y2)) ).

% divide_complex_def
tff(fact_8980_suminf__scaleR__right,axiom,
    ! [X7: fun_nat_real,R: real] :
      ( summable_real(X7)
     => ( aa_real_real(real_V1485227260804924795R_real(R),suminf_real(X7)) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_io(X7),R)) ) ) ).

% suminf_scaleR_right
tff(fact_8981_suminf__scaleR__right,axiom,
    ! [X7: fun_nat_complex,R: real] :
      ( summable_complex(X7)
     => ( aa_complex_complex(real_V2046097035970521341omplex(R),suminf_complex(X7)) = suminf_complex(aa_rea1531280491672885477omplex(aTP_Lamm_ip(X7),R)) ) ) ).

% suminf_scaleR_right
tff(fact_8982_summable__scaleR__left,axiom,
    ! [X7: fun_nat_real,X: real] :
      ( summable_real(X7)
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_iq(X7),X)) ) ).

% summable_scaleR_left
tff(fact_8983_summable__scaleR__left,axiom,
    ! [X7: fun_nat_real,X: complex] :
      ( summable_real(X7)
     => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_ir(X7),X)) ) ).

% summable_scaleR_left
tff(fact_8984_sums__scaleR__left,axiom,
    ! [X7: fun_nat_real,A: real,X: real] :
      ( aa_real_o(sums_real(X7),A)
     => aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_iq(X7),X)),aa_real_real(real_V1485227260804924795R_real(A),X)) ) ).

% sums_scaleR_left
tff(fact_8985_sums__scaleR__left,axiom,
    ! [X7: fun_nat_real,A: real,X: complex] :
      ( aa_real_o(sums_real(X7),A)
     => aa_complex_o(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_ir(X7),X)),aa_complex_complex(real_V2046097035970521341omplex(A),X)) ) ).

% sums_scaleR_left
tff(fact_8986_scaleR__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V1485227260804924795R_real(A),C)),aa_real_real(real_V1485227260804924795R_real(B),C)) ) ) ).

% scaleR_right_mono_neg
tff(fact_8987_scaleR__right__mono,axiom,
    ! [A: real,B: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V1485227260804924795R_real(A),X)),aa_real_real(real_V1485227260804924795R_real(B),X)) ) ) ).

% scaleR_right_mono
tff(fact_8988_scaleR__le__cancel__left__pos,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V1485227260804924795R_real(C),A)),aa_real_real(real_V1485227260804924795R_real(C),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% scaleR_le_cancel_left_pos
tff(fact_8989_scaleR__le__cancel__left__neg,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V1485227260804924795R_real(C),A)),aa_real_real(real_V1485227260804924795R_real(C),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ).

% scaleR_le_cancel_left_neg
tff(fact_8990_scaleR__le__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V1485227260804924795R_real(C),A)),aa_real_real(real_V1485227260804924795R_real(C),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ) ).

% scaleR_le_cancel_left
tff(fact_8991_scaleR__left__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V1485227260804924795R_real(C),A)),aa_real_real(real_V1485227260804924795R_real(C),B)) ) ) ).

% scaleR_left_mono_neg
tff(fact_8992_scaleR__left__mono,axiom,
    ! [X: real,Y2: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V1485227260804924795R_real(A),X)),aa_real_real(real_V1485227260804924795R_real(A),Y2)) ) ) ).

% scaleR_left_mono
tff(fact_8993_Real__Vector__Spaces_Ole__add__iff2,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_real_real(real_V1485227260804924795R_real(A),E)),C)),aa_real_real(plus_plus_real(aa_real_real(real_V1485227260804924795R_real(B),E)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),aa_real_real(plus_plus_real(aa_real_real(real_V1485227260804924795R_real(aa_real_real(minus_minus_real(B),A)),E)),D)) ) ).

% Real_Vector_Spaces.le_add_iff2
tff(fact_8994_Real__Vector__Spaces_Ole__add__iff1,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_real_real(real_V1485227260804924795R_real(A),E)),C)),aa_real_real(plus_plus_real(aa_real_real(real_V1485227260804924795R_real(B),E)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_real_real(real_V1485227260804924795R_real(aa_real_real(minus_minus_real(A),B)),E)),C)),D) ) ).

% Real_Vector_Spaces.le_add_iff1
tff(fact_8995_cosh__real__nonpos__less__iff,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y2),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cosh_real,X)),aa_real_real(cosh_real,Y2))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y2),X) ) ) ) ).

% cosh_real_nonpos_less_iff
tff(fact_8996_cosh__real__nonneg__less__iff,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cosh_real,X)),aa_real_real(cosh_real,Y2))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2) ) ) ) ).

% cosh_real_nonneg_less_iff
tff(fact_8997_cosh__real__strict__mono,axiom,
    ! [X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cosh_real,X)),aa_real_real(cosh_real,Y2)) ) ) ).

% cosh_real_strict_mono
tff(fact_8998_cosh__square__eq,axiom,
    ! [X: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cosh_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sinh_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real) ).

% cosh_square_eq
tff(fact_8999_cosh__square__eq,axiom,
    ! [X: complex] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(cosh_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(plus_plus_complex(aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(sinh_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_complex) ).

% cosh_square_eq
tff(fact_9000_sinh__square__eq,axiom,
    ! [X: complex] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(sinh_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(minus_minus_complex(aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(cosh_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_complex) ).

% sinh_square_eq
tff(fact_9001_sinh__square__eq,axiom,
    ! [X: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sinh_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cosh_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real) ).

% sinh_square_eq
tff(fact_9002_hyperbolic__pythagoras,axiom,
    ! [X: complex] : aa_complex_complex(minus_minus_complex(aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(cosh_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(sinh_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_complex ).

% hyperbolic_pythagoras
tff(fact_9003_hyperbolic__pythagoras,axiom,
    ! [X: real] : aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cosh_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sinh_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real ).

% hyperbolic_pythagoras
tff(fact_9004_xor__nat__def,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Ma),Nb) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_nat_int(semiri1314217659103216013at_int,Ma)),aa_nat_int(semiri1314217659103216013at_int,Nb))) ).

% xor_nat_def
tff(fact_9005_arcosh__cosh__real,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_real(arcosh_real,aa_real_real(cosh_real,X)) = X ) ) ).

% arcosh_cosh_real
tff(fact_9006_cosh__def,axiom,
    ! [X: real] : aa_real_real(cosh_real,X) = aa_real_real(real_V1485227260804924795R_real(aa_real_real(inverse_inverse_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_real_real(plus_plus_real(aa_real_real(exp_real,X)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,X)))) ).

% cosh_def
tff(fact_9007_cosh__def,axiom,
    ! [X: complex] : aa_complex_complex(cosh_complex,X) = aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(inverse_inverse_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_complex_complex(plus_plus_complex(aa_complex_complex(exp_complex,X)),aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,X)))) ).

% cosh_def
tff(fact_9008_sinh__def,axiom,
    ! [X: real] : aa_real_real(sinh_real,X) = aa_real_real(real_V1485227260804924795R_real(aa_real_real(inverse_inverse_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_real_real(minus_minus_real(aa_real_real(exp_real,X)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,X)))) ).

% sinh_def
tff(fact_9009_sinh__def,axiom,
    ! [X: complex] : aa_complex_complex(sinh_complex,X) = aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(inverse_inverse_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_complex_complex(minus_minus_complex(aa_complex_complex(exp_complex,X)),aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,X)))) ).

% sinh_def
tff(fact_9010_cosh__double,axiom,
    ! [X: complex] : aa_complex_complex(cosh_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),X)) = aa_complex_complex(plus_plus_complex(aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(cosh_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(sinh_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cosh_double
tff(fact_9011_cosh__double,axiom,
    ! [X: real] : aa_real_real(cosh_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X)) = aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cosh_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sinh_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cosh_double
tff(fact_9012_suminf__scaleR__left,axiom,
    ! [X7: fun_nat_real,X: real] :
      ( summable_real(X7)
     => ( aa_real_real(real_V1485227260804924795R_real(suminf_real(X7)),X) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_iq(X7),X)) ) ) ).

% suminf_scaleR_left
tff(fact_9013_suminf__scaleR__left,axiom,
    ! [X7: fun_nat_real,X: complex] :
      ( summable_real(X7)
     => ( aa_complex_complex(real_V2046097035970521341omplex(suminf_real(X7)),X) = suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_ir(X7),X)) ) ) ).

% suminf_scaleR_left
tff(fact_9014_zero__le__scaleR__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(real_V1485227260804924795R_real(A),B))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) )
        | ( A = zero_zero_real ) ) ) ).

% zero_le_scaleR_iff
tff(fact_9015_scaleR__le__0__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V1485227260804924795R_real(A),B)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) )
        | ( A = zero_zero_real ) ) ) ).

% scaleR_le_0_iff
tff(fact_9016_scaleR__mono,axiom,
    ! [A: real,B: real,X: real,Y2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V1485227260804924795R_real(A),X)),aa_real_real(real_V1485227260804924795R_real(B),Y2)) ) ) ) ) ).

% scaleR_mono
tff(fact_9017_scaleR__mono_H,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V1485227260804924795R_real(A),C)),aa_real_real(real_V1485227260804924795R_real(B),D)) ) ) ) ) ).

% scaleR_mono'
tff(fact_9018_split__scaleR__neg__le,axiom,
    ! [A: real,X: real] :
      ( ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V1485227260804924795R_real(A),X)),zero_zero_real) ) ).

% split_scaleR_neg_le
tff(fact_9019_split__scaleR__pos__le,axiom,
    ! [A: real,B: real] :
      ( ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(real_V1485227260804924795R_real(A),B)) ) ).

% split_scaleR_pos_le
tff(fact_9020_scaleR__nonneg__nonneg,axiom,
    ! [A: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(real_V1485227260804924795R_real(A),X)) ) ) ).

% scaleR_nonneg_nonneg
tff(fact_9021_scaleR__nonneg__nonpos,axiom,
    ! [A: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V1485227260804924795R_real(A),X)),zero_zero_real) ) ) ).

% scaleR_nonneg_nonpos
tff(fact_9022_scaleR__nonpos__nonneg,axiom,
    ! [A: real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V1485227260804924795R_real(A),X)),zero_zero_real) ) ) ).

% scaleR_nonpos_nonneg
tff(fact_9023_scaleR__nonpos__nonpos,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(real_V1485227260804924795R_real(A),B)) ) ) ).

% scaleR_nonpos_nonpos
tff(fact_9024_scaleR__left__le__one__le,axiom,
    ! [X: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V1485227260804924795R_real(A),X)),X) ) ) ).

% scaleR_left_le_one_le
tff(fact_9025_scaleR__2,axiom,
    ! [X: real] : aa_real_real(real_V1485227260804924795R_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X) = aa_real_real(plus_plus_real(X),X) ).

% scaleR_2
tff(fact_9026_scaleR__2,axiom,
    ! [X: complex] : aa_complex_complex(real_V2046097035970521341omplex(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),X) = aa_complex_complex(plus_plus_complex(X),X) ).

% scaleR_2
tff(fact_9027_real__vector__eq__affinity,axiom,
    ! [Ma: real,Y2: real,X: real,C: real] :
      ( ( Ma != zero_zero_real )
     => ( ( Y2 = aa_real_real(plus_plus_real(aa_real_real(real_V1485227260804924795R_real(Ma),X)),C) )
      <=> ( aa_real_real(minus_minus_real(aa_real_real(real_V1485227260804924795R_real(aa_real_real(inverse_inverse_real,Ma)),Y2)),aa_real_real(real_V1485227260804924795R_real(aa_real_real(inverse_inverse_real,Ma)),C)) = X ) ) ) ).

% real_vector_eq_affinity
tff(fact_9028_real__vector__eq__affinity,axiom,
    ! [Ma: real,Y2: complex,X: complex,C: complex] :
      ( ( Ma != zero_zero_real )
     => ( ( Y2 = aa_complex_complex(plus_plus_complex(aa_complex_complex(real_V2046097035970521341omplex(Ma),X)),C) )
      <=> ( aa_complex_complex(minus_minus_complex(aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(inverse_inverse_real,Ma)),Y2)),aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(inverse_inverse_real,Ma)),C)) = X ) ) ) ).

% real_vector_eq_affinity
tff(fact_9029_real__vector__affinity__eq,axiom,
    ! [Ma: real,X: real,C: real,Y2: real] :
      ( ( Ma != zero_zero_real )
     => ( ( aa_real_real(plus_plus_real(aa_real_real(real_V1485227260804924795R_real(Ma),X)),C) = Y2 )
      <=> ( X = aa_real_real(minus_minus_real(aa_real_real(real_V1485227260804924795R_real(aa_real_real(inverse_inverse_real,Ma)),Y2)),aa_real_real(real_V1485227260804924795R_real(aa_real_real(inverse_inverse_real,Ma)),C)) ) ) ) ).

% real_vector_affinity_eq
tff(fact_9030_real__vector__affinity__eq,axiom,
    ! [Ma: real,X: complex,C: complex,Y2: complex] :
      ( ( Ma != zero_zero_real )
     => ( ( aa_complex_complex(plus_plus_complex(aa_complex_complex(real_V2046097035970521341omplex(Ma),X)),C) = Y2 )
      <=> ( X = aa_complex_complex(minus_minus_complex(aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(inverse_inverse_real,Ma)),Y2)),aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(inverse_inverse_real,Ma)),C)) ) ) ) ).

% real_vector_affinity_eq
tff(fact_9031_pos__divideR__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V1485227260804924795R_real(aa_real_real(inverse_inverse_real,C)),B)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(real_V1485227260804924795R_real(C),A)) ) ) ).

% pos_divideR_le_eq
tff(fact_9032_pos__le__divideR__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(real_V1485227260804924795R_real(aa_real_real(inverse_inverse_real,C)),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V1485227260804924795R_real(C),A)),B) ) ) ).

% pos_le_divideR_eq
tff(fact_9033_neg__divideR__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V1485227260804924795R_real(aa_real_real(inverse_inverse_real,C)),B)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V1485227260804924795R_real(C),A)),B) ) ) ).

% neg_divideR_le_eq
tff(fact_9034_neg__le__divideR__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(real_V1485227260804924795R_real(aa_real_real(inverse_inverse_real,C)),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(real_V1485227260804924795R_real(C),A)) ) ) ).

% neg_le_divideR_eq
tff(fact_9035_pos__divideR__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V1485227260804924795R_real(aa_real_real(inverse_inverse_real,C)),B)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(real_V1485227260804924795R_real(C),A)) ) ) ).

% pos_divideR_less_eq
tff(fact_9036_pos__less__divideR__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(real_V1485227260804924795R_real(aa_real_real(inverse_inverse_real,C)),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V1485227260804924795R_real(C),A)),B) ) ) ).

% pos_less_divideR_eq
tff(fact_9037_neg__divideR__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V1485227260804924795R_real(aa_real_real(inverse_inverse_real,C)),B)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V1485227260804924795R_real(C),A)),B) ) ) ).

% neg_divideR_less_eq
tff(fact_9038_neg__less__divideR__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(real_V1485227260804924795R_real(aa_real_real(inverse_inverse_real,C)),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(real_V1485227260804924795R_real(C),A)) ) ) ).

% neg_less_divideR_eq
tff(fact_9039_summable__exp__generic,axiom,
    ! [X: real] : summable_real(aa_real_fun_nat_real(aTP_Lamm_is,X)) ).

% summable_exp_generic
tff(fact_9040_summable__exp__generic,axiom,
    ! [X: complex] : summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_it,X)) ).

% summable_exp_generic
tff(fact_9041_sin__converges,axiom,
    ! [X: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_iu,X)),aa_real_real(sin_real,X)) ).

% sin_converges
tff(fact_9042_sin__converges,axiom,
    ! [X: complex] : aa_complex_o(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_iv,X)),aa_complex_complex(sin_complex,X)) ).

% sin_converges
tff(fact_9043_sin__def,axiom,
    ! [X2: real] : aa_real_real(sin_real,X2) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_iu,X2)) ).

% sin_def
tff(fact_9044_sin__def,axiom,
    ! [X2: complex] : aa_complex_complex(sin_complex,X2) = suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_iv,X2)) ).

% sin_def
tff(fact_9045_cos__converges,axiom,
    ! [X: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_iw,X)),aa_real_real(cos_real,X)) ).

% cos_converges
tff(fact_9046_cos__converges,axiom,
    ! [X: complex] : aa_complex_o(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_ix,X)),aa_complex_complex(cos_complex,X)) ).

% cos_converges
tff(fact_9047_cos__def,axiom,
    ! [X2: real] : aa_real_real(cos_real,X2) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_iw,X2)) ).

% cos_def
tff(fact_9048_cos__def,axiom,
    ! [X2: complex] : aa_complex_complex(cos_complex,X2) = suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_ix,X2)) ).

% cos_def
tff(fact_9049_summable__norm__sin,axiom,
    ! [X: real] : summable_real(aa_real_fun_nat_real(aTP_Lamm_iy,X)) ).

% summable_norm_sin
tff(fact_9050_summable__norm__sin,axiom,
    ! [X: complex] : summable_real(aa_com20334259897932645t_real(aTP_Lamm_iz,X)) ).

% summable_norm_sin
tff(fact_9051_summable__norm__cos,axiom,
    ! [X: real] : summable_real(aa_real_fun_nat_real(aTP_Lamm_ja,X)) ).

% summable_norm_cos
tff(fact_9052_summable__norm__cos,axiom,
    ! [X: complex] : summable_real(aa_com20334259897932645t_real(aTP_Lamm_jb,X)) ).

% summable_norm_cos
tff(fact_9053_cosh__converges,axiom,
    ! [X: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_jc,X)),aa_real_real(cosh_real,X)) ).

% cosh_converges
tff(fact_9054_cosh__converges,axiom,
    ! [X: complex] : aa_complex_o(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_jd,X)),aa_complex_complex(cosh_complex,X)) ).

% cosh_converges
tff(fact_9055_neg__minus__divideR__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(real_V1485227260804924795R_real(aa_real_real(inverse_inverse_real,C)),B))),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V1485227260804924795R_real(C),A)),aa_real_real(uminus_uminus_real,B)) ) ) ).

% neg_minus_divideR_le_eq
tff(fact_9056_neg__le__minus__divideR__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,aa_real_real(real_V1485227260804924795R_real(aa_real_real(inverse_inverse_real,C)),B)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(real_V1485227260804924795R_real(C),A)) ) ) ).

% neg_le_minus_divideR_eq
tff(fact_9057_pos__minus__divideR__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(real_V1485227260804924795R_real(aa_real_real(inverse_inverse_real,C)),B))),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(real_V1485227260804924795R_real(C),A)) ) ) ).

% pos_minus_divideR_le_eq
tff(fact_9058_pos__le__minus__divideR__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,aa_real_real(real_V1485227260804924795R_real(aa_real_real(inverse_inverse_real,C)),B)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V1485227260804924795R_real(C),A)),aa_real_real(uminus_uminus_real,B)) ) ) ).

% pos_le_minus_divideR_eq
tff(fact_9059_neg__minus__divideR__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(real_V1485227260804924795R_real(aa_real_real(inverse_inverse_real,C)),B))),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V1485227260804924795R_real(C),A)),aa_real_real(uminus_uminus_real,B)) ) ) ).

% neg_minus_divideR_less_eq
tff(fact_9060_neg__less__minus__divideR__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(uminus_uminus_real,aa_real_real(real_V1485227260804924795R_real(aa_real_real(inverse_inverse_real,C)),B)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(real_V1485227260804924795R_real(C),A)) ) ) ).

% neg_less_minus_divideR_eq
tff(fact_9061_pos__minus__divideR__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(real_V1485227260804924795R_real(aa_real_real(inverse_inverse_real,C)),B))),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B)),aa_real_real(real_V1485227260804924795R_real(C),A)) ) ) ).

% pos_minus_divideR_less_eq
tff(fact_9062_pos__less__minus__divideR__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(uminus_uminus_real,aa_real_real(real_V1485227260804924795R_real(aa_real_real(inverse_inverse_real,C)),B)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V1485227260804924795R_real(C),A)),aa_real_real(uminus_uminus_real,B)) ) ) ).

% pos_less_minus_divideR_eq
tff(fact_9063_sinh__converges,axiom,
    ! [X: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_je,X)),aa_real_real(sinh_real,X)) ).

% sinh_converges
tff(fact_9064_sinh__converges,axiom,
    ! [X: complex] : aa_complex_o(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_jf,X)),aa_complex_complex(sinh_complex,X)) ).

% sinh_converges
tff(fact_9065_exp__converges,axiom,
    ! [X: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_is,X)),aa_real_real(exp_real,X)) ).

% exp_converges
tff(fact_9066_exp__converges,axiom,
    ! [X: complex] : aa_complex_o(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_it,X)),aa_complex_complex(exp_complex,X)) ).

% exp_converges
tff(fact_9067_exp__def,axiom,
    ! [X2: real] : aa_real_real(exp_real,X2) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_is,X2)) ).

% exp_def
tff(fact_9068_exp__def,axiom,
    ! [X2: complex] : aa_complex_complex(exp_complex,X2) = suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_it,X2)) ).

% exp_def
tff(fact_9069_summable__norm__exp,axiom,
    ! [X: real] : summable_real(aa_real_fun_nat_real(aTP_Lamm_jg,X)) ).

% summable_norm_exp
tff(fact_9070_summable__norm__exp,axiom,
    ! [X: complex] : summable_real(aa_com20334259897932645t_real(aTP_Lamm_jh,X)) ).

% summable_norm_exp
tff(fact_9071_sin__minus__converges,axiom,
    ! [X: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_ji,X)),aa_real_real(sin_real,X)) ).

% sin_minus_converges
tff(fact_9072_sin__minus__converges,axiom,
    ! [X: complex] : aa_complex_o(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_jj,X)),aa_complex_complex(sin_complex,X)) ).

% sin_minus_converges
tff(fact_9073_cos__minus__converges,axiom,
    ! [X: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_jk,X)),aa_real_real(cos_real,X)) ).

% cos_minus_converges
tff(fact_9074_cos__minus__converges,axiom,
    ! [X: complex] : aa_complex_o(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_jl,X)),aa_complex_complex(cos_complex,X)) ).

% cos_minus_converges
tff(fact_9075_tanh__add,axiom,
    ! [X: complex,Y2: complex] :
      ( ( aa_complex_complex(cosh_complex,X) != zero_zero_complex )
     => ( ( aa_complex_complex(cosh_complex,Y2) != zero_zero_complex )
       => ( aa_complex_complex(tanh_complex,aa_complex_complex(plus_plus_complex(X),Y2)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(tanh_complex,X)),aa_complex_complex(tanh_complex,Y2))),aa_complex_complex(plus_plus_complex(one_one_complex),aa_complex_complex(times_times_complex(aa_complex_complex(tanh_complex,X)),aa_complex_complex(tanh_complex,Y2)))) ) ) ) ).

% tanh_add
tff(fact_9076_tanh__add,axiom,
    ! [X: real,Y2: real] :
      ( ( aa_real_real(cosh_real,X) != zero_zero_real )
     => ( ( aa_real_real(cosh_real,Y2) != zero_zero_real )
       => ( aa_real_real(tanh_real,aa_real_real(plus_plus_real(X),Y2)) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(tanh_real,X)),aa_real_real(tanh_real,Y2))),aa_real_real(plus_plus_real(one_one_real),aa_real_real(times_times_real(aa_real_real(tanh_real,X)),aa_real_real(tanh_real,Y2)))) ) ) ) ).

% tanh_add
tff(fact_9077_XOR__upper,axiom,
    ! [X: int,Nb: nat,Y2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y2),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,X),Y2)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ) ) ) ).

% XOR_upper
tff(fact_9078_cosh__field__def,axiom,
    ! [Z2: real] : aa_real_real(cosh_real,Z2) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(exp_real,Z2)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,Z2)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% cosh_field_def
tff(fact_9079_cosh__field__def,axiom,
    ! [Z2: complex] : aa_complex_complex(cosh_complex,Z2) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(exp_complex,Z2)),aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,Z2)))),numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% cosh_field_def
tff(fact_9080_complex__inverse,axiom,
    ! [A: real,B: real] : aa_complex_complex(invers8013647133539491842omplex,aa_real_complex(complex2(A),B)) = aa_real_complex(complex2(aa_real_real(divide_divide_real(A),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,B)),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% complex_inverse
tff(fact_9081_xor__int__rec,axiom,
    ! [K2: int,L: int] :
      aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K2),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,
          aa_o_int(zero_n2684676970156552555ol_int,
            ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K2) != ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% xor_int_rec
tff(fact_9082_sinh__field__def,axiom,
    ! [Z2: real] : aa_real_real(sinh_real,Z2) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(exp_real,Z2)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,Z2)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% sinh_field_def
tff(fact_9083_sinh__field__def,axiom,
    ! [Z2: complex] : aa_complex_complex(sinh_complex,Z2) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(exp_complex,Z2)),aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,Z2)))),numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% sinh_field_def
tff(fact_9084_exp__first__term,axiom,
    ! [X: real] : aa_real_real(exp_real,X) = aa_real_real(plus_plus_real(one_one_real),suminf_real(aa_real_fun_nat_real(aTP_Lamm_jm,X))) ).

% exp_first_term
tff(fact_9085_exp__first__term,axiom,
    ! [X: complex] : aa_complex_complex(exp_complex,X) = aa_complex_complex(plus_plus_complex(one_one_complex),suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_jn,X))) ).

% exp_first_term
tff(fact_9086_cosh__zero__iff,axiom,
    ! [X: real] :
      ( ( aa_real_real(cosh_real,X) = zero_zero_real )
    <=> ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(exp_real,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(uminus_uminus_real,one_one_real) ) ) ).

% cosh_zero_iff
tff(fact_9087_cosh__zero__iff,axiom,
    ! [X: complex] :
      ( ( aa_complex_complex(cosh_complex,X) = zero_zero_complex )
    <=> ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(exp_complex,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ).

% cosh_zero_iff
tff(fact_9088_some__sym__eq__trivial,axiom,
    ! [X: real] : fChoice_real(aa_real_fun_real_o(fequal_real,X)) = X ).

% some_sym_eq_trivial
tff(fact_9089_some__eq__trivial,axiom,
    ! [X: real] : fChoice_real(aa_real_fun_real_o(aTP_Lamm_ht,X)) = X ).

% some_eq_trivial
tff(fact_9090_some__equality,axiom,
    ! [P: fun_real_o,A: real] :
      ( aa_real_o(P,A)
     => ( ! [X4: real] :
            ( aa_real_o(P,X4)
           => ( X4 = A ) )
       => ( fChoice_real(P) = A ) ) ) ).

% some_equality
tff(fact_9091_xor__int__unfold,axiom,
    ! [K2: int,L: int] :
      aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K2),L) = $ite(
        K2 = aa_int_int(uminus_uminus_int,one_one_int),
        aa_int_int(bit_ri7919022796975470100ot_int,L),
        $ite(
          L = aa_int_int(uminus_uminus_int,one_one_int),
          aa_int_int(bit_ri7919022796975470100ot_int,K2),
          $ite(
            K2 = zero_zero_int,
            L,
            $ite(L = zero_zero_int,K2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))))) ) ) ) ).

% xor_int_unfold
tff(fact_9092_bit_Odouble__compl,axiom,
    ! [X: int] : aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_ri7919022796975470100ot_int,X)) = X ).

% bit.double_compl
tff(fact_9093_bit_Ocompl__eq__compl__iff,axiom,
    ! [X: int,Y2: int] :
      ( ( aa_int_int(bit_ri7919022796975470100ot_int,X) = aa_int_int(bit_ri7919022796975470100ot_int,Y2) )
    <=> ( X = Y2 ) ) ).

% bit.compl_eq_compl_iff
tff(fact_9094_bit_Oxor__compl__left,axiom,
    ! [X: int,Y2: int] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_int_int(bit_ri7919022796975470100ot_int,X)),Y2) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,X),Y2)) ).

% bit.xor_compl_left
tff(fact_9095_bit_Oxor__compl__right,axiom,
    ! [X: int,Y2: int] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,X),aa_int_int(bit_ri7919022796975470100ot_int,Y2)) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,X),Y2)) ).

% bit.xor_compl_right
tff(fact_9096_Eps__case__prod__eq,axiom,
    ! [X: code_integer,Y2: $o] : fChoic166683996008689692eger_o(produc7828578312038201481er_o_o(aa_o_f2459543391104474489un_o_o(aTP_Lamm_hv(X),(Y2)))) = aa_o_P4192039665577170892eger_o(aa_Cod3473573201699832512eger_o(produc6677183202524767010eger_o,X),(Y2)) ).

% Eps_case_prod_eq
tff(fact_9097_Eps__case__prod__eq,axiom,
    ! [X: num,Y2: num] : fChoic5817513213647635945um_num(produc5703948589228662326_num_o(aa_num5226579579167915582_num_o(aTP_Lamm_hw(X),Y2))) = aa_num3156681643174609981um_num(aa_num8681986212680877308um_num(product_Pair_num_num,X),Y2) ).

% Eps_case_prod_eq
tff(fact_9098_Eps__case__prod__eq,axiom,
    ! [X: nat,Y2: num] : fChoic7687182810340166559at_num(produc4927758841916487424_num_o(aa_num3446304307616776968_num_o(aTP_Lamm_hx(X),Y2))) = aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,X),Y2) ).

% Eps_case_prod_eq
tff(fact_9099_Eps__case__prod__eq,axiom,
    ! [X: nat,Y2: nat] : fChoic6978938873391328853at_nat(produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_hy(X),Y2))) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X),Y2) ).

% Eps_case_prod_eq
tff(fact_9100_Eps__case__prod__eq,axiom,
    ! [X: int,Y2: int] : fChoic3800441565783186701nt_int(produc4947309494688390418_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_hz(X),Y2))) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Y2) ).

% Eps_case_prod_eq
tff(fact_9101_bit_Oconj__cancel__right,axiom,
    ! [X: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),aa_int_int(bit_ri7919022796975470100ot_int,X)) = zero_zero_int ).

% bit.conj_cancel_right
tff(fact_9102_bit_Oconj__cancel__left,axiom,
    ! [X: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(bit_ri7919022796975470100ot_int,X)),X) = zero_zero_int ).

% bit.conj_cancel_left
tff(fact_9103_bit_Ocompl__one,axiom,
    bit_ri7632146776885996613nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% bit.compl_one
tff(fact_9104_bit_Ocompl__one,axiom,
    aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(uminus_uminus_int,one_one_int)) = zero_zero_int ).

% bit.compl_one
tff(fact_9105_bit_Ocompl__zero,axiom,
    bit_ri7632146776885996613nteger(zero_z3403309356797280102nteger) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% bit.compl_zero
tff(fact_9106_bit_Ocompl__zero,axiom,
    aa_int_int(bit_ri7919022796975470100ot_int,zero_zero_int) = aa_int_int(uminus_uminus_int,one_one_int) ).

% bit.compl_zero
tff(fact_9107_bit_Oxor__one__left,axiom,
    ! [X: code_integer] : bit_se3222712562003087583nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),X) = bit_ri7632146776885996613nteger(X) ).

% bit.xor_one_left
tff(fact_9108_bit_Oxor__one__left,axiom,
    ! [X: int] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_int_int(uminus_uminus_int,one_one_int)),X) = aa_int_int(bit_ri7919022796975470100ot_int,X) ).

% bit.xor_one_left
tff(fact_9109_bit_Oxor__one__right,axiom,
    ! [X: code_integer] : bit_se3222712562003087583nteger(X,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = bit_ri7632146776885996613nteger(X) ).

% bit.xor_one_right
tff(fact_9110_bit_Oxor__one__right,axiom,
    ! [X: int] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,X),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,X) ).

% bit.xor_one_right
tff(fact_9111_bit_Oxor__cancel__left,axiom,
    ! [X: code_integer] : bit_se3222712562003087583nteger(bit_ri7632146776885996613nteger(X),X) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% bit.xor_cancel_left
tff(fact_9112_bit_Oxor__cancel__left,axiom,
    ! [X: int] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_int_int(bit_ri7919022796975470100ot_int,X)),X) = aa_int_int(uminus_uminus_int,one_one_int) ).

% bit.xor_cancel_left
tff(fact_9113_bit_Oxor__cancel__right,axiom,
    ! [X: code_integer] : bit_se3222712562003087583nteger(X,bit_ri7632146776885996613nteger(X)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% bit.xor_cancel_right
tff(fact_9114_bit_Oxor__cancel__right,axiom,
    ! [X: int] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,X),aa_int_int(bit_ri7919022796975470100ot_int,X)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% bit.xor_cancel_right
tff(fact_9115_not__nonnegative__int__iff,axiom,
    ! [K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_ri7919022796975470100ot_int,K2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int) ) ).

% not_nonnegative_int_iff
tff(fact_9116_not__negative__int__iff,axiom,
    ! [K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_ri7919022796975470100ot_int,K2)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2) ) ).

% not_negative_int_iff
tff(fact_9117_minus__not__numeral__eq,axiom,
    ! [Nb: num] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,bit_ri7632146776885996613nteger(numera6620942414471956472nteger(Nb))) = numera6620942414471956472nteger(aa_num_num(inc,Nb)) ).

% minus_not_numeral_eq
tff(fact_9118_minus__not__numeral__eq,axiom,
    ! [Nb: num] : aa_int_int(uminus_uminus_int,aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))) = aa_num_int(numeral_numeral_int,aa_num_num(inc,Nb)) ).

% minus_not_numeral_eq
tff(fact_9119_even__not__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_ri7632146776885996613nteger(A))
    <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A) ) ).

% even_not_iff
tff(fact_9120_even__not__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri7919022796975470100ot_int,A))
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) ) ).

% even_not_iff
tff(fact_9121_not__one__eq,axiom,
    bit_ri7632146776885996613nteger(one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% not_one_eq
tff(fact_9122_not__one__eq,axiom,
    aa_int_int(bit_ri7919022796975470100ot_int,one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% not_one_eq
tff(fact_9123_of__int__not__numeral,axiom,
    ! [K2: num] : aa_int_int(ring_1_of_int_int,aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,K2))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,K2)) ).

% of_int_not_numeral
tff(fact_9124_take__bit__not__take__bit,axiom,
    ! [Nb: nat,A: int] : aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_se2923211474154528505it_int(Nb),A))) = aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(bit_ri7919022796975470100ot_int,A)) ).

% take_bit_not_take_bit
tff(fact_9125_take__bit__not__iff,axiom,
    ! [Nb: nat,A: int,B: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(bit_ri7919022796975470100ot_int,A)) = aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(bit_ri7919022796975470100ot_int,B)) )
    <=> ( aa_int_int(bit_se2923211474154528505it_int(Nb),A) = aa_int_int(bit_se2923211474154528505it_int(Nb),B) ) ) ).

% take_bit_not_iff
tff(fact_9126_bit__not__int__iff,axiom,
    ! [K2: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(bit_ri7919022796975470100ot_int,K2)),Nb)
    <=> ~ aa_nat_o(bit_se1146084159140164899it_int(K2),Nb) ) ).

% bit_not_int_iff
tff(fact_9127_of__int__not__eq,axiom,
    ! [K2: int] : aa_int_int(ring_1_of_int_int,aa_int_int(bit_ri7919022796975470100ot_int,K2)) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(ring_1_of_int_int,K2)) ).

% of_int_not_eq
tff(fact_9128_split__paired__Eps,axiom,
    ! [P: fun_Pr3290071492984375169er_o_o] : fChoic166683996008689692eger_o(P) = fChoic166683996008689692eger_o(produc7828578312038201481er_o_o(aTP_Lamm_jo(P))) ).

% split_paired_Eps
tff(fact_9129_split__paired__Eps,axiom,
    ! [P: fun_Pr6331313853612757382_num_o] : fChoic5817513213647635945um_num(P) = fChoic5817513213647635945um_num(produc5703948589228662326_num_o(aTP_Lamm_jp(P))) ).

% split_paired_Eps
tff(fact_9130_split__paired__Eps,axiom,
    ! [P: fun_Pr5555124106300582480_num_o] : fChoic7687182810340166559at_num(P) = fChoic7687182810340166559at_num(produc4927758841916487424_num_o(aTP_Lamm_jq(P))) ).

% split_paired_Eps
tff(fact_9131_split__paired__Eps,axiom,
    ! [P: fun_Pr4796943543331071514_nat_o] : fChoic6978938873391328853at_nat(P) = fChoic6978938873391328853at_nat(produc6081775807080527818_nat_o(aTP_Lamm_jr(P))) ).

% split_paired_Eps
tff(fact_9132_split__paired__Eps,axiom,
    ! [P: fun_Pr3238374608701176162_int_o] : fChoic3800441565783186701nt_int(P) = fChoic3800441565783186701nt_int(produc4947309494688390418_int_o(aTP_Lamm_di(P))) ).

% split_paired_Eps
tff(fact_9133_not__diff__distrib,axiom,
    ! [A: int,B: int] : aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(bit_ri7919022796975470100ot_int,A)),B) ).

% not_diff_distrib
tff(fact_9134_not__add__distrib,axiom,
    ! [A: int,B: int] : aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(bit_ri7919022796975470100ot_int,A)),B) ).

% not_add_distrib
tff(fact_9135_minus__eq__not__plus__1,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = plus_p5714425477246183910nteger(bit_ri7632146776885996613nteger(A),one_one_Code_integer) ).

% minus_eq_not_plus_1
tff(fact_9136_minus__eq__not__plus__1,axiom,
    ! [A: int] : aa_int_int(uminus_uminus_int,A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(bit_ri7919022796975470100ot_int,A)),one_one_int) ).

% minus_eq_not_plus_1
tff(fact_9137_not__eq__complement,axiom,
    ! [A: code_integer] : bit_ri7632146776885996613nteger(A) = minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),one_one_Code_integer) ).

% not_eq_complement
tff(fact_9138_not__eq__complement,axiom,
    ! [A: int] : aa_int_int(bit_ri7919022796975470100ot_int,A) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,A)),one_one_int) ).

% not_eq_complement
tff(fact_9139_minus__eq__not__minus__1,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = bit_ri7632146776885996613nteger(minus_8373710615458151222nteger(A,one_one_Code_integer)) ).

% minus_eq_not_minus_1
tff(fact_9140_minus__eq__not__minus__1,axiom,
    ! [A: int] : aa_int_int(uminus_uminus_int,A) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),one_one_int)) ).

% minus_eq_not_minus_1
tff(fact_9141_not__int__def,axiom,
    ! [K2: int] : aa_int_int(bit_ri7919022796975470100ot_int,K2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,K2)),one_one_int) ).

% not_int_def
tff(fact_9142_and__not__numerals_I1_J,axiom,
    aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = zero_zero_int ).

% and_not_numerals(1)
tff(fact_9143_disjunctive__diff,axiom,
    ! [B: int,A: int] :
      ( ! [N: nat] :
          ( aa_nat_o(bit_se1146084159140164899it_int(B),N)
         => aa_nat_o(bit_se1146084159140164899it_int(A),N) )
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),aa_int_int(bit_ri7919022796975470100ot_int,B)) ) ) ).

% disjunctive_diff
tff(fact_9144_take__bit__not__eq__mask__diff,axiom,
    ! [Nb: nat,A: int] : aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(bit_ri7919022796975470100ot_int,A)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(bit_se2000444600071755411sk_int,Nb)),aa_int_int(bit_se2923211474154528505it_int(Nb),A)) ).

% take_bit_not_eq_mask_diff
tff(fact_9145_minus__numeral__inc__eq,axiom,
    ! [Nb: num] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(inc,Nb))) = bit_ri7632146776885996613nteger(numera6620942414471956472nteger(Nb)) ).

% minus_numeral_inc_eq
tff(fact_9146_minus__numeral__inc__eq,axiom,
    ! [Nb: num] : aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,Nb))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb)) ).

% minus_numeral_inc_eq
tff(fact_9147_not__int__div__2,axiom,
    ! [K2: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(bit_ri7919022796975470100ot_int,K2)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% not_int_div_2
tff(fact_9148_even__not__iff__int,axiom,
    ! [K2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri7919022796975470100ot_int,K2))
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K2) ) ).

% even_not_iff_int
tff(fact_9149_and__not__numerals_I4_J,axiom,
    ! [Ma: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Ma))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,Ma)) ).

% and_not_numerals(4)
tff(fact_9150_and__not__numerals_I2_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = one_one_int ).

% and_not_numerals(2)
tff(fact_9151_not__numeral__Bit0__eq,axiom,
    ! [Nb: num] : bit_ri7632146776885996613nteger(numera6620942414471956472nteger(aa_num_num(bit0,Nb))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(bit1,Nb))) ).

% not_numeral_Bit0_eq
tff(fact_9152_not__numeral__Bit0__eq,axiom,
    ! [Nb: num] : aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))) ).

% not_numeral_Bit0_eq
tff(fact_9153_bit__minus__int__iff,axiom,
    ! [K2: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,K2)),Nb)
    <=> aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,K2),one_one_int))),Nb) ) ).

% bit_minus_int_iff
tff(fact_9154_not__numeral__BitM__eq,axiom,
    ! [Nb: num] : bit_ri7632146776885996613nteger(numera6620942414471956472nteger(aa_num_num(bitM,Nb))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(bit0,Nb))) ).

% not_numeral_BitM_eq
tff(fact_9155_not__numeral__BitM__eq,axiom,
    ! [Nb: num] : aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bitM,Nb))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))) ).

% not_numeral_BitM_eq
tff(fact_9156_take__bit__not__mask__eq__0,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => ( aa_int_int(bit_se2923211474154528505it_int(Ma),aa_int_int(bit_ri7919022796975470100ot_int,aa_nat_int(bit_se2000444600071755411sk_int,Nb))) = zero_zero_int ) ) ).

% take_bit_not_mask_eq_0
tff(fact_9157_and__not__numerals_I5_J,axiom,
    ! [Ma: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Ma))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,Ma)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb)))) ).

% and_not_numerals(5)
tff(fact_9158_and__not__numerals_I7_J,axiom,
    ! [Ma: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Ma))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,Ma)) ).

% and_not_numerals(7)
tff(fact_9159_and__not__numerals_I3_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = zero_zero_int ).

% and_not_numerals(3)
tff(fact_9160_someI2,axiom,
    ! [P: fun_real_o,A: real,Q2: fun_real_o] :
      ( aa_real_o(P,A)
     => ( ! [X4: real] :
            ( aa_real_o(P,X4)
           => aa_real_o(Q2,X4) )
       => aa_real_o(Q2,fChoice_real(P)) ) ) ).

% someI2
tff(fact_9161_someI__ex,axiom,
    ! [P: fun_real_o] :
      ( ? [X_1: real] : aa_real_o(P,X_1)
     => aa_real_o(P,fChoice_real(P)) ) ).

% someI_ex
tff(fact_9162_someI2__ex,axiom,
    ! [P: fun_real_o,Q2: fun_real_o] :
      ( ? [X_1: real] : aa_real_o(P,X_1)
     => ( ! [X4: real] :
            ( aa_real_o(P,X4)
           => aa_real_o(Q2,X4) )
       => aa_real_o(Q2,fChoice_real(P)) ) ) ).

% someI2_ex
tff(fact_9163_someI2__bex,axiom,
    ! [A2: set_complex,P: fun_complex_o,Q2: fun_complex_o] :
      ( ? [X2: complex] :
          ( member_complex(X2,A2)
          & aa_complex_o(P,X2) )
     => ( ! [X4: complex] :
            ( ( member_complex(X4,A2)
              & aa_complex_o(P,X4) )
           => aa_complex_o(Q2,X4) )
       => aa_complex_o(Q2,fChoice_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_ba(A2),P))) ) ) ).

% someI2_bex
tff(fact_9164_someI2__bex,axiom,
    ! [A2: set_set_nat,P: fun_set_nat_o,Q2: fun_set_nat_o] :
      ( ? [X2: set_nat] :
          ( member_set_nat(X2,A2)
          & aa_set_nat_o(P,X2) )
     => ( ! [X4: set_nat] :
            ( ( member_set_nat(X4,A2)
              & aa_set_nat_o(P,X4) )
           => aa_set_nat_o(Q2,X4) )
       => aa_set_nat_o(Q2,fChoice_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_bc(A2),P))) ) ) ).

% someI2_bex
tff(fact_9165_someI2__bex,axiom,
    ! [A2: set_nat,P: fun_nat_o,Q2: fun_nat_o] :
      ( ? [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A2)
          & aa_nat_o(P,X2) )
     => ( ! [X4: nat] :
            ( ( aa_set_nat_o(member_nat(X4),A2)
              & aa_nat_o(P,X4) )
           => aa_nat_o(Q2,X4) )
       => aa_nat_o(Q2,fChoice_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_bd(A2),P))) ) ) ).

% someI2_bex
tff(fact_9166_someI2__bex,axiom,
    ! [A2: set_int,P: fun_int_o,Q2: fun_int_o] :
      ( ? [X2: int] :
          ( member_int(X2,A2)
          & aa_int_o(P,X2) )
     => ( ! [X4: int] :
            ( ( member_int(X4,A2)
              & aa_int_o(P,X4) )
           => aa_int_o(Q2,X4) )
       => aa_int_o(Q2,fChoice_int(aa_fun6243517787477317628_int_o(aTP_Lamm_be(A2),P))) ) ) ).

% someI2_bex
tff(fact_9167_someI2__bex,axiom,
    ! [A2: set_real,P: fun_real_o,Q2: fun_real_o] :
      ( ? [X2: real] :
          ( member_real(X2,A2)
          & aa_real_o(P,X2) )
     => ( ! [X4: real] :
            ( ( member_real(X4,A2)
              & aa_real_o(P,X4) )
           => aa_real_o(Q2,X4) )
       => aa_real_o(Q2,fChoice_real(aa_fun5868358341814070780real_o(aTP_Lamm_az(A2),P))) ) ) ).

% someI2_bex
tff(fact_9168_some__eq__ex,axiom,
    ! [P: fun_real_o] :
      ( aa_real_o(P,fChoice_real(P))
    <=> ? [X_12: real] : aa_real_o(P,X_12) ) ).

% some_eq_ex
tff(fact_9169_some1__equality,axiom,
    ! [P: fun_real_o,A: real] :
      ( ? [X2: real] :
          ( aa_real_o(P,X2)
          & ! [Y4: real] :
              ( aa_real_o(P,Y4)
             => ( Y4 = X2 ) ) )
     => ( aa_real_o(P,A)
       => ( fChoice_real(P) = A ) ) ) ).

% some1_equality
tff(fact_9170_and__not__numerals_I9_J,axiom,
    ! [Ma: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Ma))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,Ma)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb)))) ).

% and_not_numerals(9)
tff(fact_9171_and__not__numerals_I6_J,axiom,
    ! [Ma: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Ma))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,Ma)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb)))) ).

% and_not_numerals(6)
tff(fact_9172_bit__not__iff__eq,axiom,
    ! [A: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(bit_ri7919022796975470100ot_int,A)),Nb)
    <=> ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb) != zero_zero_int )
        & ~ aa_nat_o(bit_se1146084159140164899it_int(A),Nb) ) ) ).

% bit_not_iff_eq
tff(fact_9173_minus__exp__eq__not__mask,axiom,
    ! [Nb: nat] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)) = bit_ri7632146776885996613nteger(aa_nat_Code_integer(bit_se2119862282449309892nteger,Nb)) ).

% minus_exp_eq_not_mask
tff(fact_9174_minus__exp__eq__not__mask,axiom,
    ! [Nb: nat] : aa_int_int(uminus_uminus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_int_int(bit_ri7919022796975470100ot_int,aa_nat_int(bit_se2000444600071755411sk_int,Nb)) ).

% minus_exp_eq_not_mask
tff(fact_9175_and__not__numerals_I8_J,axiom,
    ! [Ma: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Ma))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,Ma)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))))) ).

% and_not_numerals(8)
tff(fact_9176_not__int__rec,axiom,
    ! [K2: int] : aa_int_int(bit_ri7919022796975470100ot_int,K2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_o_int(zero_n2684676970156552555ol_int,aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% not_int_rec
tff(fact_9177_sin__x__sin__y,axiom,
    ! [X: real,Y2: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_jt(X),Y2)),aa_real_real(times_times_real(aa_real_real(sin_real,X)),aa_real_real(sin_real,Y2))) ).

% sin_x_sin_y
tff(fact_9178_sin__x__sin__y,axiom,
    ! [X: complex,Y2: complex] : aa_complex_o(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_jv(X),Y2)),aa_complex_complex(times_times_complex(aa_complex_complex(sin_complex,X)),aa_complex_complex(sin_complex,Y2))) ).

% sin_x_sin_y
tff(fact_9179_sums__cos__x__plus__y,axiom,
    ! [X: real,Y2: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_jx(X),Y2)),aa_real_real(cos_real,aa_real_real(plus_plus_real(X),Y2))) ).

% sums_cos_x_plus_y
tff(fact_9180_sums__cos__x__plus__y,axiom,
    ! [X: complex,Y2: complex] : aa_complex_o(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_jz(X),Y2)),aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(X),Y2))) ).

% sums_cos_x_plus_y
tff(fact_9181_cos__x__cos__y,axiom,
    ! [X: real,Y2: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_kb(X),Y2)),aa_real_real(times_times_real(aa_real_real(cos_real,X)),aa_real_real(cos_real,Y2))) ).

% cos_x_cos_y
tff(fact_9182_cos__x__cos__y,axiom,
    ! [X: complex,Y2: complex] : aa_complex_o(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_kd(X),Y2)),aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,X)),aa_complex_complex(cos_complex,Y2))) ).

% cos_x_cos_y
tff(fact_9183_rat__inverse__code,axiom,
    ! [P2: rat] : quotient_of(aa_rat_rat(inverse_inverse_rat,P2)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_ke),quotient_of(P2)) ).

% rat_inverse_code
tff(fact_9184_atMost__iff,axiom,
    ! [I2: real,K2: real] :
      ( member_real(I2,set_ord_atMost_real(K2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I2),K2) ) ).

% atMost_iff
tff(fact_9185_atMost__iff,axiom,
    ! [I2: set_nat,K2: set_nat] :
      ( member_set_nat(I2,set_or4236626031148496127et_nat(K2))
    <=> aa_set_nat_o(ord_less_eq_set_nat(I2),K2) ) ).

% atMost_iff
tff(fact_9186_atMost__iff,axiom,
    ! [I2: set_int,K2: set_int] :
      ( member_set_int(I2,set_or58775011639299419et_int(K2))
    <=> ord_less_eq_set_int(I2,K2) ) ).

% atMost_iff
tff(fact_9187_atMost__iff,axiom,
    ! [I2: rat,K2: rat] :
      ( member_rat(I2,set_ord_atMost_rat(K2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I2),K2) ) ).

% atMost_iff
tff(fact_9188_atMost__iff,axiom,
    ! [I2: num,K2: num] :
      ( member_num(I2,set_ord_atMost_num(K2))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,I2),K2) ) ).

% atMost_iff
tff(fact_9189_atMost__iff,axiom,
    ! [I2: nat,K2: nat] :
      ( aa_set_nat_o(member_nat(I2),aa_nat_set_nat(set_ord_atMost_nat,K2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),K2) ) ).

% atMost_iff
tff(fact_9190_atMost__iff,axiom,
    ! [I2: int,K2: int] :
      ( member_int(I2,set_ord_atMost_int(K2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),K2) ) ).

% atMost_iff
tff(fact_9191_of__nat__sum,axiom,
    ! [F: fun_complex_nat,A2: set_complex] : aa_nat_complex(semiri8010041392384452111omplex,groups5693394587270226106ex_nat(F,A2)) = groups7754918857620584856omplex(aTP_Lamm_kf(F),A2) ).

% of_nat_sum
tff(fact_9192_of__nat__sum,axiom,
    ! [F: fun_int_nat,A2: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups4541462559716669496nt_nat(F,A2)) = groups4538972089207619220nt_int(aTP_Lamm_kg(F),A2) ).

% of_nat_sum
tff(fact_9193_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = groups3539618377306564664at_int(aTP_Lamm_kh(F),A2) ).

% of_nat_sum
tff(fact_9194_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_Extended_enat(semiri4216267220026989637d_enat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = groups7108830773950497114d_enat(aTP_Lamm_ki(F),A2) ).

% of_nat_sum
tff(fact_9195_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_kj(F)),A2) ).

% of_nat_sum
tff(fact_9196_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A2: set_nat] : aa_nat_real(semiri5074537144036343181t_real,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_kk(F)),A2) ).

% of_nat_sum
tff(fact_9197_of__int__sum,axiom,
    ! [F: fun_complex_int,A2: set_complex] : aa_int_complex(ring_17405671764205052669omplex,groups5690904116761175830ex_int(F,A2)) = groups7754918857620584856omplex(aTP_Lamm_kl(F),A2) ).

% of_int_sum
tff(fact_9198_of__int__sum,axiom,
    ! [F: fun_nat_int,A2: set_nat] : aa_int_real(ring_1_of_int_real,groups3539618377306564664at_int(F,A2)) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_km(F)),A2) ).

% of_int_sum
tff(fact_9199_of__int__sum,axiom,
    ! [F: fun_int_int,A2: set_int] : aa_int_real(ring_1_of_int_real,groups4538972089207619220nt_int(F,A2)) = groups8778361861064173332t_real(aTP_Lamm_kn(F),A2) ).

% of_int_sum
tff(fact_9200_of__int__sum,axiom,
    ! [F: fun_int_int,A2: set_int] : aa_int_rat(ring_1_of_int_rat,groups4538972089207619220nt_int(F,A2)) = groups3906332499630173760nt_rat(aTP_Lamm_ko(F),A2) ).

% of_int_sum
tff(fact_9201_of__int__sum,axiom,
    ! [F: fun_int_int,A2: set_int] : aa_int_int(ring_1_of_int_int,groups4538972089207619220nt_int(F,A2)) = groups4538972089207619220nt_int(aTP_Lamm_kp(F),A2) ).

% of_int_sum
tff(fact_9202_of__real__sum,axiom,
    ! [F: fun_complex_real,S: set_complex] : aa_real_complex(real_V4546457046886955230omplex,groups5808333547571424918x_real(F,S)) = groups7754918857620584856omplex(aTP_Lamm_kq(F),S) ).

% of_real_sum
tff(fact_9203_of__real__sum,axiom,
    ! [F: fun_nat_real,S: set_nat] : aa_real_complex(real_V4546457046886955230omplex,aa_set_nat_real(groups6591440286371151544t_real(F),S)) = aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_fz(F)),S) ).

% of_real_sum
tff(fact_9204_of__real__sum,axiom,
    ! [F: fun_nat_real,S: set_nat] : aa_real_real(real_V1803761363581548252l_real,aa_set_nat_real(groups6591440286371151544t_real(F),S)) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ga(F)),S) ).

% of_real_sum
tff(fact_9205_atMost__subset__iff,axiom,
    ! [X: set_int,Y2: set_int] :
      ( ord_le4403425263959731960et_int(set_or58775011639299419et_int(X),set_or58775011639299419et_int(Y2))
    <=> ord_less_eq_set_int(X,Y2) ) ).

% atMost_subset_iff
tff(fact_9206_atMost__subset__iff,axiom,
    ! [X: rat,Y2: rat] :
      ( ord_less_eq_set_rat(set_ord_atMost_rat(X),set_ord_atMost_rat(Y2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y2) ) ).

% atMost_subset_iff
tff(fact_9207_atMost__subset__iff,axiom,
    ! [X: num,Y2: num] :
      ( ord_less_eq_set_num(set_ord_atMost_num(X),set_ord_atMost_num(Y2))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y2) ) ).

% atMost_subset_iff
tff(fact_9208_atMost__subset__iff,axiom,
    ! [X: nat,Y2: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(set_ord_atMost_nat,X)),aa_nat_set_nat(set_ord_atMost_nat,Y2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y2) ) ).

% atMost_subset_iff
tff(fact_9209_atMost__subset__iff,axiom,
    ! [X: int,Y2: int] :
      ( ord_less_eq_set_int(set_ord_atMost_int(X),set_ord_atMost_int(Y2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y2) ) ).

% atMost_subset_iff
tff(fact_9210_sum_OatMost__Suc,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,Nb))),aa_nat_rat(G,aa_nat_nat(suc,Nb))) ).

% sum.atMost_Suc
tff(fact_9211_sum_OatMost__Suc,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_atMost_nat,Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb))) ).

% sum.atMost_Suc
tff(fact_9212_sum_OatMost__Suc,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb))) ).

% sum.atMost_Suc
tff(fact_9213_sum_OatMost__Suc,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_atMost_nat,Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb))) ).

% sum.atMost_Suc
tff(fact_9214_quotient__of__number_I3_J,axiom,
    ! [K2: num] : quotient_of(aa_num_rat(numeral_numeral_rat,K2)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_num_int(numeral_numeral_int,K2)),one_one_int) ).

% quotient_of_number(3)
tff(fact_9215_rat__one__code,axiom,
    quotient_of(one_one_rat) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),one_one_int) ).

% rat_one_code
tff(fact_9216_rat__zero__code,axiom,
    quotient_of(zero_zero_rat) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int) ).

% rat_zero_code
tff(fact_9217_quotient__of__number_I5_J,axiom,
    ! [K2: num] : quotient_of(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,K2))) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))),one_one_int) ).

% quotient_of_number(5)
tff(fact_9218_quotient__of__number_I4_J,axiom,
    quotient_of(aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) ).

% quotient_of_number(4)
tff(fact_9219_divide__rat__def,axiom,
    ! [Q: rat,R: rat] : aa_rat_rat(divide_divide_rat(Q),R) = aa_rat_rat(times_times_rat(Q),aa_rat_rat(inverse_inverse_rat,R)) ).

% divide_rat_def
tff(fact_9220_scaleR__left_Osum,axiom,
    ! [G: fun_nat_real,A2: set_nat,X: real] : aa_real_real(real_V1485227260804924795R_real(aa_set_nat_real(groups6591440286371151544t_real(G),A2)),X) = aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_iq(G),X)),A2) ).

% scaleR_left.sum
tff(fact_9221_scaleR__left_Osum,axiom,
    ! [G: fun_complex_real,A2: set_complex,X: complex] : aa_complex_complex(real_V2046097035970521341omplex(groups5808333547571424918x_real(G,A2)),X) = groups7754918857620584856omplex(aa_com3342990543769683525omplex(aTP_Lamm_kr(G),X),A2) ).

% scaleR_left.sum
tff(fact_9222_scaleR__left_Osum,axiom,
    ! [G: fun_nat_real,A2: set_nat,X: complex] : aa_complex_complex(real_V2046097035970521341omplex(aa_set_nat_real(groups6591440286371151544t_real(G),A2)),X) = aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_ir(G),X)),A2) ).

% scaleR_left.sum
tff(fact_9223_scaleR__sum__left,axiom,
    ! [F: fun_nat_real,A2: set_nat,X: real] : aa_real_real(real_V1485227260804924795R_real(aa_set_nat_real(groups6591440286371151544t_real(F),A2)),X) = aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_iq(F),X)),A2) ).

% scaleR_sum_left
tff(fact_9224_scaleR__sum__left,axiom,
    ! [F: fun_complex_real,A2: set_complex,X: complex] : aa_complex_complex(real_V2046097035970521341omplex(groups5808333547571424918x_real(F,A2)),X) = groups7754918857620584856omplex(aa_com3342990543769683525omplex(aTP_Lamm_kr(F),X),A2) ).

% scaleR_sum_left
tff(fact_9225_scaleR__sum__left,axiom,
    ! [F: fun_nat_real,A2: set_nat,X: complex] : aa_complex_complex(real_V2046097035970521341omplex(aa_set_nat_real(groups6591440286371151544t_real(F),A2)),X) = aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_ir(F),X)),A2) ).

% scaleR_sum_left
tff(fact_9226_mod__sum__eq,axiom,
    ! [F: fun_nat_nat,A: nat,A2: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ks(F),A)),A2)),A) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A2)),A) ).

% mod_sum_eq
tff(fact_9227_mod__sum__eq,axiom,
    ! [F: fun_int_int,A: int,A2: set_int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,groups4538972089207619220nt_int(aa_int_fun_int_int(aTP_Lamm_kt(F),A),A2)),A) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,groups4538972089207619220nt_int(F,A2)),A) ).

% mod_sum_eq
tff(fact_9228_scaleR__sum__right,axiom,
    ! [A: real,F: fun_nat_real,A2: set_nat] : aa_real_real(real_V1485227260804924795R_real(A),aa_set_nat_real(groups6591440286371151544t_real(F),A2)) = aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_ku(A),F)),A2) ).

% scaleR_sum_right
tff(fact_9229_scaleR__sum__right,axiom,
    ! [A: real,F: fun_complex_complex,A2: set_complex] : aa_complex_complex(real_V2046097035970521341omplex(A),groups7754918857620584856omplex(F,A2)) = groups7754918857620584856omplex(aa_fun3187328805393345264omplex(aTP_Lamm_kv(A),F),A2) ).

% scaleR_sum_right
tff(fact_9230_scaleR__right_Osum,axiom,
    ! [A: real,G: fun_nat_real,A2: set_nat] : aa_real_real(real_V1485227260804924795R_real(A),aa_set_nat_real(groups6591440286371151544t_real(G),A2)) = aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_ku(A),G)),A2) ).

% scaleR_right.sum
tff(fact_9231_scaleR__right_Osum,axiom,
    ! [A: real,G: fun_complex_complex,A2: set_complex] : aa_complex_complex(real_V2046097035970521341omplex(A),groups7754918857620584856omplex(G,A2)) = groups7754918857620584856omplex(aa_fun3187328805393345264omplex(aTP_Lamm_kv(A),G),A2) ).

% scaleR_right.sum
tff(fact_9232_sum__norm__le,axiom,
    ! [S3: set_real,F: fun_real_complex,G: fun_real_real] :
      ( ! [X4: real] :
          ( member_real(X4,S3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_real_complex(F,X4))),aa_real_real(G,X4)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups5754745047067104278omplex(F,S3))),groups8097168146408367636l_real(G,S3)) ) ).

% sum_norm_le
tff(fact_9233_sum__norm__le,axiom,
    ! [S3: set_set_nat,F: fun_set_nat_complex,G: fun_set_nat_real] :
      ( ! [X4: set_nat] :
          ( member_set_nat(X4,S3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_set_nat_complex(F,X4))),aa_set_nat_real(G,X4)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups8255218700646806128omplex(F,S3))),groups5107569545109728110t_real(G,S3)) ) ).

% sum_norm_le
tff(fact_9234_sum__norm__le,axiom,
    ! [S3: set_int,F: fun_int_complex,G: fun_int_real] :
      ( ! [X4: int] :
          ( member_int(X4,S3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_int_complex(F,X4))),aa_int_real(G,X4)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups3049146728041665814omplex(F,S3))),groups8778361861064173332t_real(G,S3)) ) ).

% sum_norm_le
tff(fact_9235_sum__norm__le,axiom,
    ! [S3: set_nat,F: fun_nat_complex,G: fun_nat_real] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),S3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,X4))),aa_nat_real(G,X4)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_set_nat_complex(groups2073611262835488442omplex(F),S3))),aa_set_nat_real(groups6591440286371151544t_real(G),S3)) ) ).

% sum_norm_le
tff(fact_9236_sum__norm__le,axiom,
    ! [S3: set_complex,F: fun_complex_complex,G: fun_complex_real] :
      ( ! [X4: complex] :
          ( member_complex(X4,S3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(F,X4))),aa_complex_real(G,X4)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups7754918857620584856omplex(F,S3))),groups5808333547571424918x_real(G,S3)) ) ).

% sum_norm_le
tff(fact_9237_sum__norm__le,axiom,
    ! [S3: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),S3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,X4))),aa_nat_real(G,X4)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_set_nat_real(groups6591440286371151544t_real(F),S3))),aa_set_nat_real(groups6591440286371151544t_real(G),S3)) ) ).

% sum_norm_le
tff(fact_9238_norm__sum,axiom,
    ! [F: fun_nat_complex,A2: set_nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_set_nat_complex(groups2073611262835488442omplex(F),A2))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ev(F)),A2)) ).

% norm_sum
tff(fact_9239_norm__sum,axiom,
    ! [F: fun_complex_complex,A2: set_complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups7754918857620584856omplex(F,A2))),groups5808333547571424918x_real(aTP_Lamm_kw(F),A2)) ).

% norm_sum
tff(fact_9240_norm__sum,axiom,
    ! [F: fun_nat_real,A2: set_nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_set_nat_real(groups6591440286371151544t_real(F),A2))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_eu(F)),A2)) ).

% norm_sum
tff(fact_9241_sum__choose__upper,axiom,
    ! [Ma: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_kx,Ma)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),aa_nat_nat(suc,Ma)) ).

% sum_choose_upper
tff(fact_9242_summable__sum,axiom,
    ! [I5: set_complex,F: fun_co8378349014452122532t_real] :
      ( ! [I3: complex] :
          ( member_complex(I3,I5)
         => summable_real(aa_com20334259897932645t_real(F,I3)) )
     => summable_real(aa_fun471735231769099483t_real(aTP_Lamm_kz(I5),F)) ) ).

% summable_sum
tff(fact_9243_summable__sum,axiom,
    ! [I5: set_real,F: fun_re4238438383274954274t_real] :
      ( ! [I3: real] :
          ( member_real(I3,I5)
         => summable_real(aa_real_fun_nat_real(F,I3)) )
     => summable_real(aa_fun628238740774635481t_real(aTP_Lamm_lb(I5),F)) ) ).

% summable_sum
tff(fact_9244_summable__sum,axiom,
    ! [I5: set_int,F: fun_int_fun_nat_real] :
      ( ! [I3: int] :
          ( member_int(I3,I5)
         => summable_real(aa_int_fun_nat_real(F,I3)) )
     => summable_real(aa_fun2842512154077127769t_real(aTP_Lamm_ld(I5),F)) ) ).

% summable_sum
tff(fact_9245_summable__sum,axiom,
    ! [I5: set_real,F: fun_re2084383175013312036omplex] :
      ( ! [I3: real] :
          ( member_real(I3,I5)
         => summable_complex(aa_rea1531280491672885477omplex(F,I3)) )
     => summable_complex(aa_fun7632768522510439005omplex(aTP_Lamm_lf(I5),F)) ) ).

% summable_sum
tff(fact_9246_summable__sum,axiom,
    ! [I5: set_nat,F: fun_na1142814060743658952omplex] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),I5)
         => summable_complex(aa_nat6454746962644518025omplex(F,I3)) )
     => summable_complex(aa_fun4886358607025424513omplex(aTP_Lamm_lh(I5),F)) ) ).

% summable_sum
tff(fact_9247_summable__sum,axiom,
    ! [I5: set_int,F: fun_in8056368165505483812omplex] :
      ( ! [I3: int] :
          ( member_int(I3,I5)
         => summable_complex(aa_int4144929030551567077omplex(F,I3)) )
     => summable_complex(aa_fun3543561798013398749omplex(aTP_Lamm_lj(I5),F)) ) ).

% summable_sum
tff(fact_9248_summable__sum,axiom,
    ! [I5: set_nat,F: fun_nat_fun_nat_nat] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),I5)
         => summable_nat(aa_nat_fun_nat_nat(F,I3)) )
     => summable_nat(aa_fun8466562714962205125at_nat(aTP_Lamm_ll(I5),F)) ) ).

% summable_sum
tff(fact_9249_summable__sum,axiom,
    ! [I5: set_complex,F: fun_co3258106519541737638omplex] :
      ( ! [I3: complex] :
          ( member_complex(I3,I5)
         => summable_complex(aa_com8181001469867462503omplex(F,I3)) )
     => summable_complex(aa_fun5003373910569599327omplex(aTP_Lamm_ln(I5),F)) ) ).

% summable_sum
tff(fact_9250_summable__sum,axiom,
    ! [I5: set_nat,F: fun_nat_fun_nat_real] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),I5)
         => summable_real(aa_nat_fun_nat_real(F,I3)) )
     => summable_real(aa_fun7039663905532832253t_real(aTP_Lamm_lp(I5),F)) ) ).

% summable_sum
tff(fact_9251_summable__sum,axiom,
    ! [I5: set_int,F: fun_int_fun_nat_int] :
      ( ! [I3: int] :
          ( member_int(I3,I5)
         => summable_int(aa_int_fun_nat_int(F,I3)) )
     => summable_int(aa_fun8710705519803144921at_int(aTP_Lamm_lr(I5),F)) ) ).

% summable_sum
tff(fact_9252_sums__sum,axiom,
    ! [I5: set_complex,F: fun_co8378349014452122532t_real,X: fun_complex_real] :
      ( ! [I3: complex] :
          ( member_complex(I3,I5)
         => aa_real_o(sums_real(aa_com20334259897932645t_real(F,I3)),aa_complex_real(X,I3)) )
     => aa_real_o(sums_real(aa_fun471735231769099483t_real(aTP_Lamm_kz(I5),F)),groups5808333547571424918x_real(X,I5)) ) ).

% sums_sum
tff(fact_9253_sums__sum,axiom,
    ! [I5: set_real,F: fun_re4238438383274954274t_real,X: fun_real_real] :
      ( ! [I3: real] :
          ( member_real(I3,I5)
         => aa_real_o(sums_real(aa_real_fun_nat_real(F,I3)),aa_real_real(X,I3)) )
     => aa_real_o(sums_real(aa_fun628238740774635481t_real(aTP_Lamm_lb(I5),F)),groups8097168146408367636l_real(X,I5)) ) ).

% sums_sum
tff(fact_9254_sums__sum,axiom,
    ! [I5: set_int,F: fun_int_fun_nat_real,X: fun_int_real] :
      ( ! [I3: int] :
          ( member_int(I3,I5)
         => aa_real_o(sums_real(aa_int_fun_nat_real(F,I3)),aa_int_real(X,I3)) )
     => aa_real_o(sums_real(aa_fun2842512154077127769t_real(aTP_Lamm_ld(I5),F)),groups8778361861064173332t_real(X,I5)) ) ).

% sums_sum
tff(fact_9255_sums__sum,axiom,
    ! [I5: set_real,F: fun_re2084383175013312036omplex,X: fun_real_complex] :
      ( ! [I3: real] :
          ( member_real(I3,I5)
         => aa_complex_o(sums_complex(aa_rea1531280491672885477omplex(F,I3)),aa_real_complex(X,I3)) )
     => aa_complex_o(sums_complex(aa_fun7632768522510439005omplex(aTP_Lamm_lf(I5),F)),groups5754745047067104278omplex(X,I5)) ) ).

% sums_sum
tff(fact_9256_sums__sum,axiom,
    ! [I5: set_nat,F: fun_na1142814060743658952omplex,X: fun_nat_complex] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),I5)
         => aa_complex_o(sums_complex(aa_nat6454746962644518025omplex(F,I3)),aa_nat_complex(X,I3)) )
     => aa_complex_o(sums_complex(aa_fun4886358607025424513omplex(aTP_Lamm_lh(I5),F)),aa_set_nat_complex(groups2073611262835488442omplex(X),I5)) ) ).

% sums_sum
tff(fact_9257_sums__sum,axiom,
    ! [I5: set_int,F: fun_in8056368165505483812omplex,X: fun_int_complex] :
      ( ! [I3: int] :
          ( member_int(I3,I5)
         => aa_complex_o(sums_complex(aa_int4144929030551567077omplex(F,I3)),aa_int_complex(X,I3)) )
     => aa_complex_o(sums_complex(aa_fun3543561798013398749omplex(aTP_Lamm_lj(I5),F)),groups3049146728041665814omplex(X,I5)) ) ).

% sums_sum
tff(fact_9258_sums__sum,axiom,
    ! [I5: set_nat,F: fun_nat_fun_nat_nat,X: fun_nat_nat] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),I5)
         => aa_nat_o(sums_nat(aa_nat_fun_nat_nat(F,I3)),aa_nat_nat(X,I3)) )
     => aa_nat_o(sums_nat(aa_fun8466562714962205125at_nat(aTP_Lamm_ll(I5),F)),aa_set_nat_nat(groups3542108847815614940at_nat(X),I5)) ) ).

% sums_sum
tff(fact_9259_sums__sum,axiom,
    ! [I5: set_complex,F: fun_co3258106519541737638omplex,X: fun_complex_complex] :
      ( ! [I3: complex] :
          ( member_complex(I3,I5)
         => aa_complex_o(sums_complex(aa_com8181001469867462503omplex(F,I3)),aa_complex_complex(X,I3)) )
     => aa_complex_o(sums_complex(aa_fun5003373910569599327omplex(aTP_Lamm_ln(I5),F)),groups7754918857620584856omplex(X,I5)) ) ).

% sums_sum
tff(fact_9260_sums__sum,axiom,
    ! [I5: set_nat,F: fun_nat_fun_nat_real,X: fun_nat_real] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),I5)
         => aa_real_o(sums_real(aa_nat_fun_nat_real(F,I3)),aa_nat_real(X,I3)) )
     => aa_real_o(sums_real(aa_fun7039663905532832253t_real(aTP_Lamm_lp(I5),F)),aa_set_nat_real(groups6591440286371151544t_real(X),I5)) ) ).

% sums_sum
tff(fact_9261_sums__sum,axiom,
    ! [I5: set_int,F: fun_int_fun_nat_int,X: fun_int_int] :
      ( ! [I3: int] :
          ( member_int(I3,I5)
         => aa_int_o(sums_int(aa_int_fun_nat_int(F,I3)),aa_int_int(X,I3)) )
     => aa_int_o(sums_int(aa_fun8710705519803144921at_int(aTP_Lamm_lr(I5),F)),groups4538972089207619220nt_int(X,I5)) ) ).

% sums_sum
tff(fact_9262_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,zero_zero_nat)),groups2906978787729119204at_rat(aTP_Lamm_ls(G),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% sum.atMost_Suc_shift
tff(fact_9263_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,zero_zero_nat)),groups3539618377306564664at_int(aTP_Lamm_ge(G),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% sum.atMost_Suc_shift
tff(fact_9264_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,zero_zero_nat)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_gd(G)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% sum.atMost_Suc_shift
tff(fact_9265_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_real_real(plus_plus_real(aa_nat_real(G,zero_zero_nat)),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_fk(G)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% sum.atMost_Suc_shift
tff(fact_9266_sum__telescope,axiom,
    ! [F: fun_nat_complex,I2: nat] : aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_lt(F)),aa_nat_set_nat(set_ord_atMost_nat,I2)) = aa_complex_complex(minus_minus_complex(aa_nat_complex(F,zero_zero_nat)),aa_nat_complex(F,aa_nat_nat(suc,I2))) ).

% sum_telescope
tff(fact_9267_sum__telescope,axiom,
    ! [F: fun_nat_rat,I2: nat] : groups2906978787729119204at_rat(aTP_Lamm_lu(F),aa_nat_set_nat(set_ord_atMost_nat,I2)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,zero_zero_nat)),aa_nat_rat(F,aa_nat_nat(suc,I2))) ).

% sum_telescope
tff(fact_9268_sum__telescope,axiom,
    ! [F: fun_nat_int,I2: nat] : groups3539618377306564664at_int(aTP_Lamm_lv(F),aa_nat_set_nat(set_ord_atMost_nat,I2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,zero_zero_nat)),aa_nat_int(F,aa_nat_nat(suc,I2))) ).

% sum_telescope
tff(fact_9269_sum__telescope,axiom,
    ! [F: fun_nat_real,I2: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_lw(F)),aa_nat_set_nat(set_ord_atMost_nat,I2)) = aa_real_real(minus_minus_real(aa_nat_real(F,zero_zero_nat)),aa_nat_real(F,aa_nat_nat(suc,I2))) ).

% sum_telescope
tff(fact_9270_polyfun__eq__coeffs,axiom,
    ! [C: fun_nat_complex,Nb: nat,D: fun_nat_complex] :
      ( ! [X3: complex] : aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_fp(C),X3)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_fp(D),X3)),aa_nat_set_nat(set_ord_atMost_nat,Nb))
    <=> ! [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),Nb)
         => ( aa_nat_complex(C,I) = aa_nat_complex(D,I) ) ) ) ).

% polyfun_eq_coeffs
tff(fact_9271_polyfun__eq__coeffs,axiom,
    ! [C: fun_nat_real,Nb: nat,D: fun_nat_real] :
      ( ! [X3: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_fn(C),X3)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_fn(D),X3)),aa_nat_set_nat(set_ord_atMost_nat,Nb))
    <=> ! [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),Nb)
         => ( aa_nat_real(C,I) = aa_nat_real(D,I) ) ) ) ).

% polyfun_eq_coeffs
tff(fact_9272_bounded__imp__summable,axiom,
    ! [A: fun_nat_int,B3: int] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(A,N))
     => ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3539618377306564664at_int(A,aa_nat_set_nat(set_ord_atMost_nat,N))),B3)
       => summable_int(A) ) ) ).

% bounded_imp_summable
tff(fact_9273_bounded__imp__summable,axiom,
    ! [A: fun_nat_nat,B3: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(A,N))
     => ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(A),aa_nat_set_nat(set_ord_atMost_nat,N))),B3)
       => summable_nat(A) ) ) ).

% bounded_imp_summable
tff(fact_9274_bounded__imp__summable,axiom,
    ! [A: fun_nat_real,B3: real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N))
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(A),aa_nat_set_nat(set_ord_atMost_nat,N))),B3)
       => summable_real(A) ) ) ).

% bounded_imp_summable
tff(fact_9275_atMost__def,axiom,
    ! [U: set_nat] : set_or4236626031148496127et_nat(U) = collect_set_nat(aTP_Lamm_lx(U)) ).

% atMost_def
tff(fact_9276_atMost__def,axiom,
    ! [U: set_int] : set_or58775011639299419et_int(U) = collect_set_int(aTP_Lamm_ly(U)) ).

% atMost_def
tff(fact_9277_atMost__def,axiom,
    ! [U: rat] : set_ord_atMost_rat(U) = collect_rat(aa_rat_fun_rat_o(aTP_Lamm_lz,U)) ).

% atMost_def
tff(fact_9278_atMost__def,axiom,
    ! [U: num] : set_ord_atMost_num(U) = collect_num(aa_num_fun_num_o(aTP_Lamm_ma,U)) ).

% atMost_def
tff(fact_9279_atMost__def,axiom,
    ! [U: nat] : aa_nat_set_nat(set_ord_atMost_nat,U) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_am,U)) ).

% atMost_def
tff(fact_9280_atMost__def,axiom,
    ! [U: int] : set_ord_atMost_int(U) = collect_int(aa_int_fun_int_o(aTP_Lamm_mb,U)) ).

% atMost_def
tff(fact_9281_sum__choose__lower,axiom,
    ! [R: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_mc,R)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,R),Nb))),Nb) ).

% sum_choose_lower
tff(fact_9282_choose__rising__sum_I2_J,axiom,
    ! [Nb: nat,Ma: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_md,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Ma)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Ma)),one_one_nat)),Ma) ).

% choose_rising_sum(2)
tff(fact_9283_choose__rising__sum_I1_J,axiom,
    ! [Nb: nat,Ma: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_md,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Ma)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Ma)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)) ).

% choose_rising_sum(1)
tff(fact_9284_polyfun__eq__0,axiom,
    ! [C: fun_nat_complex,Nb: nat] :
      ( ! [X3: complex] : aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_fp(C),X3)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_complex
    <=> ! [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),Nb)
         => ( aa_nat_complex(C,I) = zero_zero_complex ) ) ) ).

% polyfun_eq_0
tff(fact_9285_polyfun__eq__0,axiom,
    ! [C: fun_nat_real,Nb: nat] :
      ( ! [X3: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_fn(C),X3)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_real
    <=> ! [I: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),Nb)
         => ( aa_nat_real(C,I) = zero_zero_real ) ) ) ).

% polyfun_eq_0
tff(fact_9286_zero__polynom__imp__zero__coeffs,axiom,
    ! [C: fun_nat_complex,Nb: nat,K2: nat] :
      ( ! [W2: complex] : aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_fp(C),W2)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_complex
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
       => ( aa_nat_complex(C,K2) = zero_zero_complex ) ) ) ).

% zero_polynom_imp_zero_coeffs
tff(fact_9287_zero__polynom__imp__zero__coeffs,axiom,
    ! [C: fun_nat_real,Nb: nat,K2: nat] :
      ( ! [W2: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_fn(C),W2)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_real
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
       => ( aa_nat_real(C,K2) = zero_zero_real ) ) ) ).

% zero_polynom_imp_zero_coeffs
tff(fact_9288_gbinomial__parallel__sum,axiom,
    ! [A: complex,Nb: nat] : aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_me,A)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,Nb))),one_one_complex)),Nb) ).

% gbinomial_parallel_sum
tff(fact_9289_gbinomial__parallel__sum,axiom,
    ! [A: rat,Nb: nat] : groups2906978787729119204at_rat(aTP_Lamm_mf(A),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(A),aa_nat_rat(semiri681578069525770553at_rat,Nb))),one_one_rat)),Nb) ).

% gbinomial_parallel_sum
tff(fact_9290_gbinomial__parallel__sum,axiom,
    ! [A: real,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_mg,A)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(A),aa_nat_real(semiri5074537144036343181t_real,Nb))),one_one_real)),Nb) ).

% gbinomial_parallel_sum
tff(fact_9291_sum__choose__diagonal,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_mh(Ma),Nb)),aa_nat_set_nat(set_ord_atMost_nat,Ma)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),Ma) ) ) ).

% sum_choose_diagonal
tff(fact_9292_vandermonde,axiom,
    ! [Ma: nat,Nb: nat,R: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_mi(Ma),Nb),R)),aa_nat_set_nat(set_ord_atMost_nat,R)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb)),R) ).

% vandermonde
tff(fact_9293_sum__gp__basic,axiom,
    ! [X: complex,Nb: nat] : aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(one_one_complex),X)),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(power_power_complex,X)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,X),aa_nat_nat(suc,Nb))) ).

% sum_gp_basic
tff(fact_9294_sum__gp__basic,axiom,
    ! [X: rat,Nb: nat] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(one_one_rat),X)),groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_rat_rat(minus_minus_rat(one_one_rat),aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,Nb))) ).

% sum_gp_basic
tff(fact_9295_sum__gp__basic,axiom,
    ! [X: int,Nb: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),X)),groups3539618377306564664at_int(aa_int_fun_nat_int(power_power_int,X),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_nat_int(aa_int_fun_nat_int(power_power_int,X),aa_nat_nat(suc,Nb))) ).

% sum_gp_basic
tff(fact_9296_sum__gp__basic,axiom,
    ! [X: real,Nb: nat] : aa_real_real(times_times_real(aa_real_real(minus_minus_real(one_one_real),X)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_nat(suc,Nb))) ).

% sum_gp_basic
tff(fact_9297_suminf__sum,axiom,
    ! [I5: set_complex,F: fun_co8378349014452122532t_real] :
      ( ! [I3: complex] :
          ( member_complex(I3,I5)
         => summable_real(aa_com20334259897932645t_real(F,I3)) )
     => ( suminf_real(aa_fun471735231769099483t_real(aTP_Lamm_kz(I5),F)) = groups5808333547571424918x_real(aTP_Lamm_mj(F),I5) ) ) ).

% suminf_sum
tff(fact_9298_suminf__sum,axiom,
    ! [I5: set_real,F: fun_re4238438383274954274t_real] :
      ( ! [I3: real] :
          ( member_real(I3,I5)
         => summable_real(aa_real_fun_nat_real(F,I3)) )
     => ( suminf_real(aa_fun628238740774635481t_real(aTP_Lamm_lb(I5),F)) = groups8097168146408367636l_real(aTP_Lamm_mk(F),I5) ) ) ).

% suminf_sum
tff(fact_9299_suminf__sum,axiom,
    ! [I5: set_int,F: fun_int_fun_nat_real] :
      ( ! [I3: int] :
          ( member_int(I3,I5)
         => summable_real(aa_int_fun_nat_real(F,I3)) )
     => ( suminf_real(aa_fun2842512154077127769t_real(aTP_Lamm_ld(I5),F)) = groups8778361861064173332t_real(aTP_Lamm_ml(F),I5) ) ) ).

% suminf_sum
tff(fact_9300_suminf__sum,axiom,
    ! [I5: set_real,F: fun_re2084383175013312036omplex] :
      ( ! [I3: real] :
          ( member_real(I3,I5)
         => summable_complex(aa_rea1531280491672885477omplex(F,I3)) )
     => ( suminf_complex(aa_fun7632768522510439005omplex(aTP_Lamm_lf(I5),F)) = groups5754745047067104278omplex(aTP_Lamm_mm(F),I5) ) ) ).

% suminf_sum
tff(fact_9301_suminf__sum,axiom,
    ! [I5: set_nat,F: fun_na1142814060743658952omplex] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),I5)
         => summable_complex(aa_nat6454746962644518025omplex(F,I3)) )
     => ( suminf_complex(aa_fun4886358607025424513omplex(aTP_Lamm_lh(I5),F)) = aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_mn(F)),I5) ) ) ).

% suminf_sum
tff(fact_9302_suminf__sum,axiom,
    ! [I5: set_int,F: fun_in8056368165505483812omplex] :
      ( ! [I3: int] :
          ( member_int(I3,I5)
         => summable_complex(aa_int4144929030551567077omplex(F,I3)) )
     => ( suminf_complex(aa_fun3543561798013398749omplex(aTP_Lamm_lj(I5),F)) = groups3049146728041665814omplex(aTP_Lamm_mo(F),I5) ) ) ).

% suminf_sum
tff(fact_9303_suminf__sum,axiom,
    ! [I5: set_nat,F: fun_nat_fun_nat_nat] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),I5)
         => summable_nat(aa_nat_fun_nat_nat(F,I3)) )
     => ( suminf_nat(aa_fun8466562714962205125at_nat(aTP_Lamm_ll(I5),F)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_mp(F)),I5) ) ) ).

% suminf_sum
tff(fact_9304_suminf__sum,axiom,
    ! [I5: set_complex,F: fun_co3258106519541737638omplex] :
      ( ! [I3: complex] :
          ( member_complex(I3,I5)
         => summable_complex(aa_com8181001469867462503omplex(F,I3)) )
     => ( suminf_complex(aa_fun5003373910569599327omplex(aTP_Lamm_ln(I5),F)) = groups7754918857620584856omplex(aTP_Lamm_mq(F),I5) ) ) ).

% suminf_sum
tff(fact_9305_suminf__sum,axiom,
    ! [I5: set_nat,F: fun_nat_fun_nat_real] :
      ( ! [I3: nat] :
          ( aa_set_nat_o(member_nat(I3),I5)
         => summable_real(aa_nat_fun_nat_real(F,I3)) )
     => ( suminf_real(aa_fun7039663905532832253t_real(aTP_Lamm_lp(I5),F)) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_mr(F)),I5) ) ) ).

% suminf_sum
tff(fact_9306_suminf__sum,axiom,
    ! [I5: set_int,F: fun_int_fun_nat_int] :
      ( ! [I3: int] :
          ( member_int(I3,I5)
         => summable_int(aa_int_fun_nat_int(F,I3)) )
     => ( suminf_int(aa_fun8710705519803144921at_int(aTP_Lamm_lr(I5),F)) = groups4538972089207619220nt_int(aTP_Lamm_ms(F),I5) ) ) ).

% suminf_sum
tff(fact_9307_choose__row__sum,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(binomial,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ).

% choose_row_sum
tff(fact_9308_binomial,axiom,
    ! [A: nat,B: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),Nb) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_mt(A),B),Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% binomial
tff(fact_9309_summable__Cauchy__product,axiom,
    ! [A: fun_nat_complex,B: fun_nat_complex] :
      ( summable_real(aTP_Lamm_ev(A))
     => ( summable_real(aTP_Lamm_ev(B))
       => summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_mv(A),B)) ) ) ).

% summable_Cauchy_product
tff(fact_9310_summable__Cauchy__product,axiom,
    ! [A: fun_nat_real,B: fun_nat_real] :
      ( summable_real(aTP_Lamm_eu(A))
     => ( summable_real(aTP_Lamm_eu(B))
       => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_mx(A),B)) ) ) ).

% summable_Cauchy_product
tff(fact_9311_Cauchy__product,axiom,
    ! [A: fun_nat_complex,B: fun_nat_complex] :
      ( summable_real(aTP_Lamm_ev(A))
     => ( summable_real(aTP_Lamm_ev(B))
       => ( aa_complex_complex(times_times_complex(suminf_complex(A)),suminf_complex(B)) = suminf_complex(aa_fun2955221450537884980omplex(aTP_Lamm_mv(A),B)) ) ) ) ).

% Cauchy_product
tff(fact_9312_Cauchy__product,axiom,
    ! [A: fun_nat_real,B: fun_nat_real] :
      ( summable_real(aTP_Lamm_eu(A))
     => ( summable_real(aTP_Lamm_eu(B))
       => ( aa_real_real(times_times_real(suminf_real(A)),suminf_real(B)) = suminf_real(aa_fun1839206346889086768t_real(aTP_Lamm_mx(A),B)) ) ) ) ).

% Cauchy_product
tff(fact_9313_sum_Oin__pairs__0,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) = groups2906978787729119204at_rat(aTP_Lamm_my(G),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% sum.in_pairs_0
tff(fact_9314_sum_Oin__pairs__0,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) = groups3539618377306564664at_int(aTP_Lamm_mz(G),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% sum.in_pairs_0
tff(fact_9315_sum_Oin__pairs__0,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_na(G)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% sum.in_pairs_0
tff(fact_9316_sum_Oin__pairs__0,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_nb(G)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% sum.in_pairs_0
tff(fact_9317_polynomial__product,axiom,
    ! [Ma: nat,A: fun_nat_complex,Nb: nat,B: fun_nat_complex,X: complex] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),I3)
         => ( aa_nat_complex(A,I3) = zero_zero_complex ) )
     => ( ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),J2)
           => ( aa_nat_complex(B,J2) = zero_zero_complex ) )
       => ( aa_complex_complex(times_times_complex(aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_fp(A),X)),aa_nat_set_nat(set_ord_atMost_nat,Ma))),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_fp(B),X)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_fun3721018923249493343omplex(aTP_Lamm_nc(A),B),X)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb))) ) ) ) ).

% polynomial_product
tff(fact_9318_polynomial__product,axiom,
    ! [Ma: nat,A: fun_nat_rat,Nb: nat,B: fun_nat_rat,X: rat] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),I3)
         => ( aa_nat_rat(A,I3) = zero_zero_rat ) )
     => ( ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),J2)
           => ( aa_nat_rat(B,J2) = zero_zero_rat ) )
       => ( aa_rat_rat(times_times_rat(groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_nd(A),X),aa_nat_set_nat(set_ord_atMost_nat,Ma))),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_nd(B),X),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_fun5559679822142647773at_rat(aTP_Lamm_nf(A),B),X),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb))) ) ) ) ).

% polynomial_product
tff(fact_9319_polynomial__product,axiom,
    ! [Ma: nat,A: fun_nat_int,Nb: nat,B: fun_nat_int,X: int] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),I3)
         => ( aa_nat_int(A,I3) = zero_zero_int ) )
     => ( ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),J2)
           => ( aa_nat_int(B,J2) = zero_zero_int ) )
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_ng(A),X),aa_nat_set_nat(set_ord_atMost_nat,Ma))),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_ng(B),X),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = groups3539618377306564664at_int(aa_int_fun_nat_int(aa_fun2315716387381795033at_int(aTP_Lamm_ni(A),B),X),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb))) ) ) ) ).

% polynomial_product
tff(fact_9320_polynomial__product,axiom,
    ! [Ma: nat,A: fun_nat_real,Nb: nat,B: fun_nat_real,X: real] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),I3)
         => ( aa_nat_real(A,I3) = zero_zero_real ) )
     => ( ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),J2)
           => ( aa_nat_real(B,J2) = zero_zero_real ) )
       => ( aa_real_real(times_times_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_fn(A),X)),aa_nat_set_nat(set_ord_atMost_nat,Ma))),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_fn(B),X)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_fun4732485501138722137t_real(aTP_Lamm_nj(A),B),X)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb))) ) ) ) ).

% polynomial_product
tff(fact_9321_gbinomial__sum__lower__neg,axiom,
    ! [A: complex,Ma: nat] : aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_nk,A)),aa_nat_set_nat(set_ord_atMost_nat,Ma)) = aa_complex_complex(times_times_complex(aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Ma)),aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,aa_complex_complex(minus_minus_complex(A),one_one_complex)),Ma)) ).

% gbinomial_sum_lower_neg
tff(fact_9322_gbinomial__sum__lower__neg,axiom,
    ! [A: rat,Ma: nat] : groups2906978787729119204at_rat(aTP_Lamm_nl(A),aa_nat_set_nat(set_ord_atMost_nat,Ma)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Ma)),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),one_one_rat)),Ma)) ).

% gbinomial_sum_lower_neg
tff(fact_9323_gbinomial__sum__lower__neg,axiom,
    ! [A: real,Ma: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_nm,A)),aa_nat_set_nat(set_ord_atMost_nat,Ma)) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Ma)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(minus_minus_real(A),one_one_real)),Ma)) ).

% gbinomial_sum_lower_neg
tff(fact_9324_binomial__ring,axiom,
    ! [A: complex,B: complex,Nb: nat] : aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(plus_plus_complex(A),B)),Nb) = aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_nn(A),B),Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% binomial_ring
tff(fact_9325_binomial__ring,axiom,
    ! [A: rat,B: rat,Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(A),B)),Nb) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_no(A),B),Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% binomial_ring
tff(fact_9326_binomial__ring,axiom,
    ! [A: int,B: int,Nb: nat] : aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),Nb) = groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_np(A),B),Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% binomial_ring
tff(fact_9327_binomial__ring,axiom,
    ! [A: extended_enat,B: extended_enat,Nb: nat] : aa_nat_Extended_enat(power_8040749407984259932d_enat(plus_p3455044024723400733d_enat(A,B)),Nb) = groups7108830773950497114d_enat(aa_nat2025370463774033377d_enat(aa_Ext8984936454148280526d_enat(aTP_Lamm_nq(A),B),Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% binomial_ring
tff(fact_9328_binomial__ring,axiom,
    ! [A: nat,B: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),Nb) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_mt(A),B),Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% binomial_ring
tff(fact_9329_binomial__ring,axiom,
    ! [A: real,B: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(plus_plus_real(A),B)),Nb) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_nr(A),B),Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% binomial_ring
tff(fact_9330_polynomial__product__nat,axiom,
    ! [Ma: nat,A: fun_nat_nat,Nb: nat,B: fun_nat_nat,X: nat] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),I3)
         => ( aa_nat_nat(A,I3) = zero_zero_nat ) )
     => ( ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),J2)
           => ( aa_nat_nat(B,J2) = zero_zero_nat ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ns(A),X)),aa_nat_set_nat(set_ord_atMost_nat,Ma))),aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ns(B),X)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_nu(A),B),X)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb))) ) ) ) ).

% polynomial_product_nat
tff(fact_9331_choose__square__sum,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_nv,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),Nb) ).

% choose_square_sum
tff(fact_9332_pochhammer__binomial__sum,axiom,
    ! [A: rat,B: rat,Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(A),B)),Nb) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_nw(A),B),Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% pochhammer_binomial_sum
tff(fact_9333_pochhammer__binomial__sum,axiom,
    ! [A: int,B: int,Nb: nat] : aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),Nb) = groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_nx(A),B),Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% pochhammer_binomial_sum
tff(fact_9334_pochhammer__binomial__sum,axiom,
    ! [A: real,B: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(plus_plus_real(A),B)),Nb) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ny(A),B),Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% pochhammer_binomial_sum
tff(fact_9335_Cauchy__product__sums,axiom,
    ! [A: fun_nat_complex,B: fun_nat_complex] :
      ( summable_real(aTP_Lamm_ev(A))
     => ( summable_real(aTP_Lamm_ev(B))
       => aa_complex_o(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_mv(A),B)),aa_complex_complex(times_times_complex(suminf_complex(A)),suminf_complex(B))) ) ) ).

% Cauchy_product_sums
tff(fact_9336_Cauchy__product__sums,axiom,
    ! [A: fun_nat_real,B: fun_nat_real] :
      ( summable_real(aTP_Lamm_eu(A))
     => ( summable_real(aTP_Lamm_eu(B))
       => aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_mx(A),B)),aa_real_real(times_times_real(suminf_real(A)),suminf_real(B))) ) ) ).

% Cauchy_product_sums
tff(fact_9337_sum__power__add,axiom,
    ! [X: complex,Ma: nat,I5: set_nat] : aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aTP_Lamm_nz(X),Ma)),I5) = aa_complex_complex(times_times_complex(aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,X),Ma)),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(power_power_complex,X)),I5)) ).

% sum_power_add
tff(fact_9338_sum__power__add,axiom,
    ! [X: rat,Ma: nat,I5: set_nat] : groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aTP_Lamm_oa(X),Ma),I5) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(X),Ma)),groups2906978787729119204at_rat(power_power_rat(X),I5)) ).

% sum_power_add
tff(fact_9339_sum__power__add,axiom,
    ! [X: int,Ma: nat,I5: set_nat] : groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_ob(X),Ma),I5) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,X),Ma)),groups3539618377306564664at_int(aa_int_fun_nat_int(power_power_int,X),I5)) ).

% sum_power_add
tff(fact_9340_sum__power__add,axiom,
    ! [X: real,Ma: nat,I5: set_nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_oc(X),Ma)),I5) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Ma)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X)),I5)) ).

% sum_power_add
tff(fact_9341_sum_Ozero__middle,axiom,
    ! [P2: nat,K2: nat,G: fun_nat_complex,H: fun_nat_complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),P2)
       => ( aa_set_nat_complex(groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_od(K2),G),H)),aa_nat_set_nat(set_ord_atMost_nat,P2)) = aa_set_nat_complex(groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_oe(K2),G),H)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_9342_sum_Ozero__middle,axiom,
    ! [P2: nat,K2: nat,G: fun_nat_rat,H: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),P2)
       => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_of(K2),G),H),aa_nat_set_nat(set_ord_atMost_nat,P2)) = groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_og(K2),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_9343_sum_Ozero__middle,axiom,
    ! [P2: nat,K2: nat,G: fun_nat_int,H: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),P2)
       => ( groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_oh(K2),G),H),aa_nat_set_nat(set_ord_atMost_nat,P2)) = groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_oi(K2),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_9344_sum_Ozero__middle,axiom,
    ! [P2: nat,K2: nat,G: fun_nat_nat,H: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),P2)
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_oj(K2),G),H)),aa_nat_set_nat(set_ord_atMost_nat,P2)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_ok(K2),G),H)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_9345_sum_Ozero__middle,axiom,
    ! [P2: nat,K2: nat,G: fun_nat_real,H: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),P2)
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_ol(K2),G),H)),aa_nat_set_nat(set_ord_atMost_nat,P2)) = aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_om(K2),G),H)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P2),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_9346_gbinomial__partial__sum__poly,axiom,
    ! [Ma: nat,A: complex,X: complex,Y2: complex] : aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_on(Ma),A),X),Y2)),aa_nat_set_nat(set_ord_atMost_nat,Ma)) = aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_oo(Ma),A),X),Y2)),aa_nat_set_nat(set_ord_atMost_nat,Ma)) ).

% gbinomial_partial_sum_poly
tff(fact_9347_gbinomial__partial__sum__poly,axiom,
    ! [Ma: nat,A: rat,X: rat,Y2: rat] : groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_op(Ma),A),X),Y2),aa_nat_set_nat(set_ord_atMost_nat,Ma)) = groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_oq(Ma),A),X),Y2),aa_nat_set_nat(set_ord_atMost_nat,Ma)) ).

% gbinomial_partial_sum_poly
tff(fact_9348_gbinomial__partial__sum__poly,axiom,
    ! [Ma: nat,A: real,X: real,Y2: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_or(Ma),A),X),Y2)),aa_nat_set_nat(set_ord_atMost_nat,Ma)) = aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_os(Ma),A),X),Y2)),aa_nat_set_nat(set_ord_atMost_nat,Ma)) ).

% gbinomial_partial_sum_poly
tff(fact_9349_exp__series__add__commuting,axiom,
    ! [X: real,Y2: real,Nb: nat] :
      ( ( aa_real_real(times_times_real(X),Y2) = aa_real_real(times_times_real(Y2),X) )
     => ( aa_real_real(real_V1485227260804924795R_real(aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(plus_plus_real(X),Y2)),Nb)) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ot(X),Y2),Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ) ) ).

% exp_series_add_commuting
tff(fact_9350_exp__series__add__commuting,axiom,
    ! [X: complex,Y2: complex,Nb: nat] :
      ( ( aa_complex_complex(times_times_complex(X),Y2) = aa_complex_complex(times_times_complex(Y2),X) )
     => ( aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(plus_plus_complex(X),Y2)),Nb)) = aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_ou(X),Y2),Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ) ) ).

% exp_series_add_commuting
tff(fact_9351_root__polyfun,axiom,
    ! [Nb: nat,Z2: int,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => ( ( aa_nat_int(aa_int_fun_nat_int(power_power_int,Z2),Nb) = A )
      <=> ( groups3539618377306564664at_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aTP_Lamm_ov(Nb),Z2),A),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_int ) ) ) ).

% root_polyfun
tff(fact_9352_root__polyfun,axiom,
    ! [Nb: nat,Z2: complex,A: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => ( ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Z2),Nb) = A )
      <=> ( aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aTP_Lamm_ow(Nb),Z2),A)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_complex ) ) ) ).

% root_polyfun
tff(fact_9353_root__polyfun,axiom,
    ! [Nb: nat,Z2: rat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => ( ( aa_nat_rat(power_power_rat(Z2),Nb) = A )
      <=> ( groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aTP_Lamm_ox(Nb),Z2),A),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_rat ) ) ) ).

% root_polyfun
tff(fact_9354_root__polyfun,axiom,
    ! [Nb: nat,Z2: code_integer,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => ( ( aa_nat_Code_integer(power_8256067586552552935nteger(Z2),Nb) = A )
      <=> ( groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aa_Cod6397266357770349983nteger(aTP_Lamm_oy(Nb),Z2),A),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_z3403309356797280102nteger ) ) ) ).

% root_polyfun
tff(fact_9355_root__polyfun,axiom,
    ! [Nb: nat,Z2: real,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Z2),Nb) = A )
      <=> ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_oz(Nb),Z2),A)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_real ) ) ) ).

% root_polyfun
tff(fact_9356_sum__gp0,axiom,
    ! [X: complex,Nb: nat] :
      aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(power_power_complex,X)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = $ite(X = one_one_complex,aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,X),aa_nat_nat(suc,Nb)))),aa_complex_complex(minus_minus_complex(one_one_complex),X))) ).

% sum_gp0
tff(fact_9357_sum__gp0,axiom,
    ! [X: rat,Nb: nat] :
      groups2906978787729119204at_rat(power_power_rat(X),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = $ite(X = one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(one_one_rat),aa_nat_rat(power_power_rat(X),aa_nat_nat(suc,Nb)))),aa_rat_rat(minus_minus_rat(one_one_rat),X))) ).

% sum_gp0
tff(fact_9358_sum__gp0,axiom,
    ! [X: real,Nb: nat] :
      aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,X)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = $ite(X = one_one_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_nat(suc,Nb)))),aa_real_real(minus_minus_real(one_one_real),X))) ).

% sum_gp0
tff(fact_9359_choose__alternating__linear__sum,axiom,
    ! [Nb: nat] :
      ( ( Nb != one_one_nat )
     => ( aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aTP_Lamm_pa,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_complex ) ) ).

% choose_alternating_linear_sum
tff(fact_9360_choose__alternating__linear__sum,axiom,
    ! [Nb: nat] :
      ( ( Nb != one_one_nat )
     => ( groups2906978787729119204at_rat(aTP_Lamm_pb(Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_rat ) ) ).

% choose_alternating_linear_sum
tff(fact_9361_choose__alternating__linear__sum,axiom,
    ! [Nb: nat] :
      ( ( Nb != one_one_nat )
     => ( groups7501900531339628137nteger(aTP_Lamm_pc(Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_z3403309356797280102nteger ) ) ).

% choose_alternating_linear_sum
tff(fact_9362_choose__alternating__linear__sum,axiom,
    ! [Nb: nat] :
      ( ( Nb != one_one_nat )
     => ( groups3539618377306564664at_int(aTP_Lamm_pd(Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_int ) ) ).

% choose_alternating_linear_sum
tff(fact_9363_choose__alternating__linear__sum,axiom,
    ! [Nb: nat] :
      ( ( Nb != one_one_nat )
     => ( aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_pe,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_real ) ) ).

% choose_alternating_linear_sum
tff(fact_9364_gbinomial__sum__nat__pow2,axiom,
    ! [Ma: nat] : aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aTP_Lamm_pf,Ma)),aa_nat_set_nat(set_ord_atMost_nat,Ma)) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),Ma) ).

% gbinomial_sum_nat_pow2
tff(fact_9365_gbinomial__sum__nat__pow2,axiom,
    ! [Ma: nat] : groups2906978787729119204at_rat(aTP_Lamm_pg(Ma),aa_nat_set_nat(set_ord_atMost_nat,Ma)) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Ma) ).

% gbinomial_sum_nat_pow2
tff(fact_9366_gbinomial__sum__nat__pow2,axiom,
    ! [Ma: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_ph,Ma)),aa_nat_set_nat(set_ord_atMost_nat,Ma)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Ma) ).

% gbinomial_sum_nat_pow2
tff(fact_9367_gbinomial__partial__sum__poly__xpos,axiom,
    ! [Ma: nat,A: complex,X: complex,Y2: complex] : aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_on(Ma),A),X),Y2)),aa_nat_set_nat(set_ord_atMost_nat,Ma)) = aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_pi(Ma),A),X),Y2)),aa_nat_set_nat(set_ord_atMost_nat,Ma)) ).

% gbinomial_partial_sum_poly_xpos
tff(fact_9368_gbinomial__partial__sum__poly__xpos,axiom,
    ! [Ma: nat,A: rat,X: rat,Y2: rat] : groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_op(Ma),A),X),Y2),aa_nat_set_nat(set_ord_atMost_nat,Ma)) = groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_pj(Ma),A),X),Y2),aa_nat_set_nat(set_ord_atMost_nat,Ma)) ).

% gbinomial_partial_sum_poly_xpos
tff(fact_9369_gbinomial__partial__sum__poly__xpos,axiom,
    ! [Ma: nat,A: real,X: real,Y2: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_or(Ma),A),X),Y2)),aa_nat_set_nat(set_ord_atMost_nat,Ma)) = aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_pk(Ma),A),X),Y2)),aa_nat_set_nat(set_ord_atMost_nat,Ma)) ).

% gbinomial_partial_sum_poly_xpos
tff(fact_9370_binomial__r__part__sum,axiom,
    ! [Ma: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma)),one_one_nat))),aa_nat_set_nat(set_ord_atMost_nat,Ma)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma)) ).

% binomial_r_part_sum
tff(fact_9371_choose__linear__sum,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_pl,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ).

% choose_linear_sum
tff(fact_9372_choose__alternating__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aTP_Lamm_pm,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_complex ) ) ).

% choose_alternating_sum
tff(fact_9373_choose__alternating__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( groups2906978787729119204at_rat(aTP_Lamm_pn(Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_rat ) ) ).

% choose_alternating_sum
tff(fact_9374_choose__alternating__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( groups7501900531339628137nteger(aTP_Lamm_po(Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_z3403309356797280102nteger ) ) ).

% choose_alternating_sum
tff(fact_9375_choose__alternating__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( groups3539618377306564664at_int(aTP_Lamm_pp(Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_int ) ) ).

% choose_alternating_sum
tff(fact_9376_choose__alternating__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_pq,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_real ) ) ).

% choose_alternating_sum
tff(fact_9377_polyfun__extremal__lemma,axiom,
    ! [E: real,C: fun_nat_complex,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
     => ? [M7: real] :
        ! [Z3: complex] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,M7),aa_complex_real(real_V1022390504157884413omplex,Z3))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_fp(C),Z3)),aa_nat_set_nat(set_ord_atMost_nat,Nb)))),aa_real_real(times_times_real(E),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z3)),aa_nat_nat(suc,Nb)))) ) ) ).

% polyfun_extremal_lemma
tff(fact_9378_polyfun__extremal__lemma,axiom,
    ! [E: real,C: fun_nat_real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
     => ? [M7: real] :
        ! [Z3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,M7),aa_real_real(real_V7735802525324610683m_real,Z3))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_fn(C),Z3)),aa_nat_set_nat(set_ord_atMost_nat,Nb)))),aa_real_real(times_times_real(E),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(real_V7735802525324610683m_real,Z3)),aa_nat_nat(suc,Nb)))) ) ) ).

% polyfun_extremal_lemma
tff(fact_9379_rat__abs__code,axiom,
    ! [P2: rat] : quotient_of(aa_rat_rat(abs_abs_rat,P2)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_pr),quotient_of(P2)) ).

% rat_abs_code
tff(fact_9380_gbinomial__r__part__sum,axiom,
    ! [Ma: nat] : aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(gbinomial_complex,aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_complex(semiri8010041392384452111omplex,Ma))),one_one_complex))),aa_nat_set_nat(set_ord_atMost_nat,Ma)) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma)) ).

% gbinomial_r_part_sum
tff(fact_9381_gbinomial__r__part__sum,axiom,
    ! [Ma: nat] : groups2906978787729119204at_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_nat_rat(semiri681578069525770553at_rat,Ma))),one_one_rat)),aa_nat_set_nat(set_ord_atMost_nat,Ma)) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma)) ).

% gbinomial_r_part_sum
tff(fact_9382_gbinomial__r__part__sum,axiom,
    ! [Ma: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Ma))),one_one_real))),aa_nat_set_nat(set_ord_atMost_nat,Ma)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma)) ).

% gbinomial_r_part_sum
tff(fact_9383_choose__odd__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aTP_Lamm_ps,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),Nb) ) ) ).

% choose_odd_sum
tff(fact_9384_choose__odd__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),groups2906978787729119204at_rat(aTP_Lamm_pt(Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Nb) ) ) ).

% choose_odd_sum
tff(fact_9385_choose__odd__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups3539618377306564664at_int(aTP_Lamm_pu(Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb) ) ) ).

% choose_odd_sum
tff(fact_9386_choose__odd__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_pv,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Nb) ) ) ).

% choose_odd_sum
tff(fact_9387_choose__even__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aTP_Lamm_pw,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),Nb) ) ) ).

% choose_even_sum
tff(fact_9388_choose__even__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),groups2906978787729119204at_rat(aTP_Lamm_px(Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Nb) ) ) ).

% choose_even_sum
tff(fact_9389_choose__even__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups3539618377306564664at_int(aTP_Lamm_py(Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb) ) ) ).

% choose_even_sum
tff(fact_9390_choose__even__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_pz,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Nb) ) ) ).

% choose_even_sum
tff(fact_9391_gbinomial__partial__row__sum,axiom,
    ! [A: complex,Ma: nat] : aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_qa,A)),aa_nat_set_nat(set_ord_atMost_nat,Ma)) = aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Ma)),one_one_complex)),numera6690914467698888265omplex(aa_num_num(bit0,one)))),aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),one_one_nat))) ).

% gbinomial_partial_row_sum
tff(fact_9392_gbinomial__partial__row__sum,axiom,
    ! [A: rat,Ma: nat] : groups2906978787729119204at_rat(aTP_Lamm_qb(A),aa_nat_set_nat(set_ord_atMost_nat,Ma)) = aa_rat_rat(times_times_rat(aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,Ma)),one_one_rat)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),one_one_nat))) ).

% gbinomial_partial_row_sum
tff(fact_9393_gbinomial__partial__row__sum,axiom,
    ! [A: real,Ma: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_qc,A)),aa_nat_set_nat(set_ord_atMost_nat,Ma)) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,Ma)),one_one_real)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),one_one_nat))) ).

% gbinomial_partial_row_sum
tff(fact_9394_rat__uminus__code,axiom,
    ! [P2: rat] : quotient_of(aa_rat_rat(uminus_uminus_rat,P2)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_qd),quotient_of(P2)) ).

% rat_uminus_code
tff(fact_9395_rat__less__code,axiom,
    ! [P2: rat,Q: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,P2),Q)
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aTP_Lamm_qf(Q)),quotient_of(P2)) ) ).

% rat_less_code
tff(fact_9396_rat__floor__code,axiom,
    ! [P2: rat] : aa_rat_int(archim3151403230148437115or_rat,P2) = aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(divide_divide_int),quotient_of(P2)) ).

% rat_floor_code
tff(fact_9397_rat__less__eq__code,axiom,
    ! [P2: rat,Q: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,P2),Q)
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aTP_Lamm_qh(Q)),quotient_of(P2)) ) ).

% rat_less_eq_code
tff(fact_9398_mask__eq__sum__exp,axiom,
    ! [Nb: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),one_one_int) = groups3539618377306564664at_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_qi,Nb))) ).

% mask_eq_sum_exp
tff(fact_9399_mask__eq__sum__exp,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),one_one_nat) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_qi,Nb))) ).

% mask_eq_sum_exp
tff(fact_9400_mask__eq__sum__exp__nat,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),aa_nat_nat(suc,zero_zero_nat)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_qi,Nb))) ).

% mask_eq_sum_exp_nat
tff(fact_9401_of__nat__id,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,Nb) = Nb ).

% of_nat_id
tff(fact_9402_sum__nth__roots,axiom,
    ! [Nb: nat,C: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Nb)
     => ( groups7754918857620584856omplex(aTP_Lamm_ao,collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_qj(Nb),C))) = zero_zero_complex ) ) ).

% sum_nth_roots
tff(fact_9403_sum__roots__unity,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Nb)
     => ( groups7754918857620584856omplex(aTP_Lamm_ao,collect_complex(aTP_Lamm_qk(Nb))) = zero_zero_complex ) ) ).

% sum_roots_unity
tff(fact_9404_Maclaurin__minus__cos__expansion,axiom,
    ! [Nb: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),zero_zero_real)
       => ? [T3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),T3)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,T3),zero_zero_real)
            & ( aa_real_real(cos_real,X) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ql,X)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(cos_real,aa_real_real(plus_plus_real(T3),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Nb))) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
tff(fact_9405_Maclaurin__cos__expansion2,axiom,
    ! [X: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ? [T3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T3)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,T3),X)
            & ( aa_real_real(cos_real,X) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ql,X)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(cos_real,aa_real_real(plus_plus_real(T3),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Nb))) ) ) ) ) ).

% Maclaurin_cos_expansion2
tff(fact_9406_Maclaurin__sin__expansion3,axiom,
    ! [Nb: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ? [T3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T3)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,T3),X)
            & ( aa_real_real(sin_real,X) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_qm,X)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(sin_real,aa_real_real(plus_plus_real(T3),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Nb))) ) ) ) ) ).

% Maclaurin_sin_expansion3
tff(fact_9407_Maclaurin__sin__expansion4,axiom,
    ! [X: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => ? [T3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T3)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T3),X)
          & ( aa_real_real(sin_real,X) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_qm,X)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(sin_real,aa_real_real(plus_plus_real(T3),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Nb))) ) ) ) ).

% Maclaurin_sin_expansion4
tff(fact_9408_sumr__cos__zero__one,axiom,
    ! [Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_qn),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = one_one_real ).

% sumr_cos_zero_one
tff(fact_9409_Maclaurin__lemma,axiom,
    ! [H: real,F: fun_real_real,J: fun_nat_real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H)
     => ? [B7: real] : aa_real_real(F,H) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_qo(H),J)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(B7),aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,H),Nb)),aa_nat_real(semiri2265585572941072030t_real,Nb)))) ) ).

% Maclaurin_lemma
tff(fact_9410_sum__split__even__odd,axiom,
    ! [F: fun_nat_real,G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_qp(F),G)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_qq(F)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_qr(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% sum_split_even_odd
tff(fact_9411_Maclaurin__exp__le,axiom,
    ! [X: real,Nb: nat] :
    ? [T3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T3)),aa_real_real(abs_abs_real,X))
      & ( aa_real_real(exp_real,X) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_qs,X)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(exp_real,T3)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Nb))) ) ) ).

% Maclaurin_exp_le
tff(fact_9412_Maclaurin__sin__bound,axiom,
    ! [X: real,Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_real_real(sin_real,X)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_qm,X)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))))),aa_real_real(times_times_real(aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,X)),Nb))) ).

% Maclaurin_sin_bound
tff(fact_9413_sum__pos__lt__pair,axiom,
    ! [F: fun_nat_real,K2: nat] :
      ( summable_real(F)
     => ( ! [D2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(plus_plus_real(aa_nat_real(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat))),D2)))),aa_nat_real(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat))),D2)),one_one_nat)))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,K2))),suminf_real(F)) ) ) ).

% sum_pos_lt_pair
tff(fact_9414_Maclaurin__exp__lt,axiom,
    ! [X: real,Nb: nat] :
      ( ( X != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ? [T3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T3))
            & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,T3)),aa_real_real(abs_abs_real,X))
            & ( aa_real_real(exp_real,X) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_qs,X)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(exp_real,T3)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Nb))) ) ) ) ) ).

% Maclaurin_exp_lt
tff(fact_9415_Maclaurin__sin__expansion,axiom,
    ! [X: real,Nb: nat] :
    ? [T3: real] : aa_real_real(sin_real,X) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_qm,X)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(sin_real,aa_real_real(plus_plus_real(T3),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Nb))) ).

% Maclaurin_sin_expansion
tff(fact_9416_Maclaurin__sin__expansion2,axiom,
    ! [X: real,Nb: nat] :
    ? [T3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T3)),aa_real_real(abs_abs_real,X))
      & ( aa_real_real(sin_real,X) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_qm,X)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(sin_real,aa_real_real(plus_plus_real(T3),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Nb))) ) ) ).

% Maclaurin_sin_expansion2
tff(fact_9417_Maclaurin__cos__expansion,axiom,
    ! [X: real,Nb: nat] :
    ? [T3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T3)),aa_real_real(abs_abs_real,X))
      & ( aa_real_real(cos_real,X) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ql,X)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(cos_real,aa_real_real(plus_plus_real(T3),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Nb))) ) ) ).

% Maclaurin_cos_expansion
tff(fact_9418_quotient__of__int,axiom,
    ! [A: int] : quotient_of(aa_int_rat(of_int,A)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),one_one_int) ).

% quotient_of_int
tff(fact_9419_bij__betw__roots__unity,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => bij_betw_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_qt,Nb),aa_nat_set_nat(set_ord_lessThan_nat,Nb),collect_complex(aTP_Lamm_qk(Nb))) ) ).

% bij_betw_roots_unity
tff(fact_9420_Frct__code__post_I5_J,axiom,
    ! [K2: num] : frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),aa_num_int(numeral_numeral_int,K2))) = aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,K2)) ).

% Frct_code_post(5)
tff(fact_9421_ex__nat__less,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ? [M3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M3),Nb)
          & aa_nat_o(P,M3) )
    <=> ? [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),set_or1269000886237332187st_nat(zero_zero_nat,Nb))
          & aa_nat_o(P,X3) ) ) ).

% ex_nat_less
tff(fact_9422_all__nat__less,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ! [M3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M3),Nb)
         => aa_nat_o(P,M3) )
    <=> ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),set_or1269000886237332187st_nat(zero_zero_nat,Nb))
         => aa_nat_o(P,X3) ) ) ).

% all_nat_less
tff(fact_9423_Frct__code__post_I3_J,axiom,
    frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),one_one_int)) = one_one_rat ).

% Frct_code_post(3)
tff(fact_9424_Frct__code__post_I4_J,axiom,
    ! [K2: num] : frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_num_int(numeral_numeral_int,K2)),one_one_int)) = aa_num_rat(numeral_numeral_rat,K2) ).

% Frct_code_post(4)
tff(fact_9425_gauss__sum__nat,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ai),set_or1269000886237332187st_nat(zero_zero_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(suc,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% gauss_sum_nat
tff(fact_9426_arith__series__nat,axiom,
    ! [A: nat,D: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_qu(A),D)),set_or1269000886237332187st_nat(zero_zero_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),D)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% arith_series_nat
tff(fact_9427_Sum__Icc__nat,axiom,
    ! [Ma: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ai),set_or1269000886237332187st_nat(Ma,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),one_one_nat)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Sum_Icc_nat
tff(fact_9428_rat__plus__code,axiom,
    ! [P2: rat,Q: rat] : quotient_of(aa_rat_rat(plus_plus_rat(P2),Q)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_qw(Q)),quotient_of(P2)) ).

% rat_plus_code
tff(fact_9429_rat__divide__code,axiom,
    ! [P2: rat,Q: rat] : quotient_of(aa_rat_rat(divide_divide_rat(P2),Q)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_qy(Q)),quotient_of(P2)) ).

% rat_divide_code
tff(fact_9430_rat__times__code,axiom,
    ! [P2: rat,Q: rat] : quotient_of(aa_rat_rat(times_times_rat(P2),Q)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_ra(Q)),quotient_of(P2)) ).

% rat_times_code
tff(fact_9431_normalize__denom__zero,axiom,
    ! [P2: int] : aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),zero_zero_int)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int) ).

% normalize_denom_zero
tff(fact_9432_aset_I2_J,axiom,
    ! [D3: int,A2: set_int,P: fun_int_o,Q2: fun_int_o] :
      ( ! [X4: int] :
          ( ! [Xa3: int] :
              ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D3))
             => ! [Xb2: int] :
                  ( member_int(Xb2,A2)
                 => ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa3) ) ) )
         => ( aa_int_o(P,X4)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D3)) ) )
     => ( ! [X4: int] :
            ( ! [Xa3: int] :
                ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb2: int] :
                    ( member_int(Xb2,A2)
                   => ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa3) ) ) )
           => ( aa_int_o(Q2,X4)
             => aa_int_o(Q2,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D3)) ) )
       => ! [X2: int] :
            ( ! [Xa4: int] :
                ( member_int(Xa4,set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb3: int] :
                    ( member_int(Xb3,A2)
                   => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
           => ( ( aa_int_o(P,X2)
                | aa_int_o(Q2,X2) )
             => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D3))
                | aa_int_o(Q2,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D3)) ) ) ) ) ) ).

% aset(2)
tff(fact_9433_aset_I1_J,axiom,
    ! [D3: int,A2: set_int,P: fun_int_o,Q2: fun_int_o] :
      ( ! [X4: int] :
          ( ! [Xa3: int] :
              ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D3))
             => ! [Xb2: int] :
                  ( member_int(Xb2,A2)
                 => ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa3) ) ) )
         => ( aa_int_o(P,X4)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D3)) ) )
     => ( ! [X4: int] :
            ( ! [Xa3: int] :
                ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb2: int] :
                    ( member_int(Xb2,A2)
                   => ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa3) ) ) )
           => ( aa_int_o(Q2,X4)
             => aa_int_o(Q2,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D3)) ) )
       => ! [X2: int] :
            ( ! [Xa4: int] :
                ( member_int(Xa4,set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb3: int] :
                    ( member_int(Xb3,A2)
                   => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
           => ( ( aa_int_o(P,X2)
                & aa_int_o(Q2,X2) )
             => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D3))
                & aa_int_o(Q2,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D3)) ) ) ) ) ) ).

% aset(1)
tff(fact_9434_bset_I2_J,axiom,
    ! [D3: int,B3: set_int,P: fun_int_o,Q2: fun_int_o] :
      ( ! [X4: int] :
          ( ! [Xa3: int] :
              ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D3))
             => ! [Xb2: int] :
                  ( member_int(Xb2,B3)
                 => ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa3) ) ) )
         => ( aa_int_o(P,X4)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D3)) ) )
     => ( ! [X4: int] :
            ( ! [Xa3: int] :
                ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb2: int] :
                    ( member_int(Xb2,B3)
                   => ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa3) ) ) )
           => ( aa_int_o(Q2,X4)
             => aa_int_o(Q2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D3)) ) )
       => ! [X2: int] :
            ( ! [Xa4: int] :
                ( member_int(Xa4,set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb3: int] :
                    ( member_int(Xb3,B3)
                   => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
           => ( ( aa_int_o(P,X2)
                | aa_int_o(Q2,X2) )
             => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D3))
                | aa_int_o(Q2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D3)) ) ) ) ) ) ).

% bset(2)
tff(fact_9435_bset_I1_J,axiom,
    ! [D3: int,B3: set_int,P: fun_int_o,Q2: fun_int_o] :
      ( ! [X4: int] :
          ( ! [Xa3: int] :
              ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D3))
             => ! [Xb2: int] :
                  ( member_int(Xb2,B3)
                 => ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa3) ) ) )
         => ( aa_int_o(P,X4)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D3)) ) )
     => ( ! [X4: int] :
            ( ! [Xa3: int] :
                ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb2: int] :
                    ( member_int(Xb2,B3)
                   => ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa3) ) ) )
           => ( aa_int_o(Q2,X4)
             => aa_int_o(Q2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D3)) ) )
       => ! [X2: int] :
            ( ! [Xa4: int] :
                ( member_int(Xa4,set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb3: int] :
                    ( member_int(Xb3,B3)
                   => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
           => ( ( aa_int_o(P,X2)
                & aa_int_o(Q2,X2) )
             => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D3))
                & aa_int_o(Q2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D3)) ) ) ) ) ) ).

% bset(1)
tff(fact_9436_aset_I10_J,axiom,
    ! [D: int,D3: int,A2: set_int,T: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D3)
     => ! [X2: int] :
          ( ! [Xa4: int] :
              ( member_int(Xa4,set_or1266510415728281911st_int(one_one_int,D3))
             => ! [Xb3: int] :
                  ( member_int(Xb3,A2)
                 => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
         => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),T))
           => ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D3)),T)) ) ) ) ).

% aset(10)
tff(fact_9437_aset_I9_J,axiom,
    ! [D: int,D3: int,A2: set_int,T: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D3)
     => ! [X2: int] :
          ( ! [Xa4: int] :
              ( member_int(Xa4,set_or1266510415728281911st_int(one_one_int,D3))
             => ! [Xb3: int] :
                  ( member_int(Xb3,A2)
                 => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),T))
           => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D3)),T)) ) ) ) ).

% aset(9)
tff(fact_9438_bset_I10_J,axiom,
    ! [D: int,D3: int,B3: set_int,T: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D3)
     => ! [X2: int] :
          ( ! [Xa4: int] :
              ( member_int(Xa4,set_or1266510415728281911st_int(one_one_int,D3))
             => ! [Xb3: int] :
                  ( member_int(Xb3,B3)
                 => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
         => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),T))
           => ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D3)),T)) ) ) ) ).

% bset(10)
tff(fact_9439_bset_I9_J,axiom,
    ! [D: int,D3: int,B3: set_int,T: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D3)
     => ! [X2: int] :
          ( ! [Xa4: int] :
              ( member_int(Xa4,set_or1266510415728281911st_int(one_one_int,D3))
             => ! [Xb3: int] :
                  ( member_int(Xb3,B3)
                 => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),T))
           => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D3)),T)) ) ) ) ).

% bset(9)
tff(fact_9440_periodic__finite__ex,axiom,
    ! [D: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ! [X4: int,K: int] :
            ( aa_int_o(P,X4)
          <=> aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K),D))) )
       => ( ? [X_12: int] : aa_int_o(P,X_12)
        <=> ? [X3: int] :
              ( member_int(X3,set_or1266510415728281911st_int(one_one_int,D))
              & aa_int_o(P,X3) ) ) ) ) ).

% periodic_finite_ex
tff(fact_9441_aset_I7_J,axiom,
    ! [D3: int,A2: set_int,T: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ! [X2: int] :
          ( ! [Xa4: int] :
              ( member_int(Xa4,set_or1266510415728281911st_int(one_one_int,D3))
             => ! [Xb3: int] :
                  ( member_int(Xb3,A2)
                 => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,T),X2)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,T),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D3)) ) ) ) ).

% aset(7)
tff(fact_9442_aset_I5_J,axiom,
    ! [D3: int,T: int,A2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ( member_int(T,A2)
       => ! [X2: int] :
            ( ! [Xa4: int] :
                ( member_int(Xa4,set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb3: int] :
                    ( member_int(Xb3,A2)
                   => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),T)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D3)),T) ) ) ) ) ).

% aset(5)
tff(fact_9443_aset_I4_J,axiom,
    ! [D3: int,T: int,A2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ( member_int(T,A2)
       => ! [X2: int] :
            ( ! [Xa4: int] :
                ( member_int(Xa4,set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb3: int] :
                    ( member_int(Xb3,A2)
                   => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
           => ( ( X2 != T )
             => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D3) != T ) ) ) ) ) ).

% aset(4)
tff(fact_9444_aset_I3_J,axiom,
    ! [D3: int,T: int,A2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ( member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,T),one_one_int),A2)
       => ! [X2: int] :
            ( ! [Xa4: int] :
                ( member_int(Xa4,set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb3: int] :
                    ( member_int(Xb3,A2)
                   => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
           => ( ( X2 = T )
             => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D3) = T ) ) ) ) ) ).

% aset(3)
tff(fact_9445_bset_I7_J,axiom,
    ! [D3: int,T: int,B3: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ( member_int(T,B3)
       => ! [X2: int] :
            ( ! [Xa4: int] :
                ( member_int(Xa4,set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb3: int] :
                    ( member_int(Xb3,B3)
                   => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,T),X2)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,T),aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D3)) ) ) ) ) ).

% bset(7)
tff(fact_9446_bset_I5_J,axiom,
    ! [D3: int,B3: set_int,T: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ! [X2: int] :
          ( ! [Xa4: int] :
              ( member_int(Xa4,set_or1266510415728281911st_int(one_one_int,D3))
             => ! [Xb3: int] :
                  ( member_int(Xb3,B3)
                 => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),T)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D3)),T) ) ) ) ).

% bset(5)
tff(fact_9447_bset_I4_J,axiom,
    ! [D3: int,T: int,B3: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ( member_int(T,B3)
       => ! [X2: int] :
            ( ! [Xa4: int] :
                ( member_int(Xa4,set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb3: int] :
                    ( member_int(Xb3,B3)
                   => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
           => ( ( X2 != T )
             => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D3) != T ) ) ) ) ) ).

% bset(4)
tff(fact_9448_bset_I3_J,axiom,
    ! [D3: int,T: int,B3: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ( member_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,T),one_one_int),B3)
       => ! [X2: int] :
            ( ! [Xa4: int] :
                ( member_int(Xa4,set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb3: int] :
                    ( member_int(Xb3,B3)
                   => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
           => ( ( X2 = T )
             => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D3) = T ) ) ) ) ) ).

% bset(3)
tff(fact_9449_normalize__crossproduct,axiom,
    ! [Q: int,S: int,P2: int,R: int] :
      ( ( Q != zero_zero_int )
     => ( ( S != zero_zero_int )
       => ( ( aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q)) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,R),S)) )
         => ( aa_int_int(aa_int_fun_int_int(times_times_int,P2),S) = aa_int_int(aa_int_fun_int_int(times_times_int,R),Q) ) ) ) ) ).

% normalize_crossproduct
tff(fact_9450_fact__eq__fact__times,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma)
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,Ma) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),groups708209901874060359at_nat(aTP_Lamm_ai,set_or1269000886237332187st_nat(aa_nat_nat(suc,Nb),Ma))) ) ) ).

% fact_eq_fact_times
tff(fact_9451_aset_I8_J,axiom,
    ! [D3: int,A2: set_int,T: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ! [X2: int] :
          ( ! [Xa4: int] :
              ( member_int(Xa4,set_or1266510415728281911st_int(one_one_int,D3))
             => ! [Xb3: int] :
                  ( member_int(Xb3,A2)
                 => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,T),X2)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,T),aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D3)) ) ) ) ).

% aset(8)
tff(fact_9452_aset_I6_J,axiom,
    ! [D3: int,T: int,A2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ( member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,T),one_one_int),A2)
       => ! [X2: int] :
            ( ! [Xa4: int] :
                ( member_int(Xa4,set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb3: int] :
                    ( member_int(Xb3,A2)
                   => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),T)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D3)),T) ) ) ) ) ).

% aset(6)
tff(fact_9453_bset_I8_J,axiom,
    ! [D3: int,T: int,B3: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ( member_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,T),one_one_int),B3)
       => ! [X2: int] :
            ( ! [Xa4: int] :
                ( member_int(Xa4,set_or1266510415728281911st_int(one_one_int,D3))
               => ! [Xb3: int] :
                    ( member_int(Xb3,B3)
                   => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,T),X2)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,T),aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D3)) ) ) ) ) ).

% bset(8)
tff(fact_9454_bset_I6_J,axiom,
    ! [D3: int,B3: set_int,T: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ! [X2: int] :
          ( ! [Xa4: int] :
              ( member_int(Xa4,set_or1266510415728281911st_int(one_one_int,D3))
             => ! [Xb3: int] :
                  ( member_int(Xb3,B3)
                 => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),T)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D3)),T) ) ) ) ).

% bset(6)
tff(fact_9455_cpmi,axiom,
    ! [D3: int,P: fun_int_o,P3: fun_int_o,B3: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ( ? [Z3: int] :
          ! [X4: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Z3)
           => ( aa_int_o(P,X4)
            <=> aa_int_o(P3,X4) ) )
       => ( ! [X4: int] :
              ( ! [Xa3: int] :
                  ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D3))
                 => ! [Xb2: int] :
                      ( member_int(Xb2,B3)
                     => ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa3) ) ) )
             => ( aa_int_o(P,X4)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D3)) ) )
         => ( ! [X4: int,K: int] :
                ( aa_int_o(P3,X4)
              <=> aa_int_o(P3,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K),D3))) )
           => ( ? [X_12: int] : aa_int_o(P,X_12)
            <=> ( ? [X3: int] :
                    ( member_int(X3,set_or1266510415728281911st_int(one_one_int,D3))
                    & aa_int_o(P3,X3) )
                | ? [X3: int] :
                    ( member_int(X3,set_or1266510415728281911st_int(one_one_int,D3))
                    & ? [Xa2: int] :
                        ( member_int(Xa2,B3)
                        & aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa2),X3)) ) ) ) ) ) ) ) ) ).

% cpmi
tff(fact_9456_cppi,axiom,
    ! [D3: int,P: fun_int_o,P3: fun_int_o,A2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D3)
     => ( ? [Z3: int] :
          ! [X4: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X4)
           => ( aa_int_o(P,X4)
            <=> aa_int_o(P3,X4) ) )
       => ( ! [X4: int] :
              ( ! [Xa3: int] :
                  ( member_int(Xa3,set_or1266510415728281911st_int(one_one_int,D3))
                 => ! [Xb2: int] :
                      ( member_int(Xb2,A2)
                     => ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa3) ) ) )
             => ( aa_int_o(P,X4)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D3)) ) )
         => ( ! [X4: int,K: int] :
                ( aa_int_o(P3,X4)
              <=> aa_int_o(P3,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K),D3))) )
           => ( ? [X_12: int] : aa_int_o(P,X_12)
            <=> ( ? [X3: int] :
                    ( member_int(X3,set_or1266510415728281911st_int(one_one_int,D3))
                    & aa_int_o(P3,X3) )
                | ? [X3: int] :
                    ( member_int(X3,set_or1266510415728281911st_int(one_one_int,D3))
                    & ? [Xa2: int] :
                        ( member_int(Xa2,A2)
                        & aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa2),X3)) ) ) ) ) ) ) ) ) ).

% cppi
tff(fact_9457_fact__div__fact,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Ma)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) = groups708209901874060359at_nat(aTP_Lamm_ai,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat),Ma)) ) ) ).

% fact_div_fact
tff(fact_9458_Sum__Icc__int,axiom,
    ! [Ma: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ma),Nb)
     => ( groups4538972089207619220nt_int(aTP_Lamm_ar,set_or1266510415728281911st_int(Ma,Nb)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,Nb),one_one_int))),aa_int_int(aa_int_fun_int_int(times_times_int,Ma),aa_int_int(aa_int_fun_int_int(minus_minus_int,Ma),one_one_int)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% Sum_Icc_int
tff(fact_9459_rat__minus__code,axiom,
    ! [P2: rat,Q: rat] : quotient_of(aa_rat_rat(minus_minus_rat(P2),Q)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_rc(Q)),quotient_of(P2)) ).

% rat_minus_code
tff(fact_9460_bij__betw__nth__root__unity,axiom,
    ! [C: complex,Nb: nat] :
      ( ( C != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => bij_be1856998921033663316omplex(times_times_complex(aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_nat_fun_real_real(root,Nb),aa_complex_real(real_V1022390504157884413omplex,C)))),aa_real_complex(cis,aa_real_real(divide_divide_real(aa_complex_real(arg,C)),aa_nat_real(semiri5074537144036343181t_real,Nb))))),collect_complex(aTP_Lamm_qk(Nb)),collect_complex(aa_nat_fun_complex_o(aTP_Lamm_rd(C),Nb))) ) ) ).

% bij_betw_nth_root_unity
tff(fact_9461_horner__sum__of__bool__2__less,axiom,
    ! [Bs: list_o] : aa_int_o(aa_int_fun_int_o(ord_less_int,groups9116527308978886569_o_int(zero_n2684676970156552555ol_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),Bs)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),size_size_list_o(Bs))) ).

% horner_sum_of_bool_2_less
tff(fact_9462_set__encode__def,axiom,
    nat_set_encode = groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% set_encode_def
tff(fact_9463_push__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se545348938243370406it_int(Nb),K2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2) ) ).

% push_bit_nonnegative_int_iff
tff(fact_9464_push__bit__negative__int__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se545348938243370406it_int(Nb),K2)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int) ) ).

% push_bit_negative_int_iff
tff(fact_9465_real__root__zero,axiom,
    ! [Nb: nat] : aa_real_real(aa_nat_fun_real_real(root,Nb),zero_zero_real) = zero_zero_real ).

% real_root_zero
tff(fact_9466_concat__bit__of__zero__1,axiom,
    ! [Nb: nat,L: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(Nb),zero_zero_int),L) = aa_int_int(bit_se545348938243370406it_int(Nb),L) ).

% concat_bit_of_zero_1
tff(fact_9467_real__root__Suc__0,axiom,
    ! [X: real] : aa_real_real(aa_nat_fun_real_real(root,aa_nat_nat(suc,zero_zero_nat)),X) = X ).

% real_root_Suc_0
tff(fact_9468_root__0,axiom,
    ! [X: real] : aa_real_real(aa_nat_fun_real_real(root,zero_zero_nat),X) = zero_zero_real ).

% root_0
tff(fact_9469_real__root__eq__iff,axiom,
    ! [Nb: nat,X: real,Y2: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_real_real(aa_nat_fun_real_real(root,Nb),X) = aa_real_real(aa_nat_fun_real_real(root,Nb),Y2) )
      <=> ( X = Y2 ) ) ) ).

% real_root_eq_iff
tff(fact_9470_real__root__eq__0__iff,axiom,
    ! [Nb: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_real_real(aa_nat_fun_real_real(root,Nb),X) = zero_zero_real )
      <=> ( X = zero_zero_real ) ) ) ).

% real_root_eq_0_iff
tff(fact_9471_real__root__less__iff,axiom,
    ! [Nb: nat,X: real,Y2: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Nb),X)),aa_real_real(aa_nat_fun_real_real(root,Nb),Y2))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2) ) ) ).

% real_root_less_iff
tff(fact_9472_real__root__le__iff,axiom,
    ! [Nb: nat,X: real,Y2: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Nb),X)),aa_real_real(aa_nat_fun_real_real(root,Nb),Y2))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y2) ) ) ).

% real_root_le_iff
tff(fact_9473_real__root__eq__1__iff,axiom,
    ! [Nb: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_real_real(aa_nat_fun_real_real(root,Nb),X) = one_one_real )
      <=> ( X = one_one_real ) ) ) ).

% real_root_eq_1_iff
tff(fact_9474_real__root__one,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(aa_nat_fun_real_real(root,Nb),one_one_real) = one_one_real ) ) ).

% real_root_one
tff(fact_9475_real__root__lt__0__iff,axiom,
    ! [Nb: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Nb),X)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),zero_zero_real) ) ) ).

% real_root_lt_0_iff
tff(fact_9476_real__root__gt__0__iff,axiom,
    ! [Nb: nat,Y2: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Y2))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y2) ) ) ).

% real_root_gt_0_iff
tff(fact_9477_real__root__le__0__iff,axiom,
    ! [Nb: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Nb),X)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),zero_zero_real) ) ) ).

% real_root_le_0_iff
tff(fact_9478_real__root__ge__0__iff,axiom,
    ! [Nb: nat,Y2: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Y2))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y2) ) ) ).

% real_root_ge_0_iff
tff(fact_9479_real__root__lt__1__iff,axiom,
    ! [Nb: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Nb),X)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,X),one_one_real) ) ) ).

% real_root_lt_1_iff
tff(fact_9480_real__root__gt__1__iff,axiom,
    ! [Nb: nat,Y2: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Y2))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Y2) ) ) ).

% real_root_gt_1_iff
tff(fact_9481_real__root__le__1__iff,axiom,
    ! [Nb: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Nb),X)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),one_one_real) ) ) ).

% real_root_le_1_iff
tff(fact_9482_real__root__ge__1__iff,axiom,
    ! [Nb: nat,Y2: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Y2))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Y2) ) ) ).

% real_root_ge_1_iff
tff(fact_9483_push__bit__of__Suc__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ).

% push_bit_of_Suc_0
tff(fact_9484_real__root__pow__pos2,axiom,
    ! [Nb: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),X)),Nb) = X ) ) ) ).

% real_root_pow_pos2
tff(fact_9485_real__root__inverse,axiom,
    ! [Nb: nat,X: real] : aa_real_real(aa_nat_fun_real_real(root,Nb),aa_real_real(inverse_inverse_real,X)) = aa_real_real(inverse_inverse_real,aa_real_real(aa_nat_fun_real_real(root,Nb),X)) ).

% real_root_inverse
tff(fact_9486_real__root__minus,axiom,
    ! [Nb: nat,X: real] : aa_real_real(aa_nat_fun_real_real(root,Nb),aa_real_real(uminus_uminus_real,X)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_nat_fun_real_real(root,Nb),X)) ).

% real_root_minus
tff(fact_9487_push__bit__nat__eq,axiom,
    ! [Nb: nat,K2: int] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),aa_int_nat(nat2,K2)) = aa_int_nat(nat2,aa_int_int(bit_se545348938243370406it_int(Nb),K2)) ).

% push_bit_nat_eq
tff(fact_9488_real__root__commute,axiom,
    ! [Ma: nat,Nb: nat,X: real] : aa_real_real(aa_nat_fun_real_real(root,Ma),aa_real_real(aa_nat_fun_real_real(root,Nb),X)) = aa_real_real(aa_nat_fun_real_real(root,Nb),aa_real_real(aa_nat_fun_real_real(root,Ma),X)) ).

% real_root_commute
tff(fact_9489_real__root__mult,axiom,
    ! [Nb: nat,X: real,Y2: real] : aa_real_real(aa_nat_fun_real_real(root,Nb),aa_real_real(times_times_real(X),Y2)) = aa_real_real(times_times_real(aa_real_real(aa_nat_fun_real_real(root,Nb),X)),aa_real_real(aa_nat_fun_real_real(root,Nb),Y2)) ).

% real_root_mult
tff(fact_9490_real__root__mult__exp,axiom,
    ! [Ma: nat,Nb: nat,X: real] : aa_real_real(aa_nat_fun_real_real(root,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Nb)),X) = aa_real_real(aa_nat_fun_real_real(root,Ma),aa_real_real(aa_nat_fun_real_real(root,Nb),X)) ).

% real_root_mult_exp
tff(fact_9491_real__root__divide,axiom,
    ! [Nb: nat,X: real,Y2: real] : aa_real_real(aa_nat_fun_real_real(root,Nb),aa_real_real(divide_divide_real(X),Y2)) = aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(root,Nb),X)),aa_real_real(aa_nat_fun_real_real(root,Nb),Y2)) ).

% real_root_divide
tff(fact_9492_real__root__pos__pos__le,axiom,
    ! [X: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Nb),X)) ) ).

% real_root_pos_pos_le
tff(fact_9493_prod__int__eq,axiom,
    ! [I2: nat,J: nat] : groups705719431365010083at_int(semiri1314217659103216013at_int,set_or1269000886237332187st_nat(I2,J)) = groups1705073143266064639nt_int(aTP_Lamm_ar,set_or1266510415728281911st_int(aa_nat_int(semiri1314217659103216013at_int,I2),aa_nat_int(semiri1314217659103216013at_int,J))) ).

% prod_int_eq
tff(fact_9494_flip__bit__nat__def,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,Ma),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Ma),one_one_nat)) ).

% flip_bit_nat_def
tff(fact_9495_real__root__less__mono,axiom,
    ! [Nb: nat,X: real,Y2: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Nb),X)),aa_real_real(aa_nat_fun_real_real(root,Nb),Y2)) ) ) ).

% real_root_less_mono
tff(fact_9496_real__root__le__mono,axiom,
    ! [Nb: nat,X: real,Y2: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Nb),X)),aa_real_real(aa_nat_fun_real_real(root,Nb),Y2)) ) ) ).

% real_root_le_mono
tff(fact_9497_real__root__power,axiom,
    ! [Nb: nat,X: real,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(aa_nat_fun_real_real(root,Nb),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),K2)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),X)),K2) ) ) ).

% real_root_power
tff(fact_9498_real__root__abs,axiom,
    ! [Nb: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(aa_nat_fun_real_real(root,Nb),aa_real_real(abs_abs_real,X)) = aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,Nb),X)) ) ) ).

% real_root_abs
tff(fact_9499_sgn__root,axiom,
    ! [Nb: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(sgn_sgn_real,aa_real_real(aa_nat_fun_real_real(root,Nb),X)) = aa_real_real(sgn_sgn_real,X) ) ) ).

% sgn_root
tff(fact_9500_bit__push__bit__iff__int,axiom,
    ! [Ma: nat,K2: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(bit_se545348938243370406it_int(Ma),K2)),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
        & aa_nat_o(bit_se1146084159140164899it_int(K2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Ma)) ) ) ).

% bit_push_bit_iff_int
tff(fact_9501_prod__int__plus__eq,axiom,
    ! [I2: nat,J: nat] : groups705719431365010083at_int(semiri1314217659103216013at_int,set_or1269000886237332187st_nat(I2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),J))) = groups1705073143266064639nt_int(aTP_Lamm_ar,set_or1266510415728281911st_int(aa_nat_int(semiri1314217659103216013at_int,I2),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),J)))) ).

% prod_int_plus_eq
tff(fact_9502_bit__push__bit__iff__nat,axiom,
    ! [Ma: nat,Q: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Ma),Q)),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
        & aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,Q),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Ma)) ) ) ).

% bit_push_bit_iff_nat
tff(fact_9503_concat__bit__eq,axiom,
    ! [Nb: nat,K2: int,L: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(Nb),K2),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(bit_se2923211474154528505it_int(Nb),K2)),aa_int_int(bit_se545348938243370406it_int(Nb),L)) ).

% concat_bit_eq
tff(fact_9504_flip__bit__int__def,axiom,
    ! [Nb: nat,K2: int] : aa_int_int(bit_se2159334234014336723it_int(Nb),K2) = aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K2),aa_int_int(bit_se545348938243370406it_int(Nb),one_one_int)) ).

% flip_bit_int_def
tff(fact_9505_real__root__gt__zero,axiom,
    ! [Nb: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Nb),X)) ) ) ).

% real_root_gt_zero
tff(fact_9506_real__root__strict__decreasing,axiom,
    ! [Nb: nat,N2: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N2),X)),aa_real_real(aa_nat_fun_real_real(root,Nb),X)) ) ) ) ).

% real_root_strict_decreasing
tff(fact_9507_sqrt__def,axiom,
    sqrt = aa_nat_fun_real_real(root,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% sqrt_def
tff(fact_9508_root__abs__power,axiom,
    ! [Nb: nat,Y2: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,Nb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),Nb))) = aa_real_real(abs_abs_real,Y2) ) ) ).

% root_abs_power
tff(fact_9509_unset__bit__int__def,axiom,
    ! [Nb: nat,K2: int] : aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Nb),K2) = aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K2),aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_se545348938243370406it_int(Nb),one_one_int))) ).

% unset_bit_int_def
tff(fact_9510_real__root__pos__pos,axiom,
    ! [Nb: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Nb),X)) ) ) ).

% real_root_pos_pos
tff(fact_9511_real__root__strict__increasing,axiom,
    ! [Nb: nat,N2: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),one_one_real)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Nb),X)),aa_real_real(aa_nat_fun_real_real(root,N2),X)) ) ) ) ) ).

% real_root_strict_increasing
tff(fact_9512_real__root__decreasing,axiom,
    ! [Nb: nat,N2: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),X)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N2),X)),aa_real_real(aa_nat_fun_real_real(root,Nb),X)) ) ) ) ).

% real_root_decreasing
tff(fact_9513_real__root__pow__pos,axiom,
    ! [Nb: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),X)),Nb) = X ) ) ) ).

% real_root_pow_pos
tff(fact_9514_real__root__power__cancel,axiom,
    ! [Nb: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
       => ( aa_real_real(aa_nat_fun_real_real(root,Nb),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Nb)) = X ) ) ) ).

% real_root_power_cancel
tff(fact_9515_real__root__pos__unique,axiom,
    ! [Nb: nat,Y2: real,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y2)
       => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),Nb) = X )
         => ( aa_real_real(aa_nat_fun_real_real(root,Nb),X) = Y2 ) ) ) ) ).

% real_root_pos_unique
tff(fact_9516_odd__real__root__power__cancel,axiom,
    ! [Nb: nat,X: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_real(aa_nat_fun_real_real(root,Nb),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Nb)) = X ) ) ).

% odd_real_root_power_cancel
tff(fact_9517_odd__real__root__unique,axiom,
    ! [Nb: nat,Y2: real,X: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y2),Nb) = X )
       => ( aa_real_real(aa_nat_fun_real_real(root,Nb),X) = Y2 ) ) ) ).

% odd_real_root_unique
tff(fact_9518_odd__real__root__pow,axiom,
    ! [Nb: nat,X: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),X)),Nb) = X ) ) ).

% odd_real_root_pow
tff(fact_9519_push__bit__int__def,axiom,
    ! [Nb: nat,K2: int] : aa_int_int(bit_se545348938243370406it_int(Nb),K2) = aa_int_int(aa_int_fun_int_int(times_times_int,K2),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% push_bit_int_def
tff(fact_9520_push__bit__nat__def,axiom,
    ! [Nb: nat,Ma: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),Ma) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% push_bit_nat_def
tff(fact_9521_real__root__increasing,axiom,
    ! [Nb: nat,N2: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),one_one_real)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Nb),X)),aa_real_real(aa_nat_fun_real_real(root,N2),X)) ) ) ) ) ).

% real_root_increasing
tff(fact_9522_sgn__power__root,axiom,
    ! [Nb: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,aa_real_real(aa_nat_fun_real_real(root,Nb),X))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,Nb),X))),Nb)) = X ) ) ).

% sgn_power_root
tff(fact_9523_root__sgn__power,axiom,
    ! [Nb: nat,Y2: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(aa_nat_fun_real_real(root,Nb),aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,Y2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Y2)),Nb))) = Y2 ) ) ).

% root_sgn_power
tff(fact_9524_push__bit__minus__one,axiom,
    ! [Nb: nat] : aa_int_int(bit_se545348938243370406it_int(Nb),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% push_bit_minus_one
tff(fact_9525_ln__root,axiom,
    ! [Nb: nat,B: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => ( aa_real_real(ln_ln_real,aa_real_real(aa_nat_fun_real_real(root,Nb),B)) = aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,B)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ) ).

% ln_root
tff(fact_9526_log__root,axiom,
    ! [Nb: nat,A: real,B: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => ( aa_real_real(log(B),aa_real_real(aa_nat_fun_real_real(root,Nb),A)) = aa_real_real(divide_divide_real(aa_real_real(log(B),A)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ) ).

% log_root
tff(fact_9527_log__base__root,axiom,
    ! [Nb: nat,B: real,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => ( aa_real_real(log(aa_real_real(aa_nat_fun_real_real(root,Nb),B)),X) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(log(B),X)) ) ) ) ).

% log_base_root
tff(fact_9528_split__root,axiom,
    ! [P: fun_real_o,Nb: nat,X: real] :
      ( aa_real_o(P,aa_real_real(aa_nat_fun_real_real(root,Nb),X))
    <=> ( ( ( Nb = zero_zero_nat )
         => aa_real_o(P,zero_zero_real) )
        & ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => ! [Y: real] :
              ( ( aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,Y)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Y)),Nb)) = X )
             => aa_real_o(P,Y) ) ) ) ) ).

% split_root
tff(fact_9529_root__powr__inverse,axiom,
    ! [Nb: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => ( aa_real_real(aa_nat_fun_real_real(root,Nb),X) = aa_real_real(powr_real(X),aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,Nb))) ) ) ) ).

% root_powr_inverse
tff(fact_9530_VEBT_Osize_I3_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] : aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,aa_VEB5832953974749077948T_VEBT(vEBT_Node(X11,X12,X13),X14)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_list_VEBT_VEBT(size_size_VEBT_VEBT,X13)),aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,X14))),aa_nat_nat(suc,zero_zero_nat)) ).

% VEBT.size(3)
tff(fact_9531_Sum__Ico__nat,axiom,
    ! [Ma: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ai),set_or4665077453230672383an_nat(Ma,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),one_one_nat)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Sum_Ico_nat
tff(fact_9532_sum__power2,axiom,
    ! [K2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),set_or4665077453230672383an_nat(zero_zero_nat,K2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K2)),one_one_nat) ).

% sum_power2
tff(fact_9533_length__upt,axiom,
    ! [I2: nat,J: nat] : size_size_list_nat(upt(I2,J)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I2) ).

% length_upt
tff(fact_9534_nth__upt,axiom,
    ! [I2: nat,K2: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2)),J)
     => ( aa_nat_nat(nth_nat(upt(I2,J)),K2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K2) ) ) ).

% nth_upt
tff(fact_9535_take__upt,axiom,
    ! [I2: nat,Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),Ma)),Nb)
     => ( take_nat(Ma,upt(I2,Nb)) = upt(I2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),Ma)) ) ) ).

% take_upt
tff(fact_9536_map__Suc__upt,axiom,
    ! [Ma: nat,Nb: nat] : map_nat_nat(suc,upt(Ma,Nb)) = upt(aa_nat_nat(suc,Ma),aa_nat_nat(suc,Nb)) ).

% map_Suc_upt
tff(fact_9537_atLeastLessThan__upt,axiom,
    ! [I2: nat,J: nat] : set_or4665077453230672383an_nat(I2,J) = set_nat2(upt(I2,J)) ).

% atLeastLessThan_upt
tff(fact_9538_all__nat__less__eq,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ! [M3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),Nb)
         => aa_nat_o(P,M3) )
    <=> ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),set_or4665077453230672383an_nat(zero_zero_nat,Nb))
         => aa_nat_o(P,X3) ) ) ).

% all_nat_less_eq
tff(fact_9539_ex__nat__less__eq,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ? [M3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),Nb)
          & aa_nat_o(P,M3) )
    <=> ? [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),set_or4665077453230672383an_nat(zero_zero_nat,Nb))
          & aa_nat_o(P,X3) ) ) ).

% ex_nat_less_eq
tff(fact_9540_atLeastAtMost__upt,axiom,
    ! [Nb: nat,Ma: nat] : set_or1269000886237332187st_nat(Nb,Ma) = set_nat2(upt(Nb,aa_nat_nat(suc,Ma))) ).

% atLeastAtMost_upt
tff(fact_9541_atLeast__upt,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_ord_lessThan_nat,Nb) = set_nat2(upt(zero_zero_nat,Nb)) ).

% atLeast_upt
tff(fact_9542_map__add__upt,axiom,
    ! [Nb: nat,Ma: nat] : map_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_re,Nb),upt(zero_zero_nat,Ma)) = upt(Nb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb)) ).

% map_add_upt
tff(fact_9543_atMost__upto,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_ord_atMost_nat,Nb) = set_nat2(upt(zero_zero_nat,aa_nat_nat(suc,Nb))) ).

% atMost_upto
tff(fact_9544_map__decr__upt,axiom,
    ! [Ma: nat,Nb: nat] : map_nat_nat(aTP_Lamm_rf,upt(aa_nat_nat(suc,Ma),aa_nat_nat(suc,Nb))) = upt(Ma,Nb) ).

% map_decr_upt
tff(fact_9545_Chebyshev__sum__upper__nat,axiom,
    ! [Nb: nat,A: fun_nat_nat,B: fun_nat_nat] :
      ( ! [I3: nat,J2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),J2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),Nb)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(A,I3)),aa_nat_nat(A,J2)) ) )
     => ( ! [I3: nat,J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),J2)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),Nb)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(B,J2)),aa_nat_nat(B,I3)) ) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_rg(A),B)),set_or4665077453230672383an_nat(zero_zero_nat,Nb)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(A),set_or4665077453230672383an_nat(zero_zero_nat,Nb))),aa_set_nat_nat(groups3542108847815614940at_nat(B),set_or4665077453230672383an_nat(zero_zero_nat,Nb)))) ) ) ).

% Chebyshev_sum_upper_nat
tff(fact_9546_VEBT_Osize__gen_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_size_VEBT,aa_VEB5832953974749077948T_VEBT(vEBT_Node(X11,X12,X13),X14)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_list_VEBT_VEBT(vEBT_size_VEBT,X13)),aa_VEBT_VEBT_nat(vEBT_size_VEBT,X14))),aa_nat_nat(suc,zero_zero_nat)) ).

% VEBT.size_gen(1)
tff(fact_9547_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : set_or4662586982721622107an_int(L,aa_int_int(aa_int_fun_int_int(plus_plus_int,U),one_one_int)) = set_or1266510415728281911st_int(L,U) ).

% atLeastLessThanPlusOne_atLeastAtMost_int
tff(fact_9548_VEBT_Osize__gen_I2_J,axiom,
    ! [X21: $o,X22: $o] : aa_VEBT_VEBT_nat(vEBT_size_VEBT,aa_o_VEBT_VEBT(vEBT_Leaf((X21)),(X22))) = zero_zero_nat ).

% VEBT.size_gen(2)
tff(fact_9549_Cauchy__iff2,axiom,
    ! [X7: fun_nat_real] :
      ( topolo4055970368930404560y_real(X7)
    <=> ! [J3: nat] :
        ? [M8: nat] :
        ! [M3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M8),M3)
         => ! [N4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M8),N4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_nat_real(X7,M3)),aa_nat_real(X7,N4)))),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,J3)))) ) ) ) ).

% Cauchy_iff2
tff(fact_9550_divmod__step__integer__def,axiom,
    ! [L: num,Qr: produc8923325533196201883nteger] : unique4921790084139445826nteger(L,Qr) = produc6916734918728496179nteger(aTP_Lamm_bt(L),Qr) ).

% divmod_step_integer_def
tff(fact_9551_csqrt_Osimps_I1_J,axiom,
    ! [Z2: complex] : aa_complex_real(re,aa_complex_complex(csqrt,Z2)) = aa_real_real(sqrt,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_complex_real(real_V1022390504157884413omplex,Z2)),aa_complex_real(re,Z2))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% csqrt.simps(1)
tff(fact_9552_times__integer__code_I1_J,axiom,
    ! [K2: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K2),zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% times_integer_code(1)
tff(fact_9553_times__integer__code_I2_J,axiom,
    ! [L: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_z3403309356797280102nteger),L) = zero_z3403309356797280102nteger ).

% times_integer_code(2)
tff(fact_9554_divmod__integer_H__def,axiom,
    ! [Ma: num,Nb: num] : unique3479559517661332726nteger(Ma,Nb) = produc1086072967326762835nteger(divide6298287555418463151nteger(numera6620942414471956472nteger(Ma),numera6620942414471956472nteger(Nb)),modulo364778990260209775nteger(numera6620942414471956472nteger(Ma),numera6620942414471956472nteger(Nb))) ).

% divmod_integer'_def
tff(fact_9555_less__eq__integer__code_I1_J,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),zero_z3403309356797280102nteger) ).

% less_eq_integer_code(1)
tff(fact_9556_sgn__integer__code,axiom,
    ! [K2: code_integer] :
      aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K2) = $ite(
        K2 = zero_z3403309356797280102nteger,
        zero_z3403309356797280102nteger,
        $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K2),zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),one_one_Code_integer) ) ).

% sgn_integer_code
tff(fact_9557_sums__Re,axiom,
    ! [X7: fun_nat_complex,A: complex] :
      ( aa_complex_o(sums_complex(X7),A)
     => aa_real_o(sums_real(aTP_Lamm_rh(X7)),aa_complex_real(re,A)) ) ).

% sums_Re
tff(fact_9558_complex__Re__le__cmod,axiom,
    ! [X: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(re,X)),aa_complex_real(real_V1022390504157884413omplex,X)) ).

% complex_Re_le_cmod
tff(fact_9559_one__complex_Osimps_I1_J,axiom,
    aa_complex_real(re,one_one_complex) = one_one_real ).

% one_complex.simps(1)
tff(fact_9560_scaleR__complex_Osimps_I1_J,axiom,
    ! [R: real,X: complex] : aa_complex_real(re,aa_complex_complex(real_V2046097035970521341omplex(R),X)) = aa_real_real(times_times_real(R),aa_complex_real(re,X)) ).

% scaleR_complex.simps(1)
tff(fact_9561_summable__Re,axiom,
    ! [F: fun_nat_complex] :
      ( summable_complex(F)
     => summable_real(aTP_Lamm_rh(F)) ) ).

% summable_Re
tff(fact_9562_abs__Re__le__cmod,axiom,
    ! [X: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_complex_real(re,X))),aa_complex_real(real_V1022390504157884413omplex,X)) ).

% abs_Re_le_cmod
tff(fact_9563_Re__csqrt,axiom,
    ! [Z2: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(csqrt,Z2))) ).

% Re_csqrt
tff(fact_9564_one__integer_Orsp,axiom,
    one_one_int = one_one_int ).

% one_integer.rsp
tff(fact_9565_one__natural_Orsp,axiom,
    one_one_nat = one_one_nat ).

% one_natural.rsp
tff(fact_9566_cmod__plus__Re__le__0__iff,axiom,
    ! [Z2: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_complex_real(real_V1022390504157884413omplex,Z2)),aa_complex_real(re,Z2))),zero_zero_real)
    <=> ( aa_complex_real(re,Z2) = aa_real_real(uminus_uminus_real,aa_complex_real(real_V1022390504157884413omplex,Z2)) ) ) ).

% cmod_plus_Re_le_0_iff
tff(fact_9567_cos__n__Re__cis__pow__n,axiom,
    ! [Nb: nat,A: real] : aa_real_real(cos_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),A)) = aa_complex_real(re,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_real_complex(cis,A)),Nb)) ).

% cos_n_Re_cis_pow_n
tff(fact_9568_csqrt_Ocode,axiom,
    ! [Z2: complex] :
      aa_complex_complex(csqrt,Z2) = aa_real_complex(complex2(aa_real_real(sqrt,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_complex_real(real_V1022390504157884413omplex,Z2)),aa_complex_real(re,Z2))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),
        aa_real_real(times_times_real($ite(aa_complex_real(im,Z2) = zero_zero_real,one_one_real,aa_real_real(sgn_sgn_real,aa_complex_real(im,Z2)))),aa_real_real(sqrt,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_complex_real(real_V1022390504157884413omplex,Z2)),aa_complex_real(re,Z2))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))) ).

% csqrt.code
tff(fact_9569_csqrt_Osimps_I2_J,axiom,
    ! [Z2: complex] :
      aa_complex_real(im,aa_complex_complex(csqrt,Z2)) = aa_real_real(times_times_real($ite(aa_complex_real(im,Z2) = zero_zero_real,one_one_real,aa_real_real(sgn_sgn_real,aa_complex_real(im,Z2)))),aa_real_real(sqrt,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_complex_real(real_V1022390504157884413omplex,Z2)),aa_complex_real(re,Z2))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% csqrt.simps(2)
tff(fact_9570_integer__of__int__code,axiom,
    ! [K2: int] :
      code_integer_of_int(K2) = $ite(
        aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int),
        aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,code_integer_of_int(aa_int_int(uminus_uminus_int,K2))),
        $ite(
          K2 = zero_zero_int,
          zero_z3403309356797280102nteger,
          $let(
            l: code_integer,
            l:= aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),code_integer_of_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,plus_p5714425477246183910nteger(l,one_one_Code_integer)) ) ) ) ).

% integer_of_int_code
tff(fact_9571_csqrt__of__real__nonpos,axiom,
    ! [X: complex] :
      ( ( aa_complex_real(im,X) = zero_zero_real )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(re,X)),zero_zero_real)
       => ( aa_complex_complex(csqrt,X) = aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sqrt,aa_real_real(abs_abs_real,aa_complex_real(re,X))))) ) ) ) ).

% csqrt_of_real_nonpos
tff(fact_9572_Im__i__times,axiom,
    ! [Z2: complex] : aa_complex_real(im,aa_complex_complex(times_times_complex(imaginary_unit),Z2)) = aa_complex_real(re,Z2) ).

% Im_i_times
tff(fact_9573_Re__i__times,axiom,
    ! [Z2: complex] : aa_complex_real(re,aa_complex_complex(times_times_complex(imaginary_unit),Z2)) = aa_real_real(uminus_uminus_real,aa_complex_real(im,Z2)) ).

% Re_i_times
tff(fact_9574_csqrt__of__real__nonneg,axiom,
    ! [X: complex] :
      ( ( aa_complex_real(im,X) = zero_zero_real )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(re,X))
       => ( aa_complex_complex(csqrt,X) = aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sqrt,aa_complex_real(re,X))) ) ) ) ).

% csqrt_of_real_nonneg
tff(fact_9575_csqrt__minus,axiom,
    ! [X: complex] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(im,X)),zero_zero_real)
        | ( ( aa_complex_real(im,X) = zero_zero_real )
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(re,X)) ) )
     => ( aa_complex_complex(csqrt,aa_complex_complex(uminus1482373934393186551omplex,X)) = aa_complex_complex(times_times_complex(imaginary_unit),aa_complex_complex(csqrt,X)) ) ) ).

% csqrt_minus
tff(fact_9576_Cauchy__Im,axiom,
    ! [X7: fun_nat_complex] :
      ( topolo6517432010174082258omplex(X7)
     => topolo4055970368930404560y_real(aTP_Lamm_ri(X7)) ) ).

% Cauchy_Im
tff(fact_9577_modulo__integer_Oabs__eq,axiom,
    ! [Xa: int,X: int] : modulo364778990260209775nteger(code_integer_of_int(Xa),code_integer_of_int(X)) = code_integer_of_int(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xa),X)) ).

% modulo_integer.abs_eq
tff(fact_9578_sums__Im,axiom,
    ! [X7: fun_nat_complex,A: complex] :
      ( aa_complex_o(sums_complex(X7),A)
     => aa_real_o(sums_real(aTP_Lamm_ri(X7)),aa_complex_real(im,A)) ) ).

% sums_Im
tff(fact_9579_imaginary__unit_Osimps_I2_J,axiom,
    aa_complex_real(im,imaginary_unit) = one_one_real ).

% imaginary_unit.simps(2)
tff(fact_9580_one__complex_Osimps_I2_J,axiom,
    aa_complex_real(im,one_one_complex) = zero_zero_real ).

% one_complex.simps(2)
tff(fact_9581_times__integer_Oabs__eq,axiom,
    ! [Xa: int,X: int] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,code_integer_of_int(Xa)),code_integer_of_int(X)) = code_integer_of_int(aa_int_int(aa_int_fun_int_int(times_times_int,Xa),X)) ).

% times_integer.abs_eq
tff(fact_9582_one__integer__def,axiom,
    one_one_Code_integer = code_integer_of_int(one_one_int) ).

% one_integer_def
tff(fact_9583_less__eq__integer_Oabs__eq,axiom,
    ! [Xa: int,X: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,code_integer_of_int(Xa)),code_integer_of_int(X))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),X) ) ).

% less_eq_integer.abs_eq
tff(fact_9584_scaleR__complex_Osimps_I2_J,axiom,
    ! [R: real,X: complex] : aa_complex_real(im,aa_complex_complex(real_V2046097035970521341omplex(R),X)) = aa_real_real(times_times_real(R),aa_complex_real(im,X)) ).

% scaleR_complex.simps(2)
tff(fact_9585_sums__complex__iff,axiom,
    ! [F: fun_nat_complex,X: complex] :
      ( aa_complex_o(sums_complex(F),X)
    <=> ( aa_real_o(sums_real(aTP_Lamm_rh(F)),aa_complex_real(re,X))
        & aa_real_o(sums_real(aTP_Lamm_ri(F)),aa_complex_real(im,X)) ) ) ).

% sums_complex_iff
tff(fact_9586_summable__Im,axiom,
    ! [F: fun_nat_complex] :
      ( summable_complex(F)
     => summable_real(aTP_Lamm_ri(F)) ) ).

% summable_Im
tff(fact_9587_abs__Im__le__cmod,axiom,
    ! [X: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_complex_real(im,X))),aa_complex_real(real_V1022390504157884413omplex,X)) ).

% abs_Im_le_cmod
tff(fact_9588_Cauchy__Re,axiom,
    ! [X7: fun_nat_complex] :
      ( topolo6517432010174082258omplex(X7)
     => topolo4055970368930404560y_real(aTP_Lamm_rh(X7)) ) ).

% Cauchy_Re
tff(fact_9589_summable__complex__iff,axiom,
    ! [F: fun_nat_complex] :
      ( summable_complex(F)
    <=> ( summable_real(aTP_Lamm_rh(F))
        & summable_real(aTP_Lamm_ri(F)) ) ) ).

% summable_complex_iff
tff(fact_9590_times__complex_Osimps_I2_J,axiom,
    ! [X: complex,Y2: complex] : aa_complex_real(im,aa_complex_complex(times_times_complex(X),Y2)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_complex_real(re,X)),aa_complex_real(im,Y2))),aa_real_real(times_times_real(aa_complex_real(im,X)),aa_complex_real(re,Y2))) ).

% times_complex.simps(2)
tff(fact_9591_cmod__Im__le__iff,axiom,
    ! [X: complex,Y2: complex] :
      ( ( aa_complex_real(re,X) = aa_complex_real(re,Y2) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,X)),aa_complex_real(real_V1022390504157884413omplex,Y2))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_complex_real(im,X))),aa_real_real(abs_abs_real,aa_complex_real(im,Y2))) ) ) ).

% cmod_Im_le_iff
tff(fact_9592_cmod__Re__le__iff,axiom,
    ! [X: complex,Y2: complex] :
      ( ( aa_complex_real(im,X) = aa_complex_real(im,Y2) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,X)),aa_complex_real(real_V1022390504157884413omplex,Y2))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_complex_real(re,X))),aa_real_real(abs_abs_real,aa_complex_real(re,Y2))) ) ) ).

% cmod_Re_le_iff
tff(fact_9593_times__complex_Osimps_I1_J,axiom,
    ! [X: complex,Y2: complex] : aa_complex_real(re,aa_complex_complex(times_times_complex(X),Y2)) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(aa_complex_real(re,X)),aa_complex_real(re,Y2))),aa_real_real(times_times_real(aa_complex_real(im,X)),aa_complex_real(im,Y2))) ).

% times_complex.simps(1)
tff(fact_9594_scaleR__complex_Ocode,axiom,
    ! [R: real,X: complex] : aa_complex_complex(real_V2046097035970521341omplex(R),X) = aa_real_complex(complex2(aa_real_real(times_times_real(R),aa_complex_real(re,X))),aa_real_real(times_times_real(R),aa_complex_real(im,X))) ).

% scaleR_complex.code
tff(fact_9595_csqrt__principal,axiom,
    ! [Z2: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(csqrt,Z2)))
      | ( ( aa_complex_real(re,aa_complex_complex(csqrt,Z2)) = zero_zero_real )
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(csqrt,Z2))) ) ) ).

% csqrt_principal
tff(fact_9596_cmod__le,axiom,
    ! [Z2: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z2)),aa_real_real(plus_plus_real(aa_real_real(abs_abs_real,aa_complex_real(re,Z2))),aa_real_real(abs_abs_real,aa_complex_real(im,Z2)))) ).

% cmod_le
tff(fact_9597_sin__n__Im__cis__pow__n,axiom,
    ! [Nb: nat,A: real] : aa_real_real(sin_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),A)) = aa_complex_real(im,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_real_complex(cis,A)),Nb)) ).

% sin_n_Im_cis_pow_n
tff(fact_9598_Re__exp,axiom,
    ! [Z2: complex] : aa_complex_real(re,aa_complex_complex(exp_complex,Z2)) = aa_real_real(times_times_real(aa_real_real(exp_real,aa_complex_real(re,Z2))),aa_real_real(cos_real,aa_complex_real(im,Z2))) ).

% Re_exp
tff(fact_9599_Im__exp,axiom,
    ! [Z2: complex] : aa_complex_real(im,aa_complex_complex(exp_complex,Z2)) = aa_real_real(times_times_real(aa_real_real(exp_real,aa_complex_real(re,Z2))),aa_real_real(sin_real,aa_complex_real(im,Z2))) ).

% Im_exp
tff(fact_9600_complex__eq,axiom,
    ! [A: complex] : A = aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,aa_complex_real(re,A))),aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,aa_complex_real(im,A)))) ).

% complex_eq
tff(fact_9601_times__complex_Ocode,axiom,
    ! [X: complex,Y2: complex] : aa_complex_complex(times_times_complex(X),Y2) = aa_real_complex(complex2(aa_real_real(minus_minus_real(aa_real_real(times_times_real(aa_complex_real(re,X)),aa_complex_real(re,Y2))),aa_real_real(times_times_real(aa_complex_real(im,X)),aa_complex_real(im,Y2)))),aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_complex_real(re,X)),aa_complex_real(im,Y2))),aa_real_real(times_times_real(aa_complex_real(im,X)),aa_complex_real(re,Y2)))) ).

% times_complex.code
tff(fact_9602_exp__eq__polar,axiom,
    ! [Z2: complex] : aa_complex_complex(exp_complex,Z2) = aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,aa_real_real(exp_real,aa_complex_real(re,Z2)))),aa_real_complex(cis,aa_complex_real(im,Z2))) ).

% exp_eq_polar
tff(fact_9603_cmod__power2,axiom,
    ! [Z2: complex] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Z2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cmod_power2
tff(fact_9604_Im__power2,axiom,
    ! [X: complex] : aa_complex_real(im,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_complex_real(re,X))),aa_complex_real(im,X)) ).

% Im_power2
tff(fact_9605_Re__power2,axiom,
    ! [X: complex] : aa_complex_real(re,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% Re_power2
tff(fact_9606_complex__eq__0,axiom,
    ! [Z2: complex] :
      ( ( Z2 = zero_zero_complex )
    <=> ( aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Z2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_real ) ) ).

% complex_eq_0
tff(fact_9607_norm__complex__def,axiom,
    ! [Z2: complex] : aa_complex_real(real_V1022390504157884413omplex,Z2) = aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Z2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% norm_complex_def
tff(fact_9608_inverse__complex_Osimps_I1_J,axiom,
    ! [X: complex] : aa_complex_real(re,aa_complex_complex(invers8013647133539491842omplex,X)) = aa_real_real(divide_divide_real(aa_complex_real(re,X)),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% inverse_complex.simps(1)
tff(fact_9609_complex__neq__0,axiom,
    ! [Z2: complex] :
      ( ( Z2 != zero_zero_complex )
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Z2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% complex_neq_0
tff(fact_9610_Re__divide,axiom,
    ! [X: complex,Y2: complex] : aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(X),Y2)) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_complex_real(re,X)),aa_complex_real(re,Y2))),aa_real_real(times_times_real(aa_complex_real(im,X)),aa_complex_real(im,Y2)))),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Y2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Y2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% Re_divide
tff(fact_9611_csqrt__square,axiom,
    ! [B: complex] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,B))
        | ( ( aa_complex_real(re,B) = zero_zero_real )
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(im,B)) ) )
     => ( aa_complex_complex(csqrt,aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = B ) ) ).

% csqrt_square
tff(fact_9612_csqrt__unique,axiom,
    ! [W: complex,Z2: complex] :
      ( ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,W),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Z2 )
     => ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,W))
          | ( ( aa_complex_real(re,W) = zero_zero_real )
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(im,W)) ) )
       => ( aa_complex_complex(csqrt,Z2) = W ) ) ) ).

% csqrt_unique
tff(fact_9613_inverse__complex_Osimps_I2_J,axiom,
    ! [X: complex] : aa_complex_real(im,aa_complex_complex(invers8013647133539491842omplex,X)) = aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,aa_complex_real(im,X))),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% inverse_complex.simps(2)
tff(fact_9614_Im__divide,axiom,
    ! [X: complex,Y2: complex] : aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(X),Y2)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(times_times_real(aa_complex_real(im,X)),aa_complex_real(re,Y2))),aa_real_real(times_times_real(aa_complex_real(re,X)),aa_complex_real(im,Y2)))),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Y2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Y2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% Im_divide
tff(fact_9615_complex__abs__le__norm,axiom,
    ! [Z2: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_real_real(abs_abs_real,aa_complex_real(re,Z2))),aa_real_real(abs_abs_real,aa_complex_real(im,Z2)))),aa_real_real(times_times_real(aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_complex_real(real_V1022390504157884413omplex,Z2))) ).

% complex_abs_le_norm
tff(fact_9616_complex__unit__circle,axiom,
    ! [Z2: complex] :
      ( ( Z2 != zero_zero_complex )
     => ( aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(divide_divide_real(aa_complex_real(re,Z2)),aa_complex_real(real_V1022390504157884413omplex,Z2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(divide_divide_real(aa_complex_real(im,Z2)),aa_complex_real(real_V1022390504157884413omplex,Z2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real ) ) ).

% complex_unit_circle
tff(fact_9617_inverse__complex_Ocode,axiom,
    ! [X: complex] : aa_complex_complex(invers8013647133539491842omplex,X) = aa_real_complex(complex2(aa_real_real(divide_divide_real(aa_complex_real(re,X)),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,aa_complex_real(im,X))),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,X)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% inverse_complex.code
tff(fact_9618_Complex__divide,axiom,
    ! [X: complex,Y2: complex] : aa_complex_complex(divide1717551699836669952omplex(X),Y2) = aa_real_complex(complex2(aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_complex_real(re,X)),aa_complex_real(re,Y2))),aa_real_real(times_times_real(aa_complex_real(im,X)),aa_complex_real(im,Y2)))),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Y2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Y2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(times_times_real(aa_complex_real(im,X)),aa_complex_real(re,Y2))),aa_real_real(times_times_real(aa_complex_real(re,X)),aa_complex_real(im,Y2)))),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Y2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Y2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% Complex_divide
tff(fact_9619_Im__Reals__divide,axiom,
    ! [R: complex,Z2: complex] :
      ( member_complex(R,real_V2521375963428798218omplex)
     => ( aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(R),Z2)) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_complex_real(re,R))),aa_complex_real(im,Z2))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% Im_Reals_divide
tff(fact_9620_Re__Reals__divide,axiom,
    ! [R: complex,Z2: complex] :
      ( member_complex(R,real_V2521375963428798218omplex)
     => ( aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(R),Z2)) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_complex_real(re,R)),aa_complex_real(re,Z2))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% Re_Reals_divide
tff(fact_9621_real__eq__imaginary__iff,axiom,
    ! [Y2: complex,X: complex] :
      ( member_complex(Y2,real_V2521375963428798218omplex)
     => ( member_complex(X,real_V2521375963428798218omplex)
       => ( ( X = aa_complex_complex(times_times_complex(imaginary_unit),Y2) )
        <=> ( ( X = zero_zero_complex )
            & ( Y2 = zero_zero_complex ) ) ) ) ) ).

% real_eq_imaginary_iff
tff(fact_9622_imaginary__eq__real__iff,axiom,
    ! [Y2: complex,X: complex] :
      ( member_complex(Y2,real_V2521375963428798218omplex)
     => ( member_complex(X,real_V2521375963428798218omplex)
       => ( ( aa_complex_complex(times_times_complex(imaginary_unit),Y2) = X )
        <=> ( ( X = zero_zero_complex )
            & ( Y2 = zero_zero_complex ) ) ) ) ) ).

% imaginary_eq_real_iff
tff(fact_9623_complex__mult__cnj,axiom,
    ! [Z2: complex] : aa_complex_complex(times_times_complex(Z2),aa_complex_complex(cnj,Z2)) = aa_real_complex(real_V4546457046886955230omplex,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Z2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% complex_mult_cnj
tff(fact_9624_integer__of__num_I3_J,axiom,
    ! [Nb: num] :
      code_integer_of_num(aa_num_num(bit1,Nb)) = $let(
        k: code_integer,
        k:= code_integer_of_num(Nb),
        plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(k,k),one_one_Code_integer) ) ).

% integer_of_num(3)
tff(fact_9625_complex__cnj__mult,axiom,
    ! [X: complex,Y2: complex] : aa_complex_complex(cnj,aa_complex_complex(times_times_complex(X),Y2)) = aa_complex_complex(times_times_complex(aa_complex_complex(cnj,X)),aa_complex_complex(cnj,Y2)) ).

% complex_cnj_mult
tff(fact_9626_complex__cnj__one,axiom,
    aa_complex_complex(cnj,one_one_complex) = one_one_complex ).

% complex_cnj_one
tff(fact_9627_complex__cnj__one__iff,axiom,
    ! [Z2: complex] :
      ( ( aa_complex_complex(cnj,Z2) = one_one_complex )
    <=> ( Z2 = one_one_complex ) ) ).

% complex_cnj_one_iff
tff(fact_9628_complex__In__mult__cnj__zero,axiom,
    ! [Z2: complex] : aa_complex_real(im,aa_complex_complex(times_times_complex(Z2),aa_complex_complex(cnj,Z2))) = zero_zero_real ).

% complex_In_mult_cnj_zero
tff(fact_9629_sums__cnj,axiom,
    ! [F: fun_nat_complex,L: complex] :
      ( aa_complex_o(sums_complex(aTP_Lamm_rj(F)),aa_complex_complex(cnj,L))
    <=> aa_complex_o(sums_complex(F),L) ) ).

% sums_cnj
tff(fact_9630_Re__complex__div__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B)) = zero_zero_real )
    <=> ( aa_complex_real(re,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B))) = zero_zero_real ) ) ).

% Re_complex_div_eq_0
tff(fact_9631_Im__complex__div__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B)) = zero_zero_real )
    <=> ( aa_complex_real(im,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B))) = zero_zero_real ) ) ).

% Im_complex_div_eq_0
tff(fact_9632_complex__mod__sqrt__Re__mult__cnj,axiom,
    ! [Z2: complex] : aa_complex_real(real_V1022390504157884413omplex,Z2) = aa_real_real(sqrt,aa_complex_real(re,aa_complex_complex(times_times_complex(Z2),aa_complex_complex(cnj,Z2)))) ).

% complex_mod_sqrt_Re_mult_cnj
tff(fact_9633_integer__of__num__triv_I1_J,axiom,
    code_integer_of_num(one) = one_one_Code_integer ).

% integer_of_num_triv(1)
tff(fact_9634_Re__complex__div__gt__0,axiom,
    ! [A: complex,B: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))) ) ).

% Re_complex_div_gt_0
tff(fact_9635_Re__complex__div__lt__0,axiom,
    ! [A: complex,B: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B))),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(re,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))),zero_zero_real) ) ).

% Re_complex_div_lt_0
tff(fact_9636_Re__complex__div__ge__0,axiom,
    ! [A: complex,B: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))) ) ).

% Re_complex_div_ge_0
tff(fact_9637_Re__complex__div__le__0,axiom,
    ! [A: complex,B: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B))),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(re,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))),zero_zero_real) ) ).

% Re_complex_div_le_0
tff(fact_9638_Im__complex__div__gt__0,axiom,
    ! [A: complex,B: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))) ) ).

% Im_complex_div_gt_0
tff(fact_9639_Im__complex__div__lt__0,axiom,
    ! [A: complex,B: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B))),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(im,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))),zero_zero_real) ) ).

% Im_complex_div_lt_0
tff(fact_9640_Im__complex__div__ge__0,axiom,
    ! [A: complex,B: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))) ) ).

% Im_complex_div_ge_0
tff(fact_9641_Im__complex__div__le__0,axiom,
    ! [A: complex,B: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B))),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(im,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))),zero_zero_real) ) ).

% Im_complex_div_le_0
tff(fact_9642_integer__of__num_I2_J,axiom,
    ! [Nb: num] :
      code_integer_of_num(aa_num_num(bit0,Nb)) = $let(
        k: code_integer,
        k:= code_integer_of_num(Nb),
        plus_p5714425477246183910nteger(k,k) ) ).

% integer_of_num(2)
tff(fact_9643_complex__mod__mult__cnj,axiom,
    ! [Z2: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(Z2),aa_complex_complex(cnj,Z2))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% complex_mod_mult_cnj
tff(fact_9644_complex__div__gt__0,axiom,
    ! [A: complex,B: complex] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))) )
      & ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B)))) ) ) ).

% complex_div_gt_0
tff(fact_9645_integer__of__num__triv_I2_J,axiom,
    code_integer_of_num(aa_num_num(bit0,one)) = numera6620942414471956472nteger(aa_num_num(bit0,one)) ).

% integer_of_num_triv(2)
tff(fact_9646_complex__norm__square,axiom,
    ! [Z2: complex] : aa_real_complex(real_V4546457046886955230omplex,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_complex_complex(times_times_complex(Z2),aa_complex_complex(cnj,Z2)) ).

% complex_norm_square
tff(fact_9647_complex__add__cnj,axiom,
    ! [Z2: complex] : aa_complex_complex(plus_plus_complex(Z2),aa_complex_complex(cnj,Z2)) = aa_real_complex(real_V4546457046886955230omplex,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_complex_real(re,Z2))) ).

% complex_add_cnj
tff(fact_9648_complex__diff__cnj,axiom,
    ! [Z2: complex] : aa_complex_complex(minus_minus_complex(Z2),aa_complex_complex(cnj,Z2)) = aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_complex_real(im,Z2)))),imaginary_unit) ).

% complex_diff_cnj
tff(fact_9649_complex__div__cnj,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(divide1717551699836669952omplex(A),B) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B))),aa_real_complex(real_V4546457046886955230omplex,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,B)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% complex_div_cnj
tff(fact_9650_cnj__add__mult__eq__Re,axiom,
    ! [Z2: complex,W: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(Z2),aa_complex_complex(cnj,W))),aa_complex_complex(times_times_complex(aa_complex_complex(cnj,Z2)),W)) = aa_real_complex(real_V4546457046886955230omplex,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_complex_real(re,aa_complex_complex(times_times_complex(Z2),aa_complex_complex(cnj,W))))) ).

% cnj_add_mult_eq_Re
tff(fact_9651_int__of__integer__code,axiom,
    ! [K2: code_integer] :
      code_int_of_integer(K2) = $ite(
        aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K2),zero_z3403309356797280102nteger),
        aa_int_int(uminus_uminus_int,code_int_of_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K2))),
        $ite(K2 = zero_z3403309356797280102nteger,zero_zero_int,produc1553301316500091796er_int(aTP_Lamm_rk,code_divmod_integer(K2,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ) ).

% int_of_integer_code
tff(fact_9652_bit__cut__integer__def,axiom,
    ! [K2: code_integer] : code_bit_cut_integer(K2) = aa_o_P4192039665577170892eger_o(aa_Cod3473573201699832512eger_o(produc6677183202524767010eger_o,divide6298287555418463151nteger(K2,numera6620942414471956472nteger(aa_num_num(bit0,one)))),~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),K2)) ).

% bit_cut_integer_def
tff(fact_9653_times__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa: code_integer] : code_int_of_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),Xa)) = aa_int_int(aa_int_fun_int_int(times_times_int,code_int_of_integer(X)),code_int_of_integer(Xa)) ).

% times_integer.rep_eq
tff(fact_9654_one__integer_Orep__eq,axiom,
    code_int_of_integer(one_one_Code_integer) = one_one_int ).

% one_integer.rep_eq
tff(fact_9655_modulo__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa: code_integer] : code_int_of_integer(modulo364778990260209775nteger(X,Xa)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,code_int_of_integer(X)),code_int_of_integer(Xa)) ).

% modulo_integer.rep_eq
tff(fact_9656_integer__less__eq__iff,axiom,
    ! [K2: code_integer,L: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K2),L)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,code_int_of_integer(K2)),code_int_of_integer(L)) ) ).

% integer_less_eq_iff
tff(fact_9657_less__eq__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X),Xa)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,code_int_of_integer(X)),code_int_of_integer(Xa)) ) ).

% less_eq_integer.rep_eq
tff(fact_9658_divmod__integer__def,axiom,
    ! [K2: code_integer,L: code_integer] : code_divmod_integer(K2,L) = produc1086072967326762835nteger(divide6298287555418463151nteger(K2,L),modulo364778990260209775nteger(K2,L)) ).

% divmod_integer_def
tff(fact_9659_num__of__integer__code,axiom,
    ! [K2: code_integer] :
      code_num_of_integer(K2) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K2),one_one_Code_integer),one,produc7336495610019696514er_num(aTP_Lamm_rl,code_divmod_integer(K2,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ).

% num_of_integer_code
tff(fact_9660_bit__cut__integer__code,axiom,
    ! [K2: code_integer] :
      code_bit_cut_integer(K2) = $ite(K2 = zero_z3403309356797280102nteger,aa_o_P4192039665577170892eger_o(aa_Cod3473573201699832512eger_o(produc6677183202524767010eger_o,zero_z3403309356797280102nteger),$false),produc9125791028180074456eger_o(aTP_Lamm_rm(K2),code_divmod_abs(K2,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ).

% bit_cut_integer_code
tff(fact_9661_nat__of__integer__code,axiom,
    ! [K2: code_integer] :
      code_nat_of_integer(K2) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K2),zero_z3403309356797280102nteger),zero_zero_nat,produc1555791787009142072er_nat(aTP_Lamm_rn,code_divmod_integer(K2,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ).

% nat_of_integer_code
tff(fact_9662_nat__of__integer__non__positive,axiom,
    ! [K2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K2),zero_z3403309356797280102nteger)
     => ( code_nat_of_integer(K2) = zero_zero_nat ) ) ).

% nat_of_integer_non_positive
tff(fact_9663_nat__of__integer__code__post_I3_J,axiom,
    ! [K2: num] : code_nat_of_integer(numera6620942414471956472nteger(K2)) = aa_num_nat(numeral_numeral_nat,K2) ).

% nat_of_integer_code_post(3)
tff(fact_9664_nat__of__integer__code__post_I2_J,axiom,
    code_nat_of_integer(one_one_Code_integer) = one_one_nat ).

% nat_of_integer_code_post(2)
tff(fact_9665_divmod__abs__def,axiom,
    ! [K2: code_integer,L: code_integer] : code_divmod_abs(K2,L) = produc1086072967326762835nteger(divide6298287555418463151nteger(abs_abs_Code_integer(K2),abs_abs_Code_integer(L)),modulo364778990260209775nteger(abs_abs_Code_integer(K2),abs_abs_Code_integer(L))) ).

% divmod_abs_def
tff(fact_9666_divmod__integer__code,axiom,
    ! [K2: code_integer,L: code_integer] :
      code_divmod_integer(K2,L) = $ite(
        K2 = zero_z3403309356797280102nteger,
        produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger),
        $ite(
          aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),L),
          $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),K2),code_divmod_abs(K2,L),produc6916734918728496179nteger(aTP_Lamm_ro(L),code_divmod_abs(K2,L))),
          $ite(
            L = zero_z3403309356797280102nteger,
            produc1086072967326762835nteger(zero_z3403309356797280102nteger,K2),
            aa_Pro4693618923521437820nteger(aa_fun2509385170507305080nteger(produc6499014454317279255nteger,uminus1351360451143612070nteger),
              $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K2),zero_z3403309356797280102nteger),code_divmod_abs(K2,L),produc6916734918728496179nteger(aTP_Lamm_rp(L),code_divmod_abs(K2,L)))) ) ) ) ).

% divmod_integer_code
tff(fact_9667_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero_nat )
    <=> aa_nat_o(case_nat_o($false,aTP_Lamm_rq),Nat) ) ).

% nat.disc_eq_case(2)
tff(fact_9668_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero_nat )
    <=> aa_nat_o(case_nat_o($true,aTP_Lamm_rr),Nat) ) ).

% nat.disc_eq_case(1)
tff(fact_9669_less__eq__nat_Osimps_I2_J,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Ma)),Nb)
    <=> aa_nat_o(case_nat_o($false,aa_nat_fun_nat_o(ord_less_eq_nat,Ma)),Nb) ) ).

% less_eq_nat.simps(2)
tff(fact_9670_diff__Suc,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),aa_nat_nat(suc,Nb)) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_ai),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb)) ).

% diff_Suc
tff(fact_9671_or__int__rec,axiom,
    ! [K2: int,L: int] :
      aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K2),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,
          aa_o_int(zero_n2684676970156552555ol_int,
            ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K2)
            | ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% or_int_rec
tff(fact_9672_or__nonnegative__int__iff,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K2),L))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) ) ) ).

% or_nonnegative_int_iff
tff(fact_9673_or__negative__int__iff,axiom,
    ! [K2: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K2),L)),zero_zero_int)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ) ).

% or_negative_int_iff
tff(fact_9674_or__minus__numerals_I2_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))) ).

% or_minus_numerals(2)
tff(fact_9675_or__minus__numerals_I6_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))) ).

% or_minus_numerals(6)
tff(fact_9676_and__minus__minus__numerals,axiom,
    ! [Ma: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Ma))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Ma)),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Nb)),one_one_int))) ).

% and_minus_minus_numerals
tff(fact_9677_or__minus__minus__numerals,axiom,
    ! [Ma: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Ma))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Ma)),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Nb)),one_one_int))) ).

% or_minus_minus_numerals
tff(fact_9678_bit__or__int__iff,axiom,
    ! [K2: int,L: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K2),L)),Nb)
    <=> ( aa_nat_o(bit_se1146084159140164899it_int(K2),Nb)
        | aa_nat_o(bit_se1146084159140164899it_int(L),Nb) ) ) ).

% bit_or_int_iff
tff(fact_9679_OR__lower,axiom,
    ! [X: int,Y2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,X),Y2)) ) ) ).

% OR_lower
tff(fact_9680_or__greater__eq,axiom,
    ! [L: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K2),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K2),L)) ) ).

% or_greater_eq
tff(fact_9681_plus__and__or,axiom,
    ! [X: int,Y2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),Y2)),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,X),Y2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y2) ).

% plus_and_or
tff(fact_9682_or__int__def,axiom,
    ! [K2: int,L: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K2),L) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(bit_ri7919022796975470100ot_int,K2)),aa_int_int(bit_ri7919022796975470100ot_int,L))) ).

% or_int_def
tff(fact_9683_or__not__numerals_I1_J,axiom,
    aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,zero_zero_int) ).

% or_not_numerals(1)
tff(fact_9684_xor__int__def,axiom,
    ! [K2: int,L: int] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K2),L) = aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K2),aa_int_int(bit_ri7919022796975470100ot_int,L))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(bit_ri7919022796975470100ot_int,K2)),L)) ).

% xor_int_def
tff(fact_9685_concat__bit__def,axiom,
    ! [Nb: nat,K2: int,L: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(Nb),K2),L) = aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(bit_se2923211474154528505it_int(Nb),K2)),aa_int_int(bit_se545348938243370406it_int(Nb),L)) ).

% concat_bit_def
tff(fact_9686_set__bit__int__def,axiom,
    ! [Nb: nat,K2: int] : aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,Nb),K2) = aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K2),aa_int_int(bit_se545348938243370406it_int(Nb),one_one_int)) ).

% set_bit_int_def
tff(fact_9687_or__not__numerals_I2_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))) ).

% or_not_numerals(2)
tff(fact_9688_or__not__numerals_I4_J,axiom,
    ! [Ma: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Ma))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,one_one_int) ).

% or_not_numerals(4)
tff(fact_9689_or__not__numerals_I3_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))) ).

% or_not_numerals(3)
tff(fact_9690_or__not__numerals_I7_J,axiom,
    ! [Ma: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Ma))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,zero_zero_int) ).

% or_not_numerals(7)
tff(fact_9691_or__not__numerals_I6_J,axiom,
    ! [Ma: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Ma))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,Ma)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb)))) ).

% or_not_numerals(6)
tff(fact_9692_OR__upper,axiom,
    ! [X: int,Nb: nat,Y2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y2),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,X),Y2)),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ) ) ) ).

% OR_upper
tff(fact_9693_or__not__numerals_I5_J,axiom,
    ! [Ma: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Ma))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,Ma)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))))) ).

% or_not_numerals(5)
tff(fact_9694_pred__def,axiom,
    ! [Nat: nat] : aa_nat_nat(pred,Nat) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_ai),Nat) ).

% pred_def
tff(fact_9695_or__not__numerals_I9_J,axiom,
    ! [Ma: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Ma))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,Ma)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))))) ).

% or_not_numerals(9)
tff(fact_9696_or__not__numerals_I8_J,axiom,
    ! [Ma: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Ma))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,Ma)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))))) ).

% or_not_numerals(8)
tff(fact_9697_or__minus__numerals_I1_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit_or_not_num_neg(one),aa_num_num(bitM,Nb)))) ).

% or_minus_numerals(1)
tff(fact_9698_or__minus__numerals_I5_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit_or_not_num_neg(one),aa_num_num(bitM,Nb)))) ).

% or_minus_numerals(5)
tff(fact_9699_or__int__unfold,axiom,
    ! [K2: int,L: int] :
      aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K2),L) = $ite(
        ( ( K2 = aa_int_int(uminus_uminus_int,one_one_int) )
        | ( L = aa_int_int(uminus_uminus_int,one_one_int) ) ),
        aa_int_int(uminus_uminus_int,one_one_int),
        $ite(
          K2 = zero_zero_int,
          L,
          $ite(L = zero_zero_int,K2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))))) ) ) ).

% or_int_unfold
tff(fact_9700_or__nat__numerals_I2_J,axiom,
    ! [Y2: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y2))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y2)) ).

% or_nat_numerals(2)
tff(fact_9701_or__nat__numerals_I4_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X)) ).

% or_nat_numerals(4)
tff(fact_9702_or__nat__numerals_I3_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,X))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,X)) ).

% or_nat_numerals(3)
tff(fact_9703_or__nat__numerals_I1_J,axiom,
    ! [Y2: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y2))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y2)) ).

% or_nat_numerals(1)
tff(fact_9704_or__minus__numerals_I8_J,axiom,
    ! [Nb: num,Ma: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))),aa_num_int(numeral_numeral_int,Ma)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit_or_not_num_neg(Ma),aa_num_num(bit0,Nb)))) ).

% or_minus_numerals(8)
tff(fact_9705_or__minus__numerals_I4_J,axiom,
    ! [Ma: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,Ma)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit_or_not_num_neg(Ma),aa_num_num(bit0,Nb)))) ).

% or_minus_numerals(4)
tff(fact_9706_or__minus__numerals_I3_J,axiom,
    ! [Ma: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,Ma)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit_or_not_num_neg(Ma),aa_num_num(bitM,Nb)))) ).

% or_minus_numerals(3)
tff(fact_9707_or__minus__numerals_I7_J,axiom,
    ! [Nb: num,Ma: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))),aa_num_int(numeral_numeral_int,Ma)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit_or_not_num_neg(Ma),aa_num_num(bitM,Nb)))) ).

% or_minus_numerals(7)
tff(fact_9708_or__not__num__neg_Osimps_I1_J,axiom,
    aa_num_num(bit_or_not_num_neg(one),one) = one ).

% or_not_num_neg.simps(1)
tff(fact_9709_set__bit__nat__def,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,Ma),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Ma),one_one_nat)) ).

% set_bit_nat_def
tff(fact_9710_or__not__num__neg_Osimps_I4_J,axiom,
    ! [Nb: num] : aa_num_num(bit_or_not_num_neg(aa_num_num(bit0,Nb)),one) = aa_num_num(bit0,one) ).

% or_not_num_neg.simps(4)
tff(fact_9711_or__not__num__neg_Osimps_I6_J,axiom,
    ! [Nb: num,Ma: num] : aa_num_num(bit_or_not_num_neg(aa_num_num(bit0,Nb)),aa_num_num(bit1,Ma)) = aa_num_num(bit0,aa_num_num(bit_or_not_num_neg(Nb),Ma)) ).

% or_not_num_neg.simps(6)
tff(fact_9712_or__not__num__neg_Osimps_I3_J,axiom,
    ! [Ma: num] : aa_num_num(bit_or_not_num_neg(one),aa_num_num(bit1,Ma)) = aa_num_num(bit1,Ma) ).

% or_not_num_neg.simps(3)
tff(fact_9713_or__not__num__neg_Osimps_I7_J,axiom,
    ! [Nb: num] : aa_num_num(bit_or_not_num_neg(aa_num_num(bit1,Nb)),one) = one ).

% or_not_num_neg.simps(7)
tff(fact_9714_or__not__num__neg_Osimps_I5_J,axiom,
    ! [Nb: num,Ma: num] : aa_num_num(bit_or_not_num_neg(aa_num_num(bit0,Nb)),aa_num_num(bit0,Ma)) = aa_num_num(bitM,aa_num_num(bit_or_not_num_neg(Nb),Ma)) ).

% or_not_num_neg.simps(5)
tff(fact_9715_or__not__num__neg_Osimps_I9_J,axiom,
    ! [Nb: num,Ma: num] : aa_num_num(bit_or_not_num_neg(aa_num_num(bit1,Nb)),aa_num_num(bit1,Ma)) = aa_num_num(bitM,aa_num_num(bit_or_not_num_neg(Nb),Ma)) ).

% or_not_num_neg.simps(9)
tff(fact_9716_or__nat__def,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Ma),Nb) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_nat_int(semiri1314217659103216013at_int,Ma)),aa_nat_int(semiri1314217659103216013at_int,Nb))) ).

% or_nat_def
tff(fact_9717_or__not__num__neg_Osimps_I2_J,axiom,
    ! [Ma: num] : aa_num_num(bit_or_not_num_neg(one),aa_num_num(bit0,Ma)) = aa_num_num(bit1,Ma) ).

% or_not_num_neg.simps(2)
tff(fact_9718_or__not__num__neg_Osimps_I8_J,axiom,
    ! [Nb: num,Ma: num] : aa_num_num(bit_or_not_num_neg(aa_num_num(bit1,Nb)),aa_num_num(bit0,Ma)) = aa_num_num(bitM,aa_num_num(bit_or_not_num_neg(Nb),Ma)) ).

% or_not_num_neg.simps(8)
tff(fact_9719_or__not__num__neg_Oelims,axiom,
    ! [X: num,Xa: num,Y2: num] :
      ( ( aa_num_num(bit_or_not_num_neg(X),Xa) = Y2 )
     => ( ( ( X = one )
         => ( ( Xa = one )
           => ( Y2 != one ) ) )
       => ( ( ( X = one )
           => ! [M2: num] :
                ( ( Xa = aa_num_num(bit0,M2) )
               => ( Y2 != aa_num_num(bit1,M2) ) ) )
         => ( ( ( X = one )
             => ! [M2: num] :
                  ( ( Xa = aa_num_num(bit1,M2) )
                 => ( Y2 != aa_num_num(bit1,M2) ) ) )
           => ( ( ? [N: num] : X = aa_num_num(bit0,N)
               => ( ( Xa = one )
                 => ( Y2 != aa_num_num(bit0,one) ) ) )
             => ( ! [N: num] :
                    ( ( X = aa_num_num(bit0,N) )
                   => ! [M2: num] :
                        ( ( Xa = aa_num_num(bit0,M2) )
                       => ( Y2 != aa_num_num(bitM,aa_num_num(bit_or_not_num_neg(N),M2)) ) ) )
               => ( ! [N: num] :
                      ( ( X = aa_num_num(bit0,N) )
                     => ! [M2: num] :
                          ( ( Xa = aa_num_num(bit1,M2) )
                         => ( Y2 != aa_num_num(bit0,aa_num_num(bit_or_not_num_neg(N),M2)) ) ) )
                 => ( ( ? [N: num] : X = aa_num_num(bit1,N)
                     => ( ( Xa = one )
                       => ( Y2 != one ) ) )
                   => ( ! [N: num] :
                          ( ( X = aa_num_num(bit1,N) )
                         => ! [M2: num] :
                              ( ( Xa = aa_num_num(bit0,M2) )
                             => ( Y2 != aa_num_num(bitM,aa_num_num(bit_or_not_num_neg(N),M2)) ) ) )
                     => ~ ! [N: num] :
                            ( ( X = aa_num_num(bit1,N) )
                           => ! [M2: num] :
                                ( ( Xa = aa_num_num(bit1,M2) )
                               => ( Y2 != aa_num_num(bitM,aa_num_num(bit_or_not_num_neg(N),M2)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.elims
tff(fact_9720_int__numeral__or__not__num__neg,axiom,
    ! [Ma: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,Ma)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit_or_not_num_neg(Ma),Nb))) ).

% int_numeral_or_not_num_neg
tff(fact_9721_int__numeral__not__or__num__neg,axiom,
    ! [Ma: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Ma))),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit_or_not_num_neg(Nb),Ma))) ).

% int_numeral_not_or_num_neg
tff(fact_9722_numeral__or__not__num__eq,axiom,
    ! [Ma: num,Nb: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit_or_not_num_neg(Ma),Nb)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,Ma)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb)))) ).

% numeral_or_not_num_eq
tff(fact_9723_or__Suc__0__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% or_Suc_0_eq
tff(fact_9724_Suc__0__or__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% Suc_0_or_eq
tff(fact_9725_or__nat__rec,axiom,
    ! [Ma: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Ma),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
          aa_o_nat(zero_n2687167440665602831ol_nat,
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ma)
            | ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% or_nat_rec
tff(fact_9726_max__enat__simps_I4_J,axiom,
    ! [Q: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Q),extend5688581933313929465d_enat) = extend5688581933313929465d_enat ).

% max_enat_simps(4)
tff(fact_9727_max__enat__simps_I5_J,axiom,
    ! [Q: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,extend5688581933313929465d_enat),Q) = extend5688581933313929465d_enat ).

% max_enat_simps(5)
tff(fact_9728_max__enat__simps_I2_J,axiom,
    ! [Q: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Q),zero_z5237406670263579293d_enat) = Q ).

% max_enat_simps(2)
tff(fact_9729_max__enat__simps_I3_J,axiom,
    ! [Q: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,zero_z5237406670263579293d_enat),Q) = Q ).

% max_enat_simps(3)
tff(fact_9730_max__Suc__Suc,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,Ma)),aa_nat_nat(suc,Nb)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Ma),Nb)) ).

% max_Suc_Suc
tff(fact_9731_max__0R,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Nb),zero_zero_nat) = Nb ).

% max_0R
tff(fact_9732_max__0L,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),Nb) = Nb ).

% max_0L
tff(fact_9733_max__nat_Oright__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),zero_zero_nat) = A ).

% max_nat.right_neutral
tff(fact_9734_max__nat_Oneutr__eq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) )
    <=> ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% max_nat.neutr_eq_iff
tff(fact_9735_max__nat_Oleft__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),A) = A ).

% max_nat.left_neutral
tff(fact_9736_max__nat_Oeq__neutr__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% max_nat.eq_neutr_iff
tff(fact_9737_max__enat__simps_I1_J,axiom,
    ! [Ma: nat,Nb: nat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,aa_nat_Extended_enat(extended_enat2,Ma)),aa_nat_Extended_enat(extended_enat2,Nb)) = aa_nat_Extended_enat(extended_enat2,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Ma),Nb)) ).

% max_enat_simps(1)
tff(fact_9738_max__Suc__numeral,axiom,
    ! [Nb: nat,K2: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,K2)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Nb),aa_num_nat(pred_numeral,K2))) ).

% max_Suc_numeral
tff(fact_9739_max__numeral__Suc,axiom,
    ! [K2: num,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,K2)),aa_nat_nat(suc,Nb)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(pred_numeral,K2)),Nb)) ).

% max_numeral_Suc
tff(fact_9740_nat__mult__max__left,axiom,
    ! [Ma: nat,Nb: nat,Q: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Ma),Nb)),Q) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Q)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q)) ).

% nat_mult_max_left
tff(fact_9741_nat__mult__max__right,axiom,
    ! [Ma: nat,Nb: nat,Q: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Nb),Q)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ma),Q)) ).

% nat_mult_max_right
tff(fact_9742_nat__add__max__left,axiom,
    ! [Ma: nat,Nb: nat,Q: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Ma),Nb)),Q) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Q)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Q)) ).

% nat_add_max_left
tff(fact_9743_nat__add__max__right,axiom,
    ! [Ma: nat,Nb: nat,Q: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Nb),Q)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Q)) ).

% nat_add_max_right
tff(fact_9744_nat__minus__add__max,axiom,
    ! [Nb: nat,Ma: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Ma)),Ma) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Nb),Ma) ).

% nat_minus_add_max
tff(fact_9745_max__Suc1,axiom,
    ! [Nb: nat,Ma: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,Nb)),Ma) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,Nb),aa_nat_fun_nat_nat(aTP_Lamm_rs,Nb)),Ma) ).

% max_Suc1
tff(fact_9746_max__Suc2,axiom,
    ! [Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Ma),aa_nat_nat(suc,Nb)) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,Nb),aa_nat_fun_nat_nat(aTP_Lamm_rt,Nb)),Ma) ).

% max_Suc2
tff(fact_9747_or__nat__unfold,axiom,
    ! [Ma: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Ma),Nb) = $ite(
        Ma = zero_zero_nat,
        Nb,
        $ite(Nb = zero_zero_nat,Ma,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) ).

% or_nat_unfold
tff(fact_9748_or__not__num__neg_Opelims,axiom,
    ! [X: num,Xa: num,Y2: num] :
      ( ( aa_num_num(bit_or_not_num_neg(X),Xa) = Y2 )
     => ( aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),aa_num3156681643174609981um_num(aa_num8681986212680877308um_num(product_Pair_num_num,X),Xa))
       => ( ( ( X = one )
           => ( ( Xa = one )
             => ( ( Y2 = one )
               => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),aa_num3156681643174609981um_num(aa_num8681986212680877308um_num(product_Pair_num_num,one),one)) ) ) )
         => ( ( ( X = one )
             => ! [M2: num] :
                  ( ( Xa = aa_num_num(bit0,M2) )
                 => ( ( Y2 = aa_num_num(bit1,M2) )
                   => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),aa_num3156681643174609981um_num(aa_num8681986212680877308um_num(product_Pair_num_num,one),aa_num_num(bit0,M2))) ) ) )
           => ( ( ( X = one )
               => ! [M2: num] :
                    ( ( Xa = aa_num_num(bit1,M2) )
                   => ( ( Y2 = aa_num_num(bit1,M2) )
                     => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),aa_num3156681643174609981um_num(aa_num8681986212680877308um_num(product_Pair_num_num,one),aa_num_num(bit1,M2))) ) ) )
             => ( ! [N: num] :
                    ( ( X = aa_num_num(bit0,N) )
                   => ( ( Xa = one )
                     => ( ( Y2 = aa_num_num(bit0,one) )
                       => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),aa_num3156681643174609981um_num(aa_num8681986212680877308um_num(product_Pair_num_num,aa_num_num(bit0,N)),one)) ) ) )
               => ( ! [N: num] :
                      ( ( X = aa_num_num(bit0,N) )
                     => ! [M2: num] :
                          ( ( Xa = aa_num_num(bit0,M2) )
                         => ( ( Y2 = aa_num_num(bitM,aa_num_num(bit_or_not_num_neg(N),M2)) )
                           => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),aa_num3156681643174609981um_num(aa_num8681986212680877308um_num(product_Pair_num_num,aa_num_num(bit0,N)),aa_num_num(bit0,M2))) ) ) )
                 => ( ! [N: num] :
                        ( ( X = aa_num_num(bit0,N) )
                       => ! [M2: num] :
                            ( ( Xa = aa_num_num(bit1,M2) )
                           => ( ( Y2 = aa_num_num(bit0,aa_num_num(bit_or_not_num_neg(N),M2)) )
                             => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),aa_num3156681643174609981um_num(aa_num8681986212680877308um_num(product_Pair_num_num,aa_num_num(bit0,N)),aa_num_num(bit1,M2))) ) ) )
                   => ( ! [N: num] :
                          ( ( X = aa_num_num(bit1,N) )
                         => ( ( Xa = one )
                           => ( ( Y2 = one )
                             => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),aa_num3156681643174609981um_num(aa_num8681986212680877308um_num(product_Pair_num_num,aa_num_num(bit1,N)),one)) ) ) )
                     => ( ! [N: num] :
                            ( ( X = aa_num_num(bit1,N) )
                           => ! [M2: num] :
                                ( ( Xa = aa_num_num(bit0,M2) )
                               => ( ( Y2 = aa_num_num(bitM,aa_num_num(bit_or_not_num_neg(N),M2)) )
                                 => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),aa_num3156681643174609981um_num(aa_num8681986212680877308um_num(product_Pair_num_num,aa_num_num(bit1,N)),aa_num_num(bit0,M2))) ) ) )
                       => ~ ! [N: num] :
                              ( ( X = aa_num_num(bit1,N) )
                             => ! [M2: num] :
                                  ( ( Xa = aa_num_num(bit1,M2) )
                                 => ( ( Y2 = aa_num_num(bitM,aa_num_num(bit_or_not_num_neg(N),M2)) )
                                   => ~ aa_Pro7685007914727650815_num_o(accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel),aa_num3156681643174609981um_num(aa_num8681986212680877308um_num(product_Pair_num_num,aa_num_num(bit1,N)),aa_num_num(bit1,M2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.pelims
tff(fact_9749_prod__decode__aux_Oelims,axiom,
    ! [X: nat,Xa: nat,Y2: product_prod_nat_nat] :
      ( ( nat_prod_decode_aux(X,Xa) = Y2 )
     => ( Y2 = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),X),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),Xa)),nat_prod_decode_aux(aa_nat_nat(suc,X),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa),aa_nat_nat(suc,X)))) ) ) ).

% prod_decode_aux.elims
tff(fact_9750_prod__decode__aux_Osimps,axiom,
    ! [K2: nat,Ma: nat] :
      nat_prod_decode_aux(K2,Ma) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),K2),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K2),Ma)),nat_prod_decode_aux(aa_nat_nat(suc,K2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),aa_nat_nat(suc,K2)))) ).

% prod_decode_aux.simps
tff(fact_9751_prod__decode__aux_Opelims,axiom,
    ! [X: nat,Xa: nat,Y2: product_prod_nat_nat] :
      ( ( nat_prod_decode_aux(X,Xa) = Y2 )
     => ( aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(nat_pr5047031295181774490ux_rel),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X),Xa))
       => ~ ( ( Y2 = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),X),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),Xa)),nat_prod_decode_aux(aa_nat_nat(suc,X),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xa),aa_nat_nat(suc,X)))) )
           => ~ aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(nat_pr5047031295181774490ux_rel),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X),Xa)) ) ) ) ).

% prod_decode_aux.pelims
tff(fact_9752_bezw__0,axiom,
    ! [X: nat] : bezw(X,zero_zero_nat) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),zero_zero_int) ).

% bezw_0
tff(fact_9753_finite__less__ub,axiom,
    ! [F: fun_nat_nat,U: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),aa_nat_nat(F,N))
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ru(F),U))) ) ).

% finite_less_ub
tff(fact_9754_finite__M__bounded__by__nat,axiom,
    ! [P: fun_nat_o,I2: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_rv(P),I2))) ).

% finite_M_bounded_by_nat
tff(fact_9755_finite__nat__set__iff__bounded,axiom,
    ! [N2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,N2)
    <=> ? [M3: nat] :
        ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),N2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),M3) ) ) ).

% finite_nat_set_iff_bounded
tff(fact_9756_bounded__nat__set__is__finite,axiom,
    ! [N2: set_nat,Nb: nat] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),N2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Nb) )
     => aa_set_nat_o(finite_finite_nat,N2) ) ).

% bounded_nat_set_is_finite
tff(fact_9757_finite__nat__set__iff__bounded__le,axiom,
    ! [N2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,N2)
    <=> ? [M3: nat] :
        ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),N2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),M3) ) ) ).

% finite_nat_set_iff_bounded_le
tff(fact_9758_finite__divisors__nat,axiom,
    ! [Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ma)
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bg,Ma))) ) ).

% finite_divisors_nat
tff(fact_9759_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N2: set_nat,Nb: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(N2),set_or1269000886237332187st_nat(zero_zero_nat,Nb))
     => aa_set_nat_o(finite_finite_nat,N2) ) ).

% subset_eq_atLeast0_atMost_finite
tff(fact_9760_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N2: set_nat,Nb: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(N2),set_or4665077453230672383an_nat(zero_zero_nat,Nb))
     => aa_set_nat_o(finite_finite_nat,N2) ) ).

% subset_eq_atLeast0_lessThan_finite
tff(fact_9761_even__set__encode__iff,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(nat_set_encode,A2))
      <=> ~ aa_set_nat_o(member_nat(zero_zero_nat),A2) ) ) ).

% even_set_encode_iff
tff(fact_9762_finite__Collect__le__nat,axiom,
    ! [K2: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_am,K2))) ).

% finite_Collect_le_nat
tff(fact_9763_finite__Collect__less__nat,axiom,
    ! [K2: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_qi,K2))) ).

% finite_Collect_less_nat
tff(fact_9764_finite__interval__int1,axiom,
    ! [A: int,B: int] : finite_finite_int(collect_int(aa_int_fun_int_o(aTP_Lamm_rw(A),B))) ).

% finite_interval_int1
tff(fact_9765_finite__interval__int4,axiom,
    ! [A: int,B: int] : finite_finite_int(collect_int(aa_int_fun_int_o(aTP_Lamm_rx(A),B))) ).

% finite_interval_int4
tff(fact_9766_finite__interval__int3,axiom,
    ! [A: int,B: int] : finite_finite_int(collect_int(aa_int_fun_int_o(aTP_Lamm_ry(A),B))) ).

% finite_interval_int3
tff(fact_9767_finite__interval__int2,axiom,
    ! [A: int,B: int] : finite_finite_int(collect_int(aa_int_fun_int_o(aTP_Lamm_rz(A),B))) ).

% finite_interval_int2
tff(fact_9768_finite__nth__roots,axiom,
    ! [Nb: nat,C: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => finite3207457112153483333omplex(collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_qj(Nb),C))) ) ).

% finite_nth_roots
tff(fact_9769_finite__enat__bounded,axiom,
    ! [A2: set_Extended_enat,Nb: nat] :
      ( ! [Y4: extended_enat] :
          ( member_Extended_enat(Y4,A2)
         => aa_Extended_enat_o(ord_le2932123472753598470d_enat(Y4),aa_nat_Extended_enat(extended_enat2,Nb)) )
     => finite4001608067531595151d_enat(A2) ) ).

% finite_enat_bounded
tff(fact_9770_finite__divisors__int,axiom,
    ! [I2: int] :
      ( ( I2 != zero_zero_int )
     => finite_finite_int(collect_int(aa_int_fun_int_o(aTP_Lamm_bi,I2))) ) ).

% finite_divisors_int
tff(fact_9771_finite__nat__iff__bounded__le,axiom,
    ! [S3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
    <=> ? [K3: nat] : aa_set_nat_o(ord_less_eq_set_nat(S3),aa_nat_set_nat(set_ord_atMost_nat,K3)) ) ).

% finite_nat_iff_bounded_le
tff(fact_9772_finite__nat__iff__bounded,axiom,
    ! [S3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
    <=> ? [K3: nat] : aa_set_nat_o(ord_less_eq_set_nat(S3),aa_nat_set_nat(set_ord_lessThan_nat,K3)) ) ).

% finite_nat_iff_bounded
tff(fact_9773_finite__nat__bounded,axiom,
    ! [S3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ? [K: nat] : aa_set_nat_o(ord_less_eq_set_nat(S3),aa_nat_set_nat(set_ord_lessThan_nat,K)) ) ).

% finite_nat_bounded
tff(fact_9774_infinite__int__iff__unbounded__le,axiom,
    ! [S3: set_int] :
      ( ~ finite_finite_int(S3)
    <=> ! [M3: int] :
        ? [N4: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M3),aa_int_int(abs_abs_int,N4))
          & member_int(N4,S3) ) ) ).

% infinite_int_iff_unbounded_le
tff(fact_9775_unbounded__k__infinite,axiom,
    ! [K2: nat,S3: set_nat] :
      ( ! [M2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),M2)
         => ? [N7: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N7)
              & aa_set_nat_o(member_nat(N7),S3) ) )
     => ~ aa_set_nat_o(finite_finite_nat,S3) ) ).

% unbounded_k_infinite
tff(fact_9776_infinite__nat__iff__unbounded,axiom,
    ! [S3: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S3)
    <=> ! [M3: nat] :
        ? [N4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),N4)
          & aa_set_nat_o(member_nat(N4),S3) ) ) ).

% infinite_nat_iff_unbounded
tff(fact_9777_infinite__nat__iff__unbounded__le,axiom,
    ! [S3: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S3)
    <=> ! [M3: nat] :
        ? [N4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M3),N4)
          & aa_set_nat_o(member_nat(N4),S3) ) ) ).

% infinite_nat_iff_unbounded_le
tff(fact_9778_set__encode__insert,axiom,
    ! [A2: set_nat,Nb: nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ~ aa_set_nat_o(member_nat(Nb),A2)
       => ( aa_set_nat_nat(nat_set_encode,aa_set_nat_set_nat(insert_nat(Nb),A2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),aa_set_nat_nat(nat_set_encode,A2)) ) ) ) ).

% set_encode_insert
tff(fact_9779_card__Collect__less__nat,axiom,
    ! [Nb: nat] : finite_card_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_qi,Nb))) = Nb ).

% card_Collect_less_nat
tff(fact_9780_card__Collect__le__nat,axiom,
    ! [Nb: nat] : finite_card_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_am,Nb))) = aa_nat_nat(suc,Nb) ).

% card_Collect_le_nat
tff(fact_9781_card__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : finite_card_int(set_or1266510415728281911st_int(L,U)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,U),L)),one_one_int)) ).

% card_atLeastAtMost_int
tff(fact_9782_atLeastAtMost__insertL,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => ( aa_set_nat_set_nat(insert_nat(Ma),set_or1269000886237332187st_nat(aa_nat_nat(suc,Ma),Nb)) = set_or1269000886237332187st_nat(Ma,Nb) ) ) ).

% atLeastAtMost_insertL
tff(fact_9783_atLeastAtMostSuc__conv,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),aa_nat_nat(suc,Nb))
     => ( set_or1269000886237332187st_nat(Ma,aa_nat_nat(suc,Nb)) = aa_set_nat_set_nat(insert_nat(aa_nat_nat(suc,Nb)),set_or1269000886237332187st_nat(Ma,Nb)) ) ) ).

% atLeastAtMostSuc_conv
tff(fact_9784_Icc__eq__insert__lb__nat,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => ( set_or1269000886237332187st_nat(Ma,Nb) = aa_set_nat_set_nat(insert_nat(Ma),set_or1269000886237332187st_nat(aa_nat_nat(suc,Ma),Nb)) ) ) ).

% Icc_eq_insert_lb_nat
tff(fact_9785_lessThan__nat__numeral,axiom,
    ! [K2: num] : aa_nat_set_nat(set_ord_lessThan_nat,aa_num_nat(numeral_numeral_nat,K2)) = aa_set_nat_set_nat(insert_nat(aa_num_nat(pred_numeral,K2)),aa_nat_set_nat(set_ord_lessThan_nat,aa_num_nat(pred_numeral,K2))) ).

% lessThan_nat_numeral
tff(fact_9786_card__less,axiom,
    ! [M6: set_nat,I2: nat] :
      ( aa_set_nat_o(member_nat(zero_zero_nat),M6)
     => ( finite_card_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_sa(M6),I2))) != zero_zero_nat ) ) ).

% card_less
tff(fact_9787_card__less__Suc,axiom,
    ! [M6: set_nat,I2: nat] :
      ( aa_set_nat_o(member_nat(zero_zero_nat),M6)
     => ( aa_nat_nat(suc,finite_card_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_sb(M6),I2)))) = finite_card_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_sa(M6),I2))) ) ) ).

% card_less_Suc
tff(fact_9788_card__less__Suc2,axiom,
    ! [M6: set_nat,I2: nat] :
      ( ~ aa_set_nat_o(member_nat(zero_zero_nat),M6)
     => ( finite_card_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_sb(M6),I2))) = finite_card_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_sa(M6),I2))) ) ) ).

% card_less_Suc2
tff(fact_9789_atMost__nat__numeral,axiom,
    ! [K2: num] : aa_nat_set_nat(set_ord_atMost_nat,aa_num_nat(numeral_numeral_nat,K2)) = aa_set_nat_set_nat(insert_nat(aa_num_nat(numeral_numeral_nat,K2)),aa_nat_set_nat(set_ord_atMost_nat,aa_num_nat(pred_numeral,K2))) ).

% atMost_nat_numeral
tff(fact_9790_subset__card__intvl__is__intvl,axiom,
    ! [A2: set_nat,K2: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),set_or4665077453230672383an_nat(K2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),finite_card_nat(A2))))
     => ( A2 = set_or4665077453230672383an_nat(K2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K2),finite_card_nat(A2))) ) ) ).

% subset_card_intvl_is_intvl
tff(fact_9791_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N2: set_nat,Nb: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(N2),set_or4665077453230672383an_nat(zero_zero_nat,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_nat(N2)),Nb) ) ).

% subset_eq_atLeast0_lessThan_card
tff(fact_9792_card__sum__le__nat__sum,axiom,
    ! [S3: set_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ai),set_or4665077453230672383an_nat(zero_zero_nat,finite_card_nat(S3)))),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ai),S3)) ).

% card_sum_le_nat_sum
tff(fact_9793_card__nth__roots,axiom,
    ! [C: complex,Nb: nat] :
      ( ( C != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( finite_card_complex(collect_complex(aa_nat_fun_complex_o(aTP_Lamm_rd(C),Nb))) = Nb ) ) ) ).

% card_nth_roots
tff(fact_9794_card__roots__unity__eq,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( finite_card_complex(collect_complex(aTP_Lamm_qk(Nb))) = Nb ) ) ).

% card_roots_unity_eq
tff(fact_9795_set__decode__plus__power__2,axiom,
    ! [Nb: nat,Z2: nat] :
      ( ~ aa_set_nat_o(member_nat(Nb),nat_set_decode(Z2))
     => ( nat_set_decode(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),Z2)) = aa_set_nat_set_nat(insert_nat(Nb),nat_set_decode(Z2)) ) ) ).

% set_decode_plus_power_2
tff(fact_9796_sum__list__upt,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => ( groups4561878855575611511st_nat(upt(Ma,Nb)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ai),set_or4665077453230672383an_nat(Ma,Nb)) ) ) ).

% sum_list_upt
tff(fact_9797_distinct__upt,axiom,
    ! [I2: nat,J: nat] : aa_list_nat_o(distinct_nat,upt(I2,J)) ).

% distinct_upt
tff(fact_9798_atLeastAtMostPlus1__int__conv,axiom,
    ! [Ma: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ma),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Nb))
     => ( set_or1266510415728281911st_int(Ma,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Nb)) = insert_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Nb),set_or1266510415728281911st_int(Ma,Nb)) ) ) ).

% atLeastAtMostPlus1_int_conv
tff(fact_9799_card__length__sum__list__rec,axiom,
    ! [Ma: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Ma)
     => ( finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_sc(Ma),N2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_sd(Ma),N2)))),finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_se(Ma),N2)))) ) ) ).

% card_length_sum_list_rec
tff(fact_9800_card__length__sum__list,axiom,
    ! [Ma: nat,N2: nat] : finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_sc(Ma),N2))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),Ma)),one_one_nat)),N2) ).

% card_length_sum_list
tff(fact_9801_and__int_Osimps,axiom,
    ! [K2: int,L: int] :
      aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K2),L) = $ite(
        ( member_int(K2,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
        & member_int(L,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) ),
        aa_int_int(uminus_uminus_int,
          aa_o_int(zero_n2684676970156552555ol_int,
            ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K2)
            & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ))),
        aa_int_int(aa_int_fun_int_int(plus_plus_int,
            aa_o_int(zero_n2684676970156552555ol_int,
              ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K2)
              & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ).

% and_int.simps
tff(fact_9802_and__int_Oelims,axiom,
    ! [X: int,Xa: int,Y2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),Xa) = Y2 )
     => ( Y2 = $ite(
            ( member_int(X,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
            & member_int(Xa,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) ),
            aa_int_int(uminus_uminus_int,
              aa_o_int(zero_n2684676970156552555ol_int,
                ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X)
                & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa) ))),
            aa_int_int(aa_int_fun_int_int(plus_plus_int,
                aa_o_int(zero_n2684676970156552555ol_int,
                  ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X)
                  & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa) ))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ).

% and_int.elims
tff(fact_9803_and__int_Opelims,axiom,
    ! [X: int,Xa: int,Y2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),Xa) = Y2 )
     => ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Xa))
       => ~ ( ( Y2 = $ite(
                  ( member_int(X,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
                  & member_int(Xa,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) ),
                  aa_int_int(uminus_uminus_int,
                    aa_o_int(zero_n2684676970156552555ol_int,
                      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X)
                      & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa) ))),
                  aa_int_int(aa_int_fun_int_int(plus_plus_int,
                      aa_o_int(zero_n2684676970156552555ol_int,
                        ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),X)
                        & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xa) ))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) )
           => ~ aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Xa)) ) ) ) ).

% and_int.pelims
tff(fact_9804_simp__from__to,axiom,
    ! [I2: int,J: int] :
      set_or1266510415728281911st_int(I2,J) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,J),I2),bot_bot_set_int,insert_int(I2,set_or1266510415728281911st_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int),J))) ).

% simp_from_to
tff(fact_9805_and__int_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun_int_fun_int_o] :
      ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A0),A1))
     => ( ! [K: int,L2: int] :
            ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,K),L2))
           => ( ( ~ ( member_int(K,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
                    & member_int(L2,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) )
               => aa_int_o(aa_int_fun_int_o(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) )
             => aa_int_o(aa_int_fun_int_o(P,K),L2) ) )
       => aa_int_o(aa_int_fun_int_o(P,A0),A1) ) ) ).

% and_int.pinduct
tff(fact_9806_and__int_Opsimps,axiom,
    ! [K2: int,L: int] :
      ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,K2),L))
     => ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K2),L) = $ite(
            ( member_int(K2,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
            & member_int(L,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) ),
            aa_int_int(uminus_uminus_int,
              aa_o_int(zero_n2684676970156552555ol_int,
                ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K2)
                & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ))),
            aa_int_int(aa_int_fun_int_int(plus_plus_int,
                aa_o_int(zero_n2684676970156552555ol_int,
                  ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K2)
                  & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ).

% and_int.psimps
tff(fact_9807_bot__enat__def,axiom,
    bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).

% bot_enat_def
tff(fact_9808_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
tff(fact_9809_sorted__wrt__upt,axiom,
    ! [Ma: nat,Nb: nat] : aa_list_nat_o(sorted_wrt_nat(ord_less_nat),upt(Ma,Nb)) ).

% sorted_wrt_upt
tff(fact_9810_sorted__upt,axiom,
    ! [Ma: nat,Nb: nat] : aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),upt(Ma,Nb)) ).

% sorted_upt
tff(fact_9811_atLeastLessThanSuc,axiom,
    ! [Ma: nat,Nb: nat] :
      set_or4665077453230672383an_nat(Ma,aa_nat_nat(suc,Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb),aa_set_nat_set_nat(insert_nat(Nb),set_or4665077453230672383an_nat(Ma,Nb)),bot_bot_set_nat) ).

% atLeastLessThanSuc
tff(fact_9812_sorted__wrt__less__idx,axiom,
    ! [Ns: list_nat,I2: nat] :
      ( aa_list_nat_o(sorted_wrt_nat(ord_less_nat),Ns)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_nat(Ns))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),aa_nat_nat(nth_nat(Ns),I2)) ) ) ).

% sorted_wrt_less_idx
tff(fact_9813_atLeastLessThan__nat__numeral,axiom,
    ! [Ma: nat,K2: num] :
      set_or4665077453230672383an_nat(Ma,aa_num_nat(numeral_numeral_nat,K2)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),aa_num_nat(pred_numeral,K2)),aa_set_nat_set_nat(insert_nat(aa_num_nat(pred_numeral,K2)),set_or4665077453230672383an_nat(Ma,aa_num_nat(pred_numeral,K2))),bot_bot_set_nat) ).

% atLeastLessThan_nat_numeral
tff(fact_9814_sorted__list__of__set__range,axiom,
    ! [Ma: nat,Nb: nat] : linord2614967742042102400et_nat(set_or4665077453230672383an_nat(Ma,Nb)) = upt(Ma,Nb) ).

% sorted_list_of_set_range
tff(fact_9815_Suc__0__div__numeral,axiom,
    ! [K2: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,K2)) = aa_Pro4243173770044397973at_nat(product_fst_nat_nat,unique5055182867167087721od_nat(one,K2)) ).

% Suc_0_div_numeral
tff(fact_9816_Suc__0__mod__numeral,axiom,
    ! [K2: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,K2)) = aa_Pro4243173770044397973at_nat(product_snd_nat_nat,unique5055182867167087721od_nat(one,K2)) ).

% Suc_0_mod_numeral
tff(fact_9817_fst__divmod__nat,axiom,
    ! [Ma: nat,Nb: nat] : aa_Pro4243173770044397973at_nat(product_fst_nat_nat,divmod_nat(Ma,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),Nb) ).

% fst_divmod_nat
tff(fact_9818_snd__divmod__nat,axiom,
    ! [Ma: nat,Nb: nat] : aa_Pro4243173770044397973at_nat(product_snd_nat_nat,divmod_nat(Ma,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ma),Nb) ).

% snd_divmod_nat
tff(fact_9819_binomial__def,axiom,
    ! [Nb: nat,K2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K2) = finite_card_set_nat(collect_set_nat(aa_nat_fun_set_nat_o(aTP_Lamm_sf(Nb),K2))) ).

% binomial_def
tff(fact_9820_snd__divmod__integer,axiom,
    ! [K2: code_integer,L: code_integer] : produc6174133586879617921nteger(code_divmod_integer(K2,L)) = modulo364778990260209775nteger(K2,L) ).

% snd_divmod_integer
tff(fact_9821_snd__divmod__abs,axiom,
    ! [K2: code_integer,L: code_integer] : produc6174133586879617921nteger(code_divmod_abs(K2,L)) = modulo364778990260209775nteger(abs_abs_Code_integer(K2),abs_abs_Code_integer(L)) ).

% snd_divmod_abs
tff(fact_9822_bezw__non__0,axiom,
    ! [Y2: nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Y2)
     => ( bezw(X,Y2) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Y2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y2)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,bezw(Y2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y2)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Y2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y2)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y2))))) ) ) ).

% bezw_non_0
tff(fact_9823_bezw_Osimps,axiom,
    ! [X: nat,Y2: nat] :
      bezw(X,Y2) = $ite(Y2 = zero_zero_nat,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),zero_zero_int),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Y2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y2)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,bezw(Y2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y2)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Y2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y2)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y2)))))) ).

% bezw.simps
tff(fact_9824_bezw_Oelims,axiom,
    ! [X: nat,Xa: nat,Y2: product_prod_int_int] :
      ( ( bezw(X,Xa) = Y2 )
     => ( Y2 = $ite(Xa = zero_zero_nat,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),zero_zero_int),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Xa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,bezw(Xa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Xa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Xa)))))) ) ) ).

% bezw.elims
tff(fact_9825_rat__sgn__code,axiom,
    ! [P2: rat] : quotient_of(aa_rat_rat(sgn_sgn_rat,P2)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(sgn_sgn_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,quotient_of(P2)))),one_one_int) ).

% rat_sgn_code
tff(fact_9826_bezw_Opelims,axiom,
    ! [X: nat,Xa: nat,Y2: product_prod_int_int] :
      ( ( bezw(X,Xa) = Y2 )
     => ( aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(bezw_rel),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X),Xa))
       => ~ ( ( Y2 = $ite(Xa = zero_zero_nat,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),zero_zero_int),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Xa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,bezw(Xa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Xa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Xa)))))) )
           => ~ aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(bezw_rel),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X),Xa)) ) ) ) ).

% bezw.pelims
tff(fact_9827_minus__one__mod__numeral,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(aa_int_fun_int_int(adjust_mod,aa_num_int(numeral_numeral_int,Nb)),aa_Pro6799258434465214121nt_int(product_snd_int_int,unique5052692396658037445od_int(one,Nb))) ).

% minus_one_mod_numeral
tff(fact_9828_one__mod__minus__numeral,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(adjust_mod,aa_num_int(numeral_numeral_int,Nb)),aa_Pro6799258434465214121nt_int(product_snd_int_int,unique5052692396658037445od_int(one,Nb)))) ).

% one_mod_minus_numeral
tff(fact_9829_minus__numeral__mod__numeral,axiom,
    ! [Ma: num,Nb: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Ma))),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(aa_int_fun_int_int(adjust_mod,aa_num_int(numeral_numeral_int,Nb)),aa_Pro6799258434465214121nt_int(product_snd_int_int,unique5052692396658037445od_int(Ma,Nb))) ).

% minus_numeral_mod_numeral
tff(fact_9830_numeral__mod__minus__numeral,axiom,
    ! [Ma: num,Nb: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Ma)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(adjust_mod,aa_num_int(numeral_numeral_int,Nb)),aa_Pro6799258434465214121nt_int(product_snd_int_int,unique5052692396658037445od_int(Ma,Nb)))) ).

% numeral_mod_minus_numeral
tff(fact_9831_normalize__def,axiom,
    ! [P2: product_prod_int_int] :
      aa_Pro6970504127699319548nt_int(normalize,P2) = $ite(
        aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_Pro6799258434465214121nt_int(product_snd_int_int,P2)),
        $let(
          a: int,
          a:= aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,P2)),aa_Pro6799258434465214121nt_int(product_snd_int_int,P2)),
          aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,P2)),a)),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,P2)),a)) ),
        $ite(
          aa_Pro6799258434465214121nt_int(product_snd_int_int,P2) = zero_zero_int,
          aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int),
          $let(
            a: int,
            a:= aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,P2)),aa_Pro6799258434465214121nt_int(product_snd_int_int,P2))),
            aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,P2)),a)),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,P2)),a)) ) ) ) ).

% normalize_def
tff(fact_9832_drop__bit__numeral__minus__bit1,axiom,
    ! [L: num,K2: num] : aa_int_int(bit_se8568078237143864401it_int(aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K2)))) = aa_int_int(bit_se8568078237143864401it_int(aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,K2)))) ).

% drop_bit_numeral_minus_bit1
tff(fact_9833_gcd__1__int,axiom,
    ! [Ma: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Ma),one_one_int) = one_one_int ).

% gcd_1_int
tff(fact_9834_drop__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se8568078237143864401it_int(Nb),K2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K2) ) ).

% drop_bit_nonnegative_int_iff
tff(fact_9835_drop__bit__negative__int__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se8568078237143864401it_int(Nb),K2)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K2),zero_zero_int) ) ).

% drop_bit_negative_int_iff
tff(fact_9836_drop__bit__minus__one,axiom,
    ! [Nb: nat] : aa_int_int(bit_se8568078237143864401it_int(Nb),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% drop_bit_minus_one
tff(fact_9837_drop__bit__Suc__minus__bit0,axiom,
    ! [Nb: nat,K2: num] : aa_int_int(bit_se8568078237143864401it_int(aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K2)))) = aa_int_int(bit_se8568078237143864401it_int(Nb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))) ).

% drop_bit_Suc_minus_bit0
tff(fact_9838_drop__bit__numeral__minus__bit0,axiom,
    ! [L: num,K2: num] : aa_int_int(bit_se8568078237143864401it_int(aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K2)))) = aa_int_int(bit_se8568078237143864401it_int(aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K2))) ).

% drop_bit_numeral_minus_bit0
tff(fact_9839_drop__bit__Suc__minus__bit1,axiom,
    ! [Nb: nat,K2: num] : aa_int_int(bit_se8568078237143864401it_int(aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K2)))) = aa_int_int(bit_se8568078237143864401it_int(Nb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,K2)))) ).

% drop_bit_Suc_minus_bit1
tff(fact_9840_gcd__red__int,axiom,
    ! [X: int,Y2: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y2) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Y2),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X),Y2)) ).

% gcd_red_int
tff(fact_9841_gcd__ge__0__int,axiom,
    ! [X: int,Y2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y2)) ).

% gcd_ge_0_int
tff(fact_9842_bezout__int,axiom,
    ! [X: int,Y2: int] :
    ? [U2: int,V2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,U2),X)),aa_int_int(aa_int_fun_int_int(times_times_int,V2),Y2)) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y2) ).

% bezout_int
tff(fact_9843_gcd__mult__distrib__int,axiom,
    ! [K2: int,Ma: int,Nb: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,K2)),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Ma),Nb)) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(aa_int_fun_int_int(times_times_int,K2),Ma)),aa_int_int(aa_int_fun_int_int(times_times_int,K2),Nb)) ).

% gcd_mult_distrib_int
tff(fact_9844_drop__bit__push__bit__int,axiom,
    ! [Ma: nat,Nb: nat,K2: int] : aa_int_int(bit_se8568078237143864401it_int(Ma),aa_int_int(bit_se545348938243370406it_int(Nb),K2)) = aa_int_int(bit_se8568078237143864401it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb)),aa_int_int(bit_se545348938243370406it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Ma)),K2)) ).

% drop_bit_push_bit_int
tff(fact_9845_gcd__le2__int,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B)),B) ) ).

% gcd_le2_int
tff(fact_9846_gcd__le1__int,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B)),A) ) ).

% gcd_le1_int
tff(fact_9847_gcd__cases__int,axiom,
    ! [X: int,Y2: int,P: fun_int_o] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y2)
         => aa_int_o(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y2)) ) )
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y2),zero_zero_int)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),aa_int_int(uminus_uminus_int,Y2))) ) )
       => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),zero_zero_int)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y2)
             => aa_int_o(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(uminus_uminus_int,X)),Y2)) ) )
         => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),zero_zero_int)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y2),zero_zero_int)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(uminus_uminus_int,X)),aa_int_int(uminus_uminus_int,Y2))) ) )
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y2)) ) ) ) ) ).

% gcd_cases_int
tff(fact_9848_gcd__unique__int,axiom,
    ! [D: int,A: int,B: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),D)
        & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),A)
        & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),B)
        & ! [E3: int] :
            ( ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,E3),A)
              & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,E3),B) )
           => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,E3),D) ) )
    <=> ( D = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,A),B) ) ) ).

% gcd_unique_int
tff(fact_9849_gcd__non__0__int,axiom,
    ! [Y2: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Y2)
     => ( aa_int_int(aa_int_fun_int_int(gcd_gcd_int,X),Y2) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Y2),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X),Y2)) ) ) ).

% gcd_non_0_int
tff(fact_9850_gcd__code__int,axiom,
    ! [K2: int,L: int] :
      aa_int_int(aa_int_fun_int_int(gcd_gcd_int,K2),L) = aa_int_int(abs_abs_int,
        $ite(L = zero_zero_int,K2,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,L),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(abs_abs_int,K2)),aa_int_int(abs_abs_int,L))))) ).

% gcd_code_int
tff(fact_9851_drop__bit__int__def,axiom,
    ! [Nb: nat,K2: int] : aa_int_int(bit_se8568078237143864401it_int(Nb),K2) = aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% drop_bit_int_def
tff(fact_9852_nat__descend__induct,axiom,
    ! [Nb: nat,P: fun_nat_o,Ma: nat] :
      ( ! [K: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K)
         => aa_nat_o(P,K) )
     => ( ! [K: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
           => ( ! [I4: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),I4)
                 => aa_nat_o(P,I4) )
             => aa_nat_o(P,K) ) )
       => aa_nat_o(P,Ma) ) ) ).

% nat_descend_induct
tff(fact_9853_finite__enumerate,axiom,
    ! [S3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ? [R3: fun_nat_nat] :
          ( aa_set_nat_o(strict1292158309912662752at_nat(R3),aa_nat_set_nat(set_ord_lessThan_nat,finite_card_nat(S3)))
          & ! [N7: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N7),finite_card_nat(S3))
             => aa_set_nat_o(member_nat(aa_nat_nat(R3,N7)),S3) ) ) ) ).

% finite_enumerate
tff(fact_9854_divmod__integer__eq__cases,axiom,
    ! [K2: code_integer,L: code_integer] :
      code_divmod_integer(K2,L) = $ite(
        K2 = zero_z3403309356797280102nteger,
        produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger),
        $ite(
          L = zero_z3403309356797280102nteger,
          produc1086072967326762835nteger(zero_z3403309356797280102nteger,K2),
          aa_Pro4693618923521437820nteger(comp_C7848664941760151970nteger(comp_f6739083541385432444nteger(produc6499014454317279255nteger,times_3573771949741848930nteger),sgn_sgn_Code_integer,L),
            $ite(aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K2) = aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,L),code_divmod_abs(K2,L),produc6916734918728496179nteger(aTP_Lamm_sg(L),code_divmod_abs(K2,L)))) ) ) ).

% divmod_integer_eq_cases
tff(fact_9855_gcd__1__nat,axiom,
    ! [Ma: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Ma),one_one_nat) = one_one_nat ).

% gcd_1_nat
tff(fact_9856_gcd__pos__nat,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Ma),Nb))
    <=> ( ( Ma != zero_zero_nat )
        | ( Nb != zero_zero_nat ) ) ) ).

% gcd_pos_nat
tff(fact_9857_drop__bit__of__Suc__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = aa_o_nat(zero_n2687167440665602831ol_nat,Nb = zero_zero_nat) ).

% drop_bit_of_Suc_0
tff(fact_9858_gcd__mult__distrib__nat,axiom,
    ! [K2: nat,Ma: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Ma),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Ma)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K2),Nb)) ).

% gcd_mult_distrib_nat
tff(fact_9859_gcd__red__nat,axiom,
    ! [X: nat,Y2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Y2) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Y2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y2)) ).

% gcd_red_nat
tff(fact_9860_gcd__le1__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),A) ) ).

% gcd_le1_nat
tff(fact_9861_gcd__le2__nat,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)),B) ) ).

% gcd_le2_nat
tff(fact_9862_gcd__diff1__nat,axiom,
    ! [Nb: nat,Ma: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Ma)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ma),Nb)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Ma),Nb) ) ) ).

% gcd_diff1_nat
tff(fact_9863_gcd__diff2__nat,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ma),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Ma)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Ma),Nb) ) ) ).

% gcd_diff2_nat
tff(fact_9864_gcd__nat_Oelims,axiom,
    ! [X: nat,Xa: nat,Y2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Xa) = Y2 )
     => ( Y2 = $ite(Xa = zero_zero_nat,X,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa))) ) ) ).

% gcd_nat.elims
tff(fact_9865_gcd__nat_Osimps,axiom,
    ! [X: nat,Y2: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Y2) = $ite(Y2 = zero_zero_nat,X,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Y2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y2))) ).

% gcd_nat.simps
tff(fact_9866_gcd__non__0__nat,axiom,
    ! [Y2: nat,X: nat] :
      ( ( Y2 != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Y2) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Y2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Y2)) ) ) ).

% gcd_non_0_nat
tff(fact_9867_drop__bit__nat__eq,axiom,
    ! [Nb: nat,K2: int] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,Nb),aa_int_nat(nat2,K2)) = aa_int_nat(nat2,aa_int_int(bit_se8568078237143864401it_int(Nb),K2)) ).

% drop_bit_nat_eq
tff(fact_9868_bezout__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [X4: nat,Y4: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y4)),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)) ) ).

% bezout_nat
tff(fact_9869_bezout__gcd__nat_H,axiom,
    ! [B: nat,A: nat] :
    ? [X4: nat,Y4: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X4))
        & ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y4)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ) )
      | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X4))
        & ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y4)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ) ) ) ).

% bezout_gcd_nat'
tff(fact_9870_gcd__code__integer,axiom,
    ! [K2: code_integer,L: code_integer] :
      gcd_gcd_Code_integer(K2,L) = abs_abs_Code_integer($ite(L = zero_z3403309356797280102nteger,K2,gcd_gcd_Code_integer(L,modulo364778990260209775nteger(abs_abs_Code_integer(K2),abs_abs_Code_integer(L))))) ).

% gcd_code_integer
tff(fact_9871_drop__bit__nat__def,axiom,
    ! [Nb: nat,Ma: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,Nb),Ma) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% drop_bit_nat_def
tff(fact_9872_bezw__aux,axiom,
    ! [X: nat,Y2: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Y2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,bezw(X,Y2))),aa_nat_int(semiri1314217659103216013at_int,X))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(X,Y2))),aa_nat_int(semiri1314217659103216013at_int,Y2))) ).

% bezw_aux
tff(fact_9873_gcd__nat_Opelims,axiom,
    ! [X: nat,Xa: nat,Y2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,X),Xa) = Y2 )
     => ( aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(gcd_nat_rel),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X),Xa))
       => ~ ( ( Y2 = $ite(Xa = zero_zero_nat,X,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa))) )
           => ~ aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(gcd_nat_rel),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,X),Xa)) ) ) ) ).

% gcd_nat.pelims
tff(fact_9874_card__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : finite_card_int(set_or5832277885323065728an_int(L,U)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,U),aa_int_int(aa_int_fun_int_int(plus_plus_int,L),one_one_int))) ).

% card_greaterThanLessThan_int
tff(fact_9875_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : set_or4662586982721622107an_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,L),one_one_int),U) = set_or5832277885323065728an_int(L,U) ).

% atLeastPlusOneLessThan_greaterThanLessThan_int
tff(fact_9876_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [Nb: nat,J: nat,I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),aa_nat_nat(suc,I2)))
     => ( aa_nat_nat(nth_nat(linord2614967742042102400et_nat(set_or5834768355832116004an_nat(I2,J))),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),Nb)) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
tff(fact_9877_xor__minus__numerals_I2_J,axiom,
    ! [K2: int,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K2),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K2),aa_num_int(neg_numeral_sub_int(Nb),one))) ).

% xor_minus_numerals(2)
tff(fact_9878_xor__minus__numerals_I1_J,axiom,
    ! [Nb: num,K2: int] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))),K2) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_num_int(neg_numeral_sub_int(Nb),one)),K2)) ).

% xor_minus_numerals(1)
tff(fact_9879_tanh__real__bounds,axiom,
    ! [X: real] : member_real(aa_real_real(tanh_real,X),set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,one_one_real),one_one_real)) ).

% tanh_real_bounds
tff(fact_9880_greaterThanLessThan__upt,axiom,
    ! [Nb: nat,Ma: nat] : set_or5834768355832116004an_nat(Nb,Ma) = set_nat2(upt(aa_nat_nat(suc,Nb),Ma)) ).

% greaterThanLessThan_upt
tff(fact_9881_sub__BitM__One__eq,axiom,
    ! [Nb: num] : aa_num_int(neg_numeral_sub_int(aa_num_num(bitM,Nb)),one) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_int(neg_numeral_sub_int(Nb),one)) ).

% sub_BitM_One_eq
tff(fact_9882_Suc__funpow,axiom,
    ! [Nb: nat] : compow_fun_nat_nat(Nb,suc) = aa_nat_fun_nat_nat(plus_plus_nat,Nb) ).

% Suc_funpow
tff(fact_9883_max__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1623282765462674594er_nat(ord_max_nat,zero_zero_nat,aTP_Lamm_am,aTP_Lamm_qi) ).

% max_nat.semilattice_neutr_order_axioms
tff(fact_9884_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1623282765462674594er_nat(gcd_gcd_nat,zero_zero_nat,dvd_dvd_nat,aTP_Lamm_sh) ).

% gcd_nat.semilattice_neutr_order_axioms
tff(fact_9885_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [Nb: nat,J: nat,I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I2))
     => ( aa_nat_nat(nth_nat(linord2614967742042102400et_nat(set_or6659071591806873216st_nat(I2,J))),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),Nb)) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
tff(fact_9886_greaterThanAtMost__upt,axiom,
    ! [Nb: nat,Ma: nat] : set_or6659071591806873216st_nat(Nb,Ma) = set_nat2(upt(aa_nat_nat(suc,Nb),aa_nat_nat(suc,Ma))) ).

% greaterThanAtMost_upt
tff(fact_9887_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : set_or1266510415728281911st_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,L),one_one_int),U) = set_or6656581121297822940st_int(L,U) ).

% atLeastPlusOneAtMost_greaterThanAtMost_int
tff(fact_9888_times__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa)),aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_sj),Xa),X)) ).

% times_int.abs_eq
tff(fact_9889_uminus__int_Oabs__eq,axiom,
    ! [X: product_prod_nat_nat] : aa_int_int(uminus_uminus_int,aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_sk),X)) ).

% uminus_int.abs_eq
tff(fact_9890_one__int__def,axiom,
    one_one_int = aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,one_one_nat),zero_zero_nat)) ).

% one_int_def
tff(fact_9891_less__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa)),aa_Pro4240683299535347697at_int(abs_Integ,X))
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(aTP_Lamm_sm),Xa),X) ) ).

% less_int.abs_eq
tff(fact_9892_less__eq__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa)),aa_Pro4240683299535347697at_int(abs_Integ,X))
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(aTP_Lamm_so),Xa),X) ) ).

% less_eq_int.abs_eq
tff(fact_9893_plus__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa)),aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_sq),Xa),X)) ).

% plus_int.abs_eq
tff(fact_9894_minus__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro4240683299535347697at_int(abs_Integ,Xa)),aa_Pro4240683299535347697at_int(abs_Integ,X)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_ss),Xa),X)) ).

% minus_int.abs_eq
tff(fact_9895_less__eq__int_Orep__eq,axiom,
    ! [X: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Xa)
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(aTP_Lamm_so),aa_int1417691210696471063at_nat(rep_Integ,X)),aa_int1417691210696471063at_nat(rep_Integ,Xa)) ) ).

% less_eq_int.rep_eq
tff(fact_9896_less__int_Orep__eq,axiom,
    ! [X: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Xa)
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(aTP_Lamm_sm),aa_int1417691210696471063at_nat(rep_Integ,X)),aa_int1417691210696471063at_nat(rep_Integ,Xa)) ) ).

% less_int.rep_eq
tff(fact_9897_upt__conv__Nil,axiom,
    ! [J: nat,I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I2)
     => ( upt(I2,J) = nil_nat ) ) ).

% upt_conv_Nil
tff(fact_9898_upt__eq__Nil__conv,axiom,
    ! [I2: nat,J: nat] :
      ( ( upt(I2,J) = nil_nat )
    <=> ( ( J = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I2) ) ) ).

% upt_eq_Nil_conv
tff(fact_9899_upt__0,axiom,
    ! [I2: nat] : upt(I2,zero_zero_nat) = nil_nat ).

% upt_0
tff(fact_9900_hd__upt,axiom,
    ! [I2: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
     => ( hd_nat(upt(I2,J)) = I2 ) ) ).

% hd_upt
tff(fact_9901_uminus__int__def,axiom,
    uminus_uminus_int = aa_fun8399641135911395862nt_int(map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc2626176000494625587at_nat(aTP_Lamm_sk)) ).

% uminus_int_def
tff(fact_9902_bij__betw__Suc,axiom,
    ! [M6: set_nat,N2: set_nat] :
      ( aa_set_nat_o(bij_betw_nat_nat(suc,M6),N2)
    <=> ( aa_set_nat_set_nat(image_nat_nat(suc),M6) = N2 ) ) ).

% bij_betw_Suc
tff(fact_9903_image__Suc__lessThan,axiom,
    ! [Nb: nat] : aa_set_nat_set_nat(image_nat_nat(suc),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = set_or1269000886237332187st_nat(one_one_nat,Nb) ).

% image_Suc_lessThan
tff(fact_9904_image__Suc__atMost,axiom,
    ! [Nb: nat] : aa_set_nat_set_nat(image_nat_nat(suc),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = set_or1269000886237332187st_nat(one_one_nat,aa_nat_nat(suc,Nb)) ).

% image_Suc_atMost
tff(fact_9905_image__atLeastZeroLessThan__int,axiom,
    ! [U: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),U)
     => ( set_or4662586982721622107an_int(zero_zero_int,U) = image_nat_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_ord_lessThan_nat,aa_int_nat(nat2,U))) ) ) ).

% image_atLeastZeroLessThan_int
tff(fact_9906_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C: nat,X: nat,Y2: nat] :
      aa_set_nat_set_nat(image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_st,C)),set_or4665077453230672383an_nat(X,Y2)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),Y2),
        set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Y2),C)),
        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y2),aa_set_nat_set_nat(insert_nat(zero_zero_nat),bot_bot_set_nat),bot_bot_set_nat) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_9907_Inf__real__def,axiom,
    ! [X7: set_real] : comple4887499456419720421f_real(X7) = aa_real_real(uminus_uminus_real,comple1385675409528146559p_real(image_real_real(uminus_uminus_real,X7))) ).

% Inf_real_def
tff(fact_9908_finite__int__iff__bounded,axiom,
    ! [S3: set_int] :
      ( finite_finite_int(S3)
    <=> ? [K3: int] : ord_less_eq_set_int(image_int_int(abs_abs_int,S3),aa_int_set_int(set_ord_lessThan_int,K3)) ) ).

% finite_int_iff_bounded
tff(fact_9909_finite__int__iff__bounded__le,axiom,
    ! [S3: set_int] :
      ( finite_finite_int(S3)
    <=> ? [K3: int] : ord_less_eq_set_int(image_int_int(abs_abs_int,S3),set_ord_atMost_int(K3)) ) ).

% finite_int_iff_bounded_le
tff(fact_9910_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] : image_int_int(aa_int_fun_int_int(aTP_Lamm_su,L),set_or4662586982721622107an_int(zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,U),L))) = set_or4662586982721622107an_int(L,U) ).

% image_add_int_atLeastLessThan
tff(fact_9911_times__int__def,axiom,
    times_times_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_sj)) ).

% times_int_def
tff(fact_9912_minus__int__def,axiom,
    minus_minus_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_ss)) ).

% minus_int_def
tff(fact_9913_plus__int__def,axiom,
    plus_plus_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_sq)) ).

% plus_int_def
tff(fact_9914_remdups__upt,axiom,
    ! [Ma: nat,Nb: nat] : remdups_nat(upt(Ma,Nb)) = upt(Ma,Nb) ).

% remdups_upt
tff(fact_9915_suminf__eq__SUP__real,axiom,
    ! [X7: fun_nat_real] :
      ( summable_real(X7)
     => ( ! [I3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(X7,I3))
       => ( suminf_real(X7) = comple1385675409528146559p_real(image_nat_real(aTP_Lamm_sv(X7),top_top_set_nat)) ) ) ) ).

% suminf_eq_SUP_real
tff(fact_9916_UN__atMost__UNIV,axiom,
    comple7399068483239264473et_nat(image_nat_set_nat(set_ord_atMost_nat,top_top_set_nat)) = top_top_set_nat ).

% UN_atMost_UNIV
tff(fact_9917_UN__lessThan__UNIV,axiom,
    comple7399068483239264473et_nat(image_nat_set_nat(set_ord_lessThan_nat,top_top_set_nat)) = top_top_set_nat ).

% UN_lessThan_UNIV
tff(fact_9918_range__mod,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_set_nat_set_nat(image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_sw,Nb)),top_top_set_nat) = set_or4665077453230672383an_nat(zero_zero_nat,Nb) ) ) ).

% range_mod
tff(fact_9919_card__UNIV__unit,axiom,
    finite410649719033368117t_unit(top_to1996260823553986621t_unit) = one_one_nat ).

% card_UNIV_unit
tff(fact_9920_card__UNIV__bool,axiom,
    finite_card_o(top_top_set_o) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% card_UNIV_bool
tff(fact_9921_range__mult,axiom,
    ! [A: real] :
      image_real_real(times_times_real(A),top_top_set_real) = $ite(A = zero_zero_real,insert_real(zero_zero_real,bot_bot_set_real),top_top_set_real) ).

% range_mult
tff(fact_9922_top__enat__def,axiom,
    top_to3028658606643905974d_enat = extend5688581933313929465d_enat ).

% top_enat_def
tff(fact_9923_root__def,axiom,
    ! [Nb: nat,X: real] :
      aa_real_real(aa_nat_fun_real_real(root,Nb),X) = $ite(Nb = zero_zero_nat,zero_zero_real,aa_real_real(the_in5290026491893676941l_real(top_top_set_real,aa_nat_fun_real_real(aTP_Lamm_sx,Nb)),X)) ).

% root_def
tff(fact_9924_card__UNIV__char,axiom,
    finite_card_char(top_top_set_char) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))))) ).

% card_UNIV_char
tff(fact_9925_UNIV__char__of__nat,axiom,
    top_top_set_char = image_nat_char(unique3096191561947761185of_nat,set_or4665077453230672383an_nat(zero_zero_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))))))) ).

% UNIV_char_of_nat
tff(fact_9926_nat__of__char__less__256,axiom,
    ! [C: char] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_char_nat(comm_s629917340098488124ar_nat,C)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one)))))))))) ).

% nat_of_char_less_256
tff(fact_9927_range__nat__of__char,axiom,
    image_char_nat(comm_s629917340098488124ar_nat,top_top_set_char) = set_or4665077453230672383an_nat(zero_zero_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one)))))))))) ).

% range_nat_of_char
tff(fact_9928_integer__of__char__code,axiom,
    ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] : integer_of_char(aa_o_char(char2((B0),(B1),(B22),(B32),(B42),(B52),(B62)),(B72))) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_n356916108424825756nteger((B72))),numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_n356916108424825756nteger((B62)))),numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_n356916108424825756nteger((B52)))),numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_n356916108424825756nteger((B42)))),numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_n356916108424825756nteger((B32)))),numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_n356916108424825756nteger((B22)))),numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_n356916108424825756nteger((B1)))),numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_n356916108424825756nteger((B0))) ).

% integer_of_char_code
tff(fact_9929_char__of__integer__code,axiom,
    ! [K2: code_integer] : char_of_integer(K2) = produc4188289175737317920o_char(aTP_Lamm_tf,code_bit_cut_integer(K2)) ).

% char_of_integer_code
tff(fact_9930_String_Ochar__of__ascii__of,axiom,
    ! [C: char] : aa_char_nat(comm_s629917340098488124ar_nat,ascii_of(C)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,one)))),aa_char_nat(comm_s629917340098488124ar_nat,C)) ).

% String.char_of_ascii_of
tff(fact_9931_upt__rec__numeral,axiom,
    ! [Ma: num,Nb: num] :
      upt(aa_num_nat(numeral_numeral_nat,Ma),aa_num_nat(numeral_numeral_nat,Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Ma)),aa_num_nat(numeral_numeral_nat,Nb)),cons_nat(aa_num_nat(numeral_numeral_nat,Ma),upt(aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,Ma)),aa_num_nat(numeral_numeral_nat,Nb))),nil_nat) ).

% upt_rec_numeral
tff(fact_9932_upt__conv__Cons__Cons,axiom,
    ! [Ma: nat,Nb: nat,Ns: list_nat,Q: nat] :
      ( ( cons_nat(Ma,cons_nat(Nb,Ns)) = upt(Ma,Q) )
    <=> ( cons_nat(Nb,Ns) = upt(aa_nat_nat(suc,Ma),Q) ) ) ).

% upt_conv_Cons_Cons
tff(fact_9933_upt__conv__Cons,axiom,
    ! [I2: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
     => ( upt(I2,J) = cons_nat(I2,upt(aa_nat_nat(suc,I2),J)) ) ) ).

% upt_conv_Cons
tff(fact_9934_upt__eq__Cons__conv,axiom,
    ! [I2: nat,J: nat,X: nat,Xs: list_nat] :
      ( ( upt(I2,J) = cons_nat(X,Xs) )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
        & ( I2 = X )
        & ( upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),one_one_nat),J) = Xs ) ) ) ).

% upt_eq_Cons_conv
tff(fact_9935_upt__rec,axiom,
    ! [I2: nat,J: nat] :
      upt(I2,J) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J),cons_nat(I2,upt(aa_nat_nat(suc,I2),J)),nil_nat) ).

% upt_rec
tff(fact_9936_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I2: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,I2)),J)
     => ( linord2614967742042102400et_nat(set_or6659071591806873216st_nat(I2,J)) = cons_nat(aa_nat_nat(suc,I2),linord2614967742042102400et_nat(set_or6659071591806873216st_nat(aa_nat_nat(suc,I2),J))) ) ) ).

% sorted_list_of_set_greaterThanAtMost
tff(fact_9937_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I2: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I2)),J)
     => ( linord2614967742042102400et_nat(set_or5834768355832116004an_nat(I2,J)) = cons_nat(aa_nat_nat(suc,I2),linord2614967742042102400et_nat(set_or5834768355832116004an_nat(aa_nat_nat(suc,I2),J))) ) ) ).

% sorted_list_of_set_greaterThanLessThan
tff(fact_9938_tl__upt,axiom,
    ! [Ma: nat,Nb: nat] : tl_nat(upt(Ma,Nb)) = upt(aa_nat_nat(suc,Ma),Nb) ).

% tl_upt
tff(fact_9939_upto__aux__rec,axiom,
    ! [I2: int,J: int,Js: list_int] :
      upto_aux(I2,J,Js) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,J),I2),Js,upto_aux(I2,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int),cons_int(J,Js))) ).

% upto_aux_rec
tff(fact_9940_sorted__list__of__set__lessThan__Suc,axiom,
    ! [K2: nat] : linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,K2))) = append_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_lessThan_nat,K2)),cons_nat(K2,nil_nat)) ).

% sorted_list_of_set_lessThan_Suc
tff(fact_9941_sorted__list__of__set__atMost__Suc,axiom,
    ! [K2: nat] : linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,K2))) = append_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_atMost_nat,K2)),cons_nat(aa_nat_nat(suc,K2),nil_nat)) ).

% sorted_list_of_set_atMost_Suc
tff(fact_9942_sup__enat__def,axiom,
    sup_su3973961784419623482d_enat = ord_ma741700101516333627d_enat ).

% sup_enat_def
tff(fact_9943_sup__nat__def,axiom,
    sup_sup_nat = ord_max_nat ).

% sup_nat_def
tff(fact_9944_upt__add__eq__append,axiom,
    ! [I2: nat,J: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => ( upt(I2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K2)) = append_nat(upt(I2,J),upt(J,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K2))) ) ) ).

% upt_add_eq_append
tff(fact_9945_atLeastLessThan__add__Un,axiom,
    ! [I2: nat,J: nat,K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => ( set_or4665077453230672383an_nat(I2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K2)) = aa_set_nat_set_nat(sup_sup_set_nat(set_or4665077453230672383an_nat(I2,J)),set_or4665077453230672383an_nat(J,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K2))) ) ) ).

% atLeastLessThan_add_Un
tff(fact_9946_upt__Suc,axiom,
    ! [I2: nat,J: nat] :
      upt(I2,aa_nat_nat(suc,J)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J),append_nat(upt(I2,J),cons_nat(J,nil_nat)),nil_nat) ).

% upt_Suc
tff(fact_9947_upt__Suc__append,axiom,
    ! [I2: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => ( upt(I2,aa_nat_nat(suc,J)) = append_nat(upt(I2,J),cons_nat(J,nil_nat)) ) ) ).

% upt_Suc_append
tff(fact_9948_drop__upt,axiom,
    ! [Ma: nat,I2: nat,J: nat] : drop_nat(Ma,upt(I2,J)) = upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),Ma),J) ).

% drop_upt
tff(fact_9949_upto_Opsimps,axiom,
    ! [I2: int,J: int] :
      ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,I2),J))
     => ( upto(I2,J) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),J),cons_int(I2,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int),J)),nil_int) ) ) ).

% upto.psimps
tff(fact_9950_upto__empty,axiom,
    ! [J: int,I2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,J),I2)
     => ( upto(I2,J) = nil_int ) ) ).

% upto_empty
tff(fact_9951_upto__Nil2,axiom,
    ! [I2: int,J: int] :
      ( ( nil_int = upto(I2,J) )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,J),I2) ) ).

% upto_Nil2
tff(fact_9952_upto__Nil,axiom,
    ! [I2: int,J: int] :
      ( ( upto(I2,J) = nil_int )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,J),I2) ) ).

% upto_Nil
tff(fact_9953_upto__single,axiom,
    ! [I2: int] : upto(I2,I2) = cons_int(I2,nil_int) ).

% upto_single
tff(fact_9954_nth__upto,axiom,
    ! [I2: int,K2: nat,J: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),aa_nat_int(semiri1314217659103216013at_int,K2))),J)
     => ( aa_nat_int(nth_int(upto(I2,J)),K2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),aa_nat_int(semiri1314217659103216013at_int,K2)) ) ) ).

% nth_upto
tff(fact_9955_length__upto,axiom,
    ! [I2: int,J: int] : size_size_list_int(upto(I2,J)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),I2)),one_one_int)) ).

% length_upto
tff(fact_9956_upto__rec__numeral_I1_J,axiom,
    ! [Ma: num,Nb: num] :
      upto(aa_num_int(numeral_numeral_int,Ma),aa_num_int(numeral_numeral_int,Nb)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Ma)),aa_num_int(numeral_numeral_int,Nb)),cons_int(aa_num_int(numeral_numeral_int,Ma),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Ma)),one_one_int),aa_num_int(numeral_numeral_int,Nb))),nil_int) ).

% upto_rec_numeral(1)
tff(fact_9957_upto__rec__numeral_I2_J,axiom,
    ! [Ma: num,Nb: num] :
      upto(aa_num_int(numeral_numeral_int,Ma),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Ma)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))),cons_int(aa_num_int(numeral_numeral_int,Ma),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Ma)),one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)))),nil_int) ).

% upto_rec_numeral(2)
tff(fact_9958_upto__rec__numeral_I3_J,axiom,
    ! [Ma: num,Nb: num] :
      upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Ma)),aa_num_int(numeral_numeral_int,Nb)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Ma))),aa_num_int(numeral_numeral_int,Nb)),cons_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Ma)),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Ma))),one_one_int),aa_num_int(numeral_numeral_int,Nb))),nil_int) ).

% upto_rec_numeral(3)
tff(fact_9959_upto__rec__numeral_I4_J,axiom,
    ! [Ma: num,Nb: num] :
      upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Ma)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Ma))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))),cons_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Ma)),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Ma))),one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)))),nil_int) ).

% upto_rec_numeral(4)
tff(fact_9960_upto__aux__def,axiom,
    ! [I2: int,J: int,Js: list_int] : upto_aux(I2,J,Js) = append_int(upto(I2,J),Js) ).

% upto_aux_def
tff(fact_9961_sorted__upto,axiom,
    ! [Ma: int,Nb: int] : aa_list_int_o(sorted_wrt_int(ord_less_eq_int),upto(Ma,Nb)) ).

% sorted_upto
tff(fact_9962_sorted__wrt__upto,axiom,
    ! [I2: int,J: int] : aa_list_int_o(sorted_wrt_int(ord_less_int),upto(I2,J)) ).

% sorted_wrt_upto
tff(fact_9963_atLeastAtMost__upto,axiom,
    ! [I2: int,J: int] : set_or1266510415728281911st_int(I2,J) = set_int2(upto(I2,J)) ).

% atLeastAtMost_upto
tff(fact_9964_distinct__upto,axiom,
    ! [I2: int,J: int] : aa_list_int_o(distinct_int,upto(I2,J)) ).

% distinct_upto
tff(fact_9965_upto__code,axiom,
    ! [I2: int,J: int] : upto(I2,J) = upto_aux(I2,J,nil_int) ).

% upto_code
tff(fact_9966_upto__split2,axiom,
    ! [I2: int,J: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J),K2)
       => ( upto(I2,K2) = append_int(upto(I2,J),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,J),one_one_int),K2)) ) ) ) ).

% upto_split2
tff(fact_9967_upto__split1,axiom,
    ! [I2: int,J: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J),K2)
       => ( upto(I2,K2) = append_int(upto(I2,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),upto(J,K2)) ) ) ) ).

% upto_split1
tff(fact_9968_atLeastLessThan__upto,axiom,
    ! [I2: int,J: int] : set_or4662586982721622107an_int(I2,J) = set_int2(upto(I2,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int))) ).

% atLeastLessThan_upto
tff(fact_9969_greaterThanAtMost__upto,axiom,
    ! [I2: int,J: int] : set_or6656581121297822940st_int(I2,J) = set_int2(upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int),J)) ).

% greaterThanAtMost_upto
tff(fact_9970_upto__rec1,axiom,
    ! [I2: int,J: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),J)
     => ( upto(I2,J) = cons_int(I2,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int),J)) ) ) ).

% upto_rec1
tff(fact_9971_upto_Osimps,axiom,
    ! [I2: int,J: int] :
      upto(I2,J) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),J),cons_int(I2,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int),J)),nil_int) ).

% upto.simps
tff(fact_9972_upto_Oelims,axiom,
    ! [X: int,Xa: int,Y2: list_int] :
      ( ( upto(X,Xa) = Y2 )
     => ( Y2 = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Xa),cons_int(X,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,X),one_one_int),Xa)),nil_int) ) ) ).

% upto.elims
tff(fact_9973_upto__rec2,axiom,
    ! [I2: int,J: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),J)
     => ( upto(I2,J) = append_int(upto(I2,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),cons_int(J,nil_int)) ) ) ).

% upto_rec2
tff(fact_9974_greaterThanLessThan__upto,axiom,
    ! [I2: int,J: int] : set_or5832277885323065728an_int(I2,J) = set_int2(upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int))) ).

% greaterThanLessThan_upto
tff(fact_9975_upto__split3,axiom,
    ! [I2: int,J: int,K2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J),K2)
       => ( upto(I2,K2) = append_int(upto(I2,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),cons_int(J,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,J),one_one_int),K2))) ) ) ) ).

% upto_split3
tff(fact_9976_upto_Opelims,axiom,
    ! [X: int,Xa: int,Y2: list_int] :
      ( ( upto(X,Xa) = Y2 )
     => ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Xa))
       => ~ ( ( Y2 = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Xa),cons_int(X,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,X),one_one_int),Xa)),nil_int) )
           => ~ aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Xa)) ) ) ) ).

% upto.pelims
tff(fact_9977_take__bit__numeral__minus__numeral__int,axiom,
    ! [Ma: num,Nb: num] : aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,Ma)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = case_option_int_num(zero_zero_int,aTP_Lamm_tg(Ma),bit_take_bit_num(aa_num_nat(numeral_numeral_nat,Ma),Nb)) ).

% take_bit_numeral_minus_numeral_int
tff(fact_9978_take__bit__num__simps_I1_J,axiom,
    ! [Ma: num] : bit_take_bit_num(zero_zero_nat,Ma) = none_num ).

% take_bit_num_simps(1)
tff(fact_9979_take__bit__num__simps_I2_J,axiom,
    ! [Nb: nat] : bit_take_bit_num(aa_nat_nat(suc,Nb),one) = some_num(one) ).

% take_bit_num_simps(2)
tff(fact_9980_take__bit__num__simps_I5_J,axiom,
    ! [R: num] : bit_take_bit_num(aa_num_nat(numeral_numeral_nat,R),one) = some_num(one) ).

% take_bit_num_simps(5)
tff(fact_9981_take__bit__num__simps_I3_J,axiom,
    ! [Nb: nat,Ma: num] : bit_take_bit_num(aa_nat_nat(suc,Nb),aa_num_num(bit0,Ma)) = case_o6005452278849405969um_num(none_num,aTP_Lamm_th,bit_take_bit_num(Nb,Ma)) ).

% take_bit_num_simps(3)
tff(fact_9982_take__bit__num__simps_I4_J,axiom,
    ! [Nb: nat,Ma: num] : bit_take_bit_num(aa_nat_nat(suc,Nb),aa_num_num(bit1,Ma)) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(Nb,Ma))) ).

% take_bit_num_simps(4)
tff(fact_9983_take__bit__num__simps_I6_J,axiom,
    ! [R: num,Ma: num] : bit_take_bit_num(aa_num_nat(numeral_numeral_nat,R),aa_num_num(bit0,Ma)) = case_o6005452278849405969um_num(none_num,aTP_Lamm_th,bit_take_bit_num(aa_num_nat(pred_numeral,R),Ma)) ).

% take_bit_num_simps(6)
tff(fact_9984_take__bit__num__simps_I7_J,axiom,
    ! [R: num,Ma: num] : bit_take_bit_num(aa_num_nat(numeral_numeral_nat,R),aa_num_num(bit1,Ma)) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(aa_num_nat(pred_numeral,R),Ma))) ).

% take_bit_num_simps(7)
tff(fact_9985_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
    ! [Nb: nat,Ma: num] : bit_take_bit_num(Nb,aa_num_num(bit0,Ma)) = case_nat_option_num(none_num,aTP_Lamm_ti(Ma),Nb) ).

% Code_Abstract_Nat.take_bit_num_code(2)
tff(fact_9986_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
    ! [Nb: nat] : bit_take_bit_num(Nb,one) = case_nat_option_num(none_num,aTP_Lamm_tj,Nb) ).

% Code_Abstract_Nat.take_bit_num_code(1)
tff(fact_9987_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
    ! [Nb: nat,Ma: num] : bit_take_bit_num(Nb,aa_num_num(bit1,Ma)) = case_nat_option_num(none_num,aTP_Lamm_tk(Ma),Nb) ).

% Code_Abstract_Nat.take_bit_num_code(3)
tff(fact_9988_and__minus__numerals_I7_J,axiom,
    ! [Nb: num,Ma: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))),aa_num_int(numeral_numeral_int,Ma)) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(Ma,aa_num_num(bitM,Nb))) ).

% and_minus_numerals(7)
tff(fact_9989_and__minus__numerals_I3_J,axiom,
    ! [Ma: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,Ma)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(Ma,aa_num_num(bitM,Nb))) ).

% and_minus_numerals(3)
tff(fact_9990_and__minus__numerals_I4_J,axiom,
    ! [Ma: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,Ma)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(Ma,aa_num_num(bit0,Nb))) ).

% and_minus_numerals(4)
tff(fact_9991_and__minus__numerals_I8_J,axiom,
    ! [Nb: num,Ma: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))),aa_num_int(numeral_numeral_int,Ma)) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(Ma,aa_num_num(bit0,Nb))) ).

% and_minus_numerals(8)
tff(fact_9992_and__not__num_Osimps_I3_J,axiom,
    ! [Nb: num] : bit_and_not_num(one,aa_num_num(bit1,Nb)) = none_num ).

% and_not_num.simps(3)
tff(fact_9993_and__not__num_Osimps_I1_J,axiom,
    bit_and_not_num(one,one) = none_num ).

% and_not_num.simps(1)
tff(fact_9994_and__not__num_Osimps_I2_J,axiom,
    ! [Nb: num] : bit_and_not_num(one,aa_num_num(bit0,Nb)) = some_num(one) ).

% and_not_num.simps(2)
tff(fact_9995_and__not__num_Osimps_I4_J,axiom,
    ! [Ma: num] : bit_and_not_num(aa_num_num(bit0,Ma),one) = some_num(aa_num_num(bit0,Ma)) ).

% and_not_num.simps(4)
tff(fact_9996_and__not__num_Osimps_I7_J,axiom,
    ! [Ma: num] : bit_and_not_num(aa_num_num(bit1,Ma),one) = some_num(aa_num_num(bit0,Ma)) ).

% and_not_num.simps(7)
tff(fact_9997_and__not__num__eq__Some__iff,axiom,
    ! [Ma: num,Nb: num,Q: num] :
      ( ( bit_and_not_num(Ma,Nb) = some_num(Q) )
    <=> ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,Ma)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))) = aa_num_int(numeral_numeral_int,Q) ) ) ).

% and_not_num_eq_Some_iff
tff(fact_9998_and__not__num_Osimps_I8_J,axiom,
    ! [Ma: num,Nb: num] : bit_and_not_num(aa_num_num(bit1,Ma),aa_num_num(bit0,Nb)) = case_o6005452278849405969um_num(some_num(one),aTP_Lamm_tl,bit_and_not_num(Ma,Nb)) ).

% and_not_num.simps(8)
tff(fact_9999_and__not__num__eq__None__iff,axiom,
    ! [Ma: num,Nb: num] :
      ( ( bit_and_not_num(Ma,Nb) = none_num )
    <=> ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,Ma)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))) = zero_zero_int ) ) ).

% and_not_num_eq_None_iff
tff(fact_10000_int__numeral__not__and__num,axiom,
    ! [Ma: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Ma))),aa_num_int(numeral_numeral_int,Nb)) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(Nb,Ma)) ).

% int_numeral_not_and_num
tff(fact_10001_int__numeral__and__not__num,axiom,
    ! [Ma: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,Ma)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(Ma,Nb)) ).

% int_numeral_and_not_num
tff(fact_10002_Bit__Operations_Otake__bit__num__code,axiom,
    ! [Nb: nat,Ma: num] : bit_take_bit_num(Nb,Ma) = produc478579273971653890on_num(aTP_Lamm_tp,aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,Nb),Ma)) ).

% Bit_Operations.take_bit_num_code
tff(fact_10003_take__bit__num__def,axiom,
    ! [Nb: nat,Ma: num] :
      bit_take_bit_num(Nb,Ma) = $ite(aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_num_nat(numeral_numeral_nat,Ma)) = zero_zero_nat,none_num,some_num(aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_num_nat(numeral_numeral_nat,Ma))))) ).

% take_bit_num_def
tff(fact_10004_num__of__nat__numeral__eq,axiom,
    ! [Q: num] : aa_nat_num(num_of_nat,aa_num_nat(numeral_numeral_nat,Q)) = Q ).

% num_of_nat_numeral_eq
tff(fact_10005_num__of__nat_Osimps_I1_J,axiom,
    aa_nat_num(num_of_nat,zero_zero_nat) = one ).

% num_of_nat.simps(1)
tff(fact_10006_numeral__num__of__nat,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_num_nat(numeral_numeral_nat,aa_nat_num(num_of_nat,Nb)) = Nb ) ) ).

% numeral_num_of_nat
tff(fact_10007_num__of__nat__One,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),one_one_nat)
     => ( aa_nat_num(num_of_nat,Nb) = one ) ) ).

% num_of_nat_One
tff(fact_10008_num__of__nat__double,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Nb)) = aa_num_num(bit0,aa_nat_num(num_of_nat,Nb)) ) ) ).

% num_of_nat_double
tff(fact_10009_num__of__nat__plus__distrib,axiom,
    ! [Ma: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ma)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ma),Nb)) = aa_num_num(plus_plus_num(aa_nat_num(num_of_nat,Ma)),aa_nat_num(num_of_nat,Nb)) ) ) ) ).

% num_of_nat_plus_distrib
tff(fact_10010_num__of__nat_Osimps_I2_J,axiom,
    ! [Nb: nat] :
      aa_nat_num(num_of_nat,aa_nat_nat(suc,Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb),aa_num_num(inc,aa_nat_num(num_of_nat,Nb)),one) ).

% num_of_nat.simps(2)
tff(fact_10011_DERIV__real__root__generic,axiom,
    ! [Nb: nat,X: real,D3: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( X != zero_zero_real )
       => ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
             => ( D3 = aa_real_real(inverse_inverse_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) ) ) )
         => ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),zero_zero_real)
               => ( D3 = aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat)))))) ) ) )
           => ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
               => ( D3 = aa_real_real(inverse_inverse_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) ) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Nb),D3,topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ) ) ).

% DERIV_real_root_generic
tff(fact_10012_DERIV__mirror,axiom,
    ! [F: fun_real_real,Y2: real,X: real] :
      ( has_fi5821293074295781190e_real(F,Y2,topolo2177554685111907308n_real(aa_real_real(uminus_uminus_real,X),top_top_set_real))
    <=> has_fi5821293074295781190e_real(aTP_Lamm_tq(F),aa_real_real(uminus_uminus_real,Y2),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_mirror
tff(fact_10013_DERIV__const__ratio__const,axiom,
    ! [A: real,B: real,F: fun_real_real,K2: real] :
      ( ( A != B )
     => ( ! [X4: real] : has_fi5821293074295781190e_real(F,K2,topolo2177554685111907308n_real(X4,top_top_set_real))
       => ( aa_real_real(minus_minus_real(aa_real_real(F,B)),aa_real_real(F,A)) = aa_real_real(times_times_real(aa_real_real(minus_minus_real(B),A)),K2) ) ) ) ).

% DERIV_const_ratio_const
tff(fact_10014_DERIV__neg__imp__decreasing,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ! [X4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X4)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),B)
             => ? [Y5: real] :
                  ( has_fi5821293074295781190e_real(F,Y5,topolo2177554685111907308n_real(X4,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,Y5),zero_zero_real) ) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,B)),aa_real_real(F,A)) ) ) ).

% DERIV_neg_imp_decreasing
tff(fact_10015_DERIV__pos__imp__increasing,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ! [X4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X4)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),B)
             => ? [Y5: real] :
                  ( has_fi5821293074295781190e_real(F,Y5,topolo2177554685111907308n_real(X4,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y5) ) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,A)),aa_real_real(F,B)) ) ) ).

% DERIV_pos_imp_increasing
tff(fact_10016_MVT2,axiom,
    ! [A: real,B: real,F: fun_real_real,F4: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ! [X4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X4)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),B)
             => has_fi5821293074295781190e_real(F,aa_real_real(F4,X4),topolo2177554685111907308n_real(X4,top_top_set_real)) ) )
       => ? [Z: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Z)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,Z),B)
            & ( aa_real_real(minus_minus_real(aa_real_real(F,B)),aa_real_real(F,A)) = aa_real_real(times_times_real(aa_real_real(minus_minus_real(B),A)),aa_real_real(F4,Z)) ) ) ) ) ).

% MVT2
tff(fact_10017_DERIV__nonneg__imp__nondecreasing,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( ! [X4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X4)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),B)
             => ? [Y5: real] :
                  ( has_fi5821293074295781190e_real(F,Y5,topolo2177554685111907308n_real(X4,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y5) ) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,A)),aa_real_real(F,B)) ) ) ).

% DERIV_nonneg_imp_nondecreasing
tff(fact_10018_DERIV__nonpos__imp__nonincreasing,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( ! [X4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X4)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),B)
             => ? [Y5: real] :
                  ( has_fi5821293074295781190e_real(F,Y5,topolo2177554685111907308n_real(X4,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y5),zero_zero_real) ) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,B)),aa_real_real(F,A)) ) ) ).

% DERIV_nonpos_imp_nonincreasing
tff(fact_10019_deriv__nonneg__imp__mono,axiom,
    ! [A: real,B: real,G: fun_real_real,G2: fun_real_real] :
      ( ! [X4: real] :
          ( member_real(X4,set_or1222579329274155063t_real(A,B))
         => has_fi5821293074295781190e_real(G,aa_real_real(G2,X4),topolo2177554685111907308n_real(X4,top_top_set_real)) )
     => ( ! [X4: real] :
            ( member_real(X4,set_or1222579329274155063t_real(A,B))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(G2,X4)) )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(G,A)),aa_real_real(G,B)) ) ) ) ).

% deriv_nonneg_imp_mono
tff(fact_10020_DERIV__const__average,axiom,
    ! [A: real,B: real,V: fun_real_real,K2: real] :
      ( ( A != B )
     => ( ! [X4: real] : has_fi5821293074295781190e_real(V,K2,topolo2177554685111907308n_real(X4,top_top_set_real))
       => ( aa_real_real(V,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(A),B)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(V,A)),aa_real_real(V,B))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ) ).

% DERIV_const_average
tff(fact_10021_DERIV__local__max,axiom,
    ! [F: fun_real_real,L: real,X: real,D: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D)
       => ( ! [Y4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(X),Y4))),D)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,Y4)),aa_real_real(F,X)) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_max
tff(fact_10022_DERIV__local__min,axiom,
    ! [F: fun_real_real,L: real,X: real,D: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D)
       => ( ! [Y4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(X),Y4))),D)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,X)),aa_real_real(F,Y4)) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_min
tff(fact_10023_DERIV__ln__divide,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => has_fi5821293074295781190e_real(ln_ln_real,aa_real_real(divide_divide_real(one_one_real),X),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_ln_divide
tff(fact_10024_DERIV__pow,axiom,
    ! [Nb: nat,X: real,S: set_real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_tr,Nb),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat)))),topolo2177554685111907308n_real(X,S)) ).

% DERIV_pow
tff(fact_10025_DERIV__fun__pow,axiom,
    ! [G: fun_real_real,Ma: real,X: real,Nb: nat] :
      ( has_fi5821293074295781190e_real(G,Ma,topolo2177554685111907308n_real(X,top_top_set_real))
     => has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_ts(G),Nb),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(G,X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)))),Ma),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_fun_pow
tff(fact_10026_has__real__derivative__powr,axiom,
    ! [Z2: real,R: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z2)
     => has_fi5821293074295781190e_real(aTP_Lamm_tt(R),aa_real_real(times_times_real(R),aa_real_real(powr_real(Z2),aa_real_real(minus_minus_real(R),one_one_real))),topolo2177554685111907308n_real(Z2,top_top_set_real)) ) ).

% has_real_derivative_powr
tff(fact_10027_DERIV__log,axiom,
    ! [X: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => has_fi5821293074295781190e_real(log(B),aa_real_real(divide_divide_real(one_one_real),aa_real_real(times_times_real(aa_real_real(ln_ln_real,B)),X)),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_log
tff(fact_10028_DERIV__fun__powr,axiom,
    ! [G: fun_real_real,Ma: real,X: real,R: real] :
      ( has_fi5821293074295781190e_real(G,Ma,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(G,X))
       => has_fi5821293074295781190e_real(aa_rea7340605660300375871l_real(aTP_Lamm_tu(G),R),aa_real_real(times_times_real(aa_real_real(times_times_real(R),aa_real_real(powr_real(aa_real_real(G,X)),aa_real_real(minus_minus_real(R),aa_nat_real(semiri5074537144036343181t_real,one_one_nat))))),Ma),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ).

% DERIV_fun_powr
tff(fact_10029_DERIV__powr,axiom,
    ! [G: fun_real_real,Ma: real,X: real,F: fun_real_real,R: real] :
      ( has_fi5821293074295781190e_real(G,Ma,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(G,X))
       => ( has_fi5821293074295781190e_real(F,R,topolo2177554685111907308n_real(X,top_top_set_real))
         => has_fi5821293074295781190e_real(aa_fun6946558556880649704l_real(aTP_Lamm_tv(G),F),aa_real_real(times_times_real(aa_real_real(powr_real(aa_real_real(G,X)),aa_real_real(F,X))),aa_real_real(plus_plus_real(aa_real_real(times_times_real(R),aa_real_real(ln_ln_real,aa_real_real(G,X)))),aa_real_real(divide_divide_real(aa_real_real(times_times_real(Ma),aa_real_real(F,X))),aa_real_real(G,X)))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ).

% DERIV_powr
tff(fact_10030_DERIV__real__sqrt,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
     => has_fi5821293074295781190e_real(sqrt,aa_real_real(divide_divide_real(aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_real_sqrt
tff(fact_10031_DERIV__series_H,axiom,
    ! [F: fun_re4238438383274954274t_real,F4: fun_re4238438383274954274t_real,X0: real,A: real,B: real,L4: fun_nat_real] :
      ( ! [N: nat] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_la(F),N),aa_nat_real(aa_real_fun_nat_real(F4,X0),N),topolo2177554685111907308n_real(X0,top_top_set_real))
     => ( ! [X4: real] :
            ( member_real(X4,set_or1633881224788618240n_real(A,B))
           => summable_real(aa_real_fun_nat_real(F,X4)) )
       => ( member_real(X0,set_or1633881224788618240n_real(A,B))
         => ( summable_real(aa_real_fun_nat_real(F4,X0))
           => ( summable_real(L4)
             => ( ! [N: nat,X4: real,Y4: real] :
                    ( member_real(X4,set_or1633881224788618240n_real(A,B))
                   => ( member_real(Y4,set_or1633881224788618240n_real(A,B))
                     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(F,X4),N)),aa_nat_real(aa_real_fun_nat_real(F,Y4),N)))),aa_real_real(times_times_real(aa_nat_real(L4,N)),aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(X4),Y4)))) ) )
               => has_fi5821293074295781190e_real(aTP_Lamm_mk(F),suminf_real(aa_real_fun_nat_real(F4,X0)),topolo2177554685111907308n_real(X0,top_top_set_real)) ) ) ) ) ) ) ).

% DERIV_series'
tff(fact_10032_DERIV__arctan,axiom,
    ! [X: real] : has_fi5821293074295781190e_real(arctan,aa_real_real(inverse_inverse_real,aa_real_real(plus_plus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),topolo2177554685111907308n_real(X,top_top_set_real)) ).

% DERIV_arctan
tff(fact_10033_arsinh__real__has__field__derivative,axiom,
    ! [X: real,A2: set_real] : has_fi5821293074295781190e_real(arsinh_real,aa_real_real(divide_divide_real(one_one_real),aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real))),topolo2177554685111907308n_real(X,A2)) ).

% arsinh_real_has_field_derivative
tff(fact_10034_DERIV__real__sqrt__generic,axiom,
    ! [X: real,D3: real] :
      ( ( X != zero_zero_real )
     => ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
         => ( D3 = aa_real_real(divide_divide_real(aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) )
       => ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),zero_zero_real)
           => ( D3 = aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,X)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) )
         => has_fi5821293074295781190e_real(sqrt,D3,topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ).

% DERIV_real_sqrt_generic
tff(fact_10035_arcosh__real__has__field__derivative,axiom,
    ! [X: real,A2: set_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X)
     => has_fi5821293074295781190e_real(arcosh_real,aa_real_real(divide_divide_real(one_one_real),aa_real_real(sqrt,aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real))),topolo2177554685111907308n_real(X,A2)) ) ).

% arcosh_real_has_field_derivative
tff(fact_10036_artanh__real__has__field__derivative,axiom,
    ! [X: real,A2: set_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,X)),one_one_real)
     => has_fi5821293074295781190e_real(artanh_real,aa_real_real(divide_divide_real(one_one_real),aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),topolo2177554685111907308n_real(X,A2)) ) ).

% artanh_real_has_field_derivative
tff(fact_10037_DERIV__power__series_H,axiom,
    ! [R4: real,F: fun_nat_real,X0: real] :
      ( ! [X4: real] :
          ( member_real(X4,set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,R4),R4))
         => summable_real(aa_real_fun_nat_real(aTP_Lamm_tw(F),X4)) )
     => ( member_real(X0,set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,R4),R4))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R4)
         => has_fi5821293074295781190e_real(aTP_Lamm_ty(F),suminf_real(aa_real_fun_nat_real(aTP_Lamm_tw(F),X0)),topolo2177554685111907308n_real(X0,top_top_set_real)) ) ) ) ).

% DERIV_power_series'
tff(fact_10038_DERIV__real__root,axiom,
    ! [Nb: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
       => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Nb),aa_real_real(inverse_inverse_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ).

% DERIV_real_root
tff(fact_10039_DERIV__arccos,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),one_one_real)
       => has_fi5821293074295781190e_real(arccos,aa_real_real(inverse_inverse_real,aa_real_real(uminus_uminus_real,aa_real_real(sqrt,aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ).

% DERIV_arccos
tff(fact_10040_DERIV__arcsin,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),one_one_real)
       => has_fi5821293074295781190e_real(arcsin,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ).

% DERIV_arcsin
tff(fact_10041_Maclaurin__all__le,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,X: real,Nb: nat] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( ! [M2: nat,X4: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),X4),topolo2177554685111907308n_real(X4,top_top_set_real))
       => ? [T3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T3)),aa_real_real(abs_abs_real,X))
            & ( aa_real_real(F,X) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_tz(Diff),X)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Nb),T3)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Nb))) ) ) ) ) ).

% Maclaurin_all_le
tff(fact_10042_Maclaurin__all__le__objl,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,X: real,Nb: nat] :
      ( ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
        & ! [M2: nat,X4: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),X4),topolo2177554685111907308n_real(X4,top_top_set_real)) )
     => ? [T3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T3)),aa_real_real(abs_abs_real,X))
          & ( aa_real_real(F,X) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_tz(Diff),X)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Nb),T3)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Nb))) ) ) ) ).

% Maclaurin_all_le_objl
tff(fact_10043_DERIV__odd__real__root,axiom,
    ! [Nb: nat,X: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( ( X != zero_zero_real )
       => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Nb),aa_real_real(inverse_inverse_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ).

% DERIV_odd_real_root
tff(fact_10044_Maclaurin,axiom,
    ! [H: real,Nb: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
         => ( ! [M2: nat,T3: real] :
                ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Nb)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T3)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T3),H) )
               => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T3),topolo2177554685111907308n_real(T3,top_top_set_real)) )
           => ? [T3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T3)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,T3),H)
                & ( aa_real_real(F,H) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_ua(H),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Nb),T3)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),Nb))) ) ) ) ) ) ) ).

% Maclaurin
tff(fact_10045_Maclaurin2,axiom,
    ! [H: real,Diff: fun_na6724076748692969250l_real,F: fun_real_real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M2: nat,T3: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Nb)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T3)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T3),H) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T3),topolo2177554685111907308n_real(T3,top_top_set_real)) )
         => ? [T3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T3)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T3),H)
              & ( aa_real_real(F,H) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_ua(H),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Nb),T3)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),Nb))) ) ) ) ) ) ).

% Maclaurin2
tff(fact_10046_Maclaurin__minus,axiom,
    ! [H: real,Nb: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,H),zero_zero_real)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
         => ( ! [M2: nat,T3: real] :
                ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Nb)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,H),T3)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T3),zero_zero_real) )
               => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T3),topolo2177554685111907308n_real(T3,top_top_set_real)) )
           => ? [T3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,H),T3)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,T3),zero_zero_real)
                & ( aa_real_real(F,H) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_ua(H),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Nb),T3)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),Nb))) ) ) ) ) ) ) ).

% Maclaurin_minus
tff(fact_10047_Maclaurin__all__lt,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Nb: nat,X: real] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( ( X != zero_zero_real )
         => ( ! [M2: nat,X4: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),X4),topolo2177554685111907308n_real(X4,top_top_set_real))
           => ? [T3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T3))
                & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,T3)),aa_real_real(abs_abs_real,X))
                & ( aa_real_real(F,X) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_tz(Diff),X)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Nb),T3)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Nb))) ) ) ) ) ) ) ).

% Maclaurin_all_lt
tff(fact_10048_Maclaurin__bi__le,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Nb: nat,X: real] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( ! [M2: nat,T3: real] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Nb)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T3)),aa_real_real(abs_abs_real,X)) )
           => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T3),topolo2177554685111907308n_real(T3,top_top_set_real)) )
       => ? [T3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T3)),aa_real_real(abs_abs_real,X))
            & ( aa_real_real(F,X) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_tz(Diff),X)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Nb),T3)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Nb))) ) ) ) ) ).

% Maclaurin_bi_le
tff(fact_10049_Taylor,axiom,
    ! [Nb: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B: real,C: real,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M2: nat,T3: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Nb)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),T3)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T3),B) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T3),topolo2177554685111907308n_real(T3,top_top_set_real)) )
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),C)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),B)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),B)
                 => ( ( X != C )
                   => ? [T3: real] :
                        ( $ite(
                            aa_real_o(aa_real_fun_real_o(ord_less_real,X),C),
                            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),T3)
                            & aa_real_o(aa_real_fun_real_o(ord_less_real,T3),C) ),
                            ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),T3)
                            & aa_real_o(aa_real_fun_real_o(ord_less_real,T3),X) ) )
                        & ( aa_real_real(F,X) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_ub(Diff),C),X)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Nb),T3)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(X),C)),Nb))) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
tff(fact_10050_Taylor__up,axiom,
    ! [Nb: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B: real,C: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M2: nat,T3: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Nb)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),T3)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T3),B) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T3),topolo2177554685111907308n_real(T3,top_top_set_real)) )
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),C)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),B)
             => ? [T3: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),T3)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,T3),B)
                  & ( aa_real_real(F,B) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_uc(Diff),B),C)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Nb),T3)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(B),C)),Nb))) ) ) ) ) ) ) ) ).

% Taylor_up
tff(fact_10051_Taylor__down,axiom,
    ! [Nb: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B: real,C: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M2: nat,T3: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Nb)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),T3)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T3),B) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T3),topolo2177554685111907308n_real(T3,top_top_set_real)) )
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),C)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),B)
             => ? [T3: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),T3)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,T3),C)
                  & ( aa_real_real(F,A) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_uc(Diff),A),C)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Nb),T3)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(A),C)),Nb))) ) ) ) ) ) ) ) ).

% Taylor_down
tff(fact_10052_Maclaurin__lemma2,axiom,
    ! [Nb: nat,H: real,Diff: fun_na6724076748692969250l_real,K2: nat,B3: real] :
      ( ! [M2: nat,T3: real] :
          ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Nb)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T3)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T3),H) )
         => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T3),topolo2177554685111907308n_real(T3,top_top_set_real)) )
     => ( ( Nb = aa_nat_nat(suc,K2) )
       => ! [M: nat,T4: real] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T4)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),H) )
           => has_fi5821293074295781190e_real(aa_nat_fun_real_real(aa_rea1103652080130114956l_real(aa_fun2567993205388704171l_real(aTP_Lamm_ue(Nb),Diff),B3),M),aa_real_real(minus_minus_real(aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),T4)),aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_uf(Diff),M),T4)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,M))))),aa_real_real(times_times_real(B3),aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,T4),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,M)))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,M))))))),topolo2177554685111907308n_real(T4,top_top_set_real)) ) ) ) ).

% Maclaurin_lemma2
tff(fact_10053_DERIV__arctan__series,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,X)),one_one_real)
     => has_fi5821293074295781190e_real(aTP_Lamm_ug,suminf_real(aa_real_fun_nat_real(aTP_Lamm_uh,X)),topolo2177554685111907308n_real(X,top_top_set_real)) ) ).

% DERIV_arctan_series
tff(fact_10054_DERIV__even__real__root,axiom,
    ! [Nb: nat,X: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),zero_zero_real)
         => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Nb),aa_real_real(inverse_inverse_real,aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),X)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ).

% DERIV_even_real_root
tff(fact_10055_isCont__Lb__Ub,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( ! [X4: real] :
            ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X4)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),B) )
           => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X4,top_top_set_real),F) )
       => ? [L5: real,M7: real] :
            ( ! [X2: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X2)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X2),B) )
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L5),aa_real_real(F,X2))
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,X2)),M7) ) )
            & ! [Y5: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L5),Y5)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y5),M7) )
               => ? [X4: real] :
                    ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X4)
                    & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),B)
                    & ( aa_real_real(F,X4) = Y5 ) ) ) ) ) ) ).

% isCont_Lb_Ub
tff(fact_10056_isCont__real__sqrt,axiom,
    ! [X: real] : topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),sqrt) ).

% isCont_real_sqrt
tff(fact_10057_isCont__real__root,axiom,
    ! [X: real,Nb: nat] : topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),aa_nat_fun_real_real(root,Nb)) ).

% isCont_real_root
tff(fact_10058_isCont__inverse__function2,axiom,
    ! [A: real,X: real,B: real,G: fun_real_real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),B)
       => ( ! [Z: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Z)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z),B)
               => ( aa_real_real(G,aa_real_real(F,Z)) = Z ) ) )
         => ( ! [Z: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Z)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z),B)
                 => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z,top_top_set_real),F) ) )
           => topolo4422821103128117721l_real(topolo2177554685111907308n_real(aa_real_real(F,X),top_top_set_real),G) ) ) ) ) ).

% isCont_inverse_function2
tff(fact_10059_isCont__arcosh,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X)
     => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),arcosh_real) ) ).

% isCont_arcosh
tff(fact_10060_LIM__cos__div__sin,axiom,
    filterlim_real_real(aTP_Lamm_ui,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),top_top_set_real)) ).

% LIM_cos_div_sin
tff(fact_10061_isCont__arccos,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),one_one_real)
       => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),arccos) ) ) ).

% isCont_arccos
tff(fact_10062_isCont__arcsin,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),one_one_real)
       => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),arcsin) ) ) ).

% isCont_arcsin
tff(fact_10063_LIM__less__bound,axiom,
    ! [B: real,X: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),X)
     => ( ! [X4: real] :
            ( member_real(X4,set_or1633881224788618240n_real(B,X))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X4)) )
       => ( topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),F)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X)) ) ) ) ).

% LIM_less_bound
tff(fact_10064_isCont__artanh,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),one_one_real)
       => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),artanh_real) ) ) ).

% isCont_artanh
tff(fact_10065_isCont__inverse__function,axiom,
    ! [D: real,X: real,G: fun_real_real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D)
     => ( ! [Z: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(Z),X))),D)
           => ( aa_real_real(G,aa_real_real(F,Z)) = Z ) )
       => ( ! [Z: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(Z),X))),D)
             => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z,top_top_set_real),F) )
         => topolo4422821103128117721l_real(topolo2177554685111907308n_real(aa_real_real(F,X),top_top_set_real),G) ) ) ) ).

% isCont_inverse_function
tff(fact_10066_GMVT_H,axiom,
    ! [A: real,B: real,F: fun_real_real,G: fun_real_real,G2: fun_real_real,F4: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ! [Z: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Z)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z),B)
             => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z,top_top_set_real),F) ) )
       => ( ! [Z: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Z)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z),B)
               => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z,top_top_set_real),G) ) )
         => ( ! [Z: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Z)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),B)
                 => has_fi5821293074295781190e_real(G,aa_real_real(G2,Z),topolo2177554685111907308n_real(Z,top_top_set_real)) ) )
           => ( ! [Z: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Z)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),B)
                   => has_fi5821293074295781190e_real(F,aa_real_real(F4,Z),topolo2177554685111907308n_real(Z,top_top_set_real)) ) )
             => ? [C3: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),C3)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,C3),B)
                  & ( aa_real_real(times_times_real(aa_real_real(minus_minus_real(aa_real_real(F,B)),aa_real_real(F,A))),aa_real_real(G2,C3)) = aa_real_real(times_times_real(aa_real_real(minus_minus_real(aa_real_real(G,B)),aa_real_real(G,A))),aa_real_real(F4,C3)) ) ) ) ) ) ) ) ).

% GMVT'
tff(fact_10067_summable__Leibniz_I2_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(A,zero_zero_nat))
         => ! [N7: nat] : member_real(suminf_real(aTP_Lamm_uj(A)),set_or1222579329274155063t_real(aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_uj(A)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N7))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_uj(A)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N7)),one_one_nat))))) ) ) ) ).

% summable_Leibniz(2)
tff(fact_10068_summable__Leibniz_I3_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(A,zero_zero_nat)),zero_zero_real)
         => ! [N7: nat] : member_real(suminf_real(aTP_Lamm_uj(A)),set_or1222579329274155063t_real(aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_uj(A)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N7)),one_one_nat))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_uj(A)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N7))))) ) ) ) ).

% summable_Leibniz(3)
tff(fact_10069_mult__nat__right__at__top,axiom,
    ! [C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
     => filterlim_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_cc,C),at_top_nat,at_top_nat) ) ).

% mult_nat_right_at_top
tff(fact_10070_mult__nat__left__at__top,axiom,
    ! [C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C)
     => filterlim_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),at_top_nat,at_top_nat) ) ).

% mult_nat_left_at_top
tff(fact_10071_monoseq__convergent,axiom,
    ! [X7: fun_nat_real,B3: real] :
      ( topolo6980174941875973593q_real(X7)
     => ( ! [I3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_nat_real(X7,I3))),B3)
       => ~ ! [L5: real] : ~ filterlim_nat_real(X7,topolo2815343760600316023s_real(L5),at_top_nat) ) ) ).

% monoseq_convergent
tff(fact_10072_LIMSEQ__root,axiom,
    filterlim_nat_real(aTP_Lamm_uk,topolo2815343760600316023s_real(one_one_real),at_top_nat) ).

% LIMSEQ_root
tff(fact_10073_nested__sequence__unique,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N)),aa_nat_real(F,aa_nat_nat(suc,N)))
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(G,aa_nat_nat(suc,N))),aa_nat_real(G,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N)),aa_nat_real(G,N))
         => ( filterlim_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_fh(F),G),topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
           => ? [L2: real] :
                ( ! [N7: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N7)),L2)
                & filterlim_nat_real(F,topolo2815343760600316023s_real(L2),at_top_nat)
                & ! [N7: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L2),aa_nat_real(G,N7))
                & filterlim_nat_real(G,topolo2815343760600316023s_real(L2),at_top_nat) ) ) ) ) ) ).

% nested_sequence_unique
tff(fact_10074_LIMSEQ__inverse__zero,axiom,
    ! [X7: fun_nat_real] :
      ( ! [R3: real] :
        ? [N5: nat] :
        ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N5),N)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,R3),aa_nat_real(X7,N)) )
     => filterlim_nat_real(aTP_Lamm_ul(X7),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_inverse_zero
tff(fact_10075_lim__inverse__n_H,axiom,
    filterlim_nat_real(aTP_Lamm_um,topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ).

% lim_inverse_n'
tff(fact_10076_LIMSEQ__root__const,axiom,
    ! [C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_un,C),topolo2815343760600316023s_real(one_one_real),at_top_nat) ) ).

% LIMSEQ_root_const
tff(fact_10077_LIMSEQ__inverse__real__of__nat,axiom,
    filterlim_nat_real(aTP_Lamm_uo,topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat
tff(fact_10078_LIMSEQ__inverse__real__of__nat__add,axiom,
    ! [R: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_up,R),topolo2815343760600316023s_real(R),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add
tff(fact_10079_increasing__LIMSEQ,axiom,
    ! [F: fun_nat_real,L: real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N)),aa_nat_real(F,aa_nat_nat(suc,N)))
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N)),L)
       => ( ! [E2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
             => ? [N7: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L),aa_real_real(plus_plus_real(aa_nat_real(F,N7)),E2)) )
         => filterlim_nat_real(F,topolo2815343760600316023s_real(L),at_top_nat) ) ) ) ).

% increasing_LIMSEQ
tff(fact_10080_LIMSEQ__realpow__zero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),one_one_real)
       => filterlim_nat_real(aa_real_fun_nat_real(power_power_real,X),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ) ).

% LIMSEQ_realpow_zero
tff(fact_10081_LIMSEQ__divide__realpow__zero,axiom,
    ! [X: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X)
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_uq(X),A),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_divide_realpow_zero
tff(fact_10082_LIMSEQ__abs__realpow__zero,axiom,
    ! [C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,C)),one_one_real)
     => filterlim_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,C)),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_abs_realpow_zero
tff(fact_10083_LIMSEQ__abs__realpow__zero2,axiom,
    ! [C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,C)),one_one_real)
     => filterlim_nat_real(aa_real_fun_nat_real(power_power_real,C),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_abs_realpow_zero2
tff(fact_10084_LIMSEQ__inverse__realpow__zero,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),X)
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_ur,X),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_inverse_realpow_zero
tff(fact_10085_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
    ! [R: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_us,R),topolo2815343760600316023s_real(R),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add_minus
tff(fact_10086_tendsto__exp__limit__sequentially,axiom,
    ! [X: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_ut,X),topolo2815343760600316023s_real(aa_real_real(exp_real,X)),at_top_nat) ).

% tendsto_exp_limit_sequentially
tff(fact_10087_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [R: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_uu,R),topolo2815343760600316023s_real(R),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add_minus_mult
tff(fact_10088_summable__Leibniz_I1_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A)
       => summable_real(aTP_Lamm_uj(A)) ) ) ).

% summable_Leibniz(1)
tff(fact_10089_summable,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N))
         => summable_real(aTP_Lamm_uj(A)) ) ) ) ).

% summable
tff(fact_10090_cos__diff__limit__1,axiom,
    ! [Theta: fun_nat_real,Theta2: real] :
      ( filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_uv(Theta),Theta2),topolo2815343760600316023s_real(one_one_real),at_top_nat)
     => ~ ! [K: fun_nat_int] : ~ filterlim_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_uw(Theta),K),topolo2815343760600316023s_real(Theta2),at_top_nat) ) ).

% cos_diff_limit_1
tff(fact_10091_cos__limit__1,axiom,
    ! [Theta: fun_nat_real] :
      ( filterlim_nat_real(aTP_Lamm_ux(Theta),topolo2815343760600316023s_real(one_one_real),at_top_nat)
     => ? [K: fun_nat_int] : filterlim_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_uw(Theta),K),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% cos_limit_1
tff(fact_10092_summable__Leibniz_I4_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A)
       => filterlim_nat_real(aTP_Lamm_uy(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_uj(A))),at_top_nat) ) ) ).

% summable_Leibniz(4)
tff(fact_10093_zeroseq__arctan__series,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,X)),one_one_real)
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_eb,X),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% zeroseq_arctan_series
tff(fact_10094_summable__Leibniz_H_I3_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N))
         => filterlim_nat_real(aTP_Lamm_uy(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_uj(A))),at_top_nat) ) ) ) ).

% summable_Leibniz'(3)
tff(fact_10095_summable__Leibniz_H_I2_J,axiom,
    ! [A: fun_nat_real,Nb: nat] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_uj(A)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),suminf_real(aTP_Lamm_uj(A))) ) ) ) ).

% summable_Leibniz'(2)
tff(fact_10096_sums__alternating__upper__lower,axiom,
    ! [A: fun_nat_real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N))
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N))
       => ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
         => ? [L2: real] :
              ( ! [N7: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_uj(A)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N7)))),L2)
              & filterlim_nat_real(aTP_Lamm_uy(A),topolo2815343760600316023s_real(L2),at_top_nat)
              & ! [N7: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L2),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_uj(A)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N7)),one_one_nat))))
              & filterlim_nat_real(aTP_Lamm_uz(A),topolo2815343760600316023s_real(L2),at_top_nat) ) ) ) ) ).

% sums_alternating_upper_lower
tff(fact_10097_summable__Leibniz_I5_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A)
       => filterlim_nat_real(aTP_Lamm_uz(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_uj(A))),at_top_nat) ) ) ).

% summable_Leibniz(5)
tff(fact_10098_summable__Leibniz_H_I5_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N))
         => filterlim_nat_real(aTP_Lamm_uz(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_uj(A))),at_top_nat) ) ) ) ).

% summable_Leibniz'(5)
tff(fact_10099_summable__Leibniz_H_I4_J,axiom,
    ! [A: fun_nat_real,Nb: nat] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N))),aa_nat_real(A,N))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,suminf_real(aTP_Lamm_uj(A))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_uj(A)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),one_one_nat)))) ) ) ) ).

% summable_Leibniz'(4)
tff(fact_10100_real__bounded__linear,axiom,
    ! [F: fun_real_real] :
      ( real_V5970128139526366754l_real(F)
    <=> ? [C2: real] :
        ! [X3: real] : aa_real_real(F,X3) = aa_real_real(times_times_real(X3),C2) ) ).

% real_bounded_linear
tff(fact_10101_tendsto__exp__limit__at__right,axiom,
    ! [X: real] : filterlim_real_real(aTP_Lamm_va(X),topolo2815343760600316023s_real(aa_real_real(exp_real,X)),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ).

% tendsto_exp_limit_at_right
tff(fact_10102_tendsto__arcosh__at__left__1,axiom,
    filterlim_real_real(arcosh_real,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(one_one_real,set_or5849166863359141190n_real(one_one_real))) ).

% tendsto_arcosh_at_left_1
tff(fact_10103_INT__greaterThan__UNIV,axiom,
    comple7806235888213564991et_nat(image_nat_set_nat(set_or1210151606488870762an_nat,top_top_set_nat)) = bot_bot_set_nat ).

% INT_greaterThan_UNIV
tff(fact_10104_filterlim__tan__at__right,axiom,
    filterlim_real_real(tan_real,at_bot_real,topolo2177554685111907308n_real(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),set_or5849166863359141190n_real(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))))) ).

% filterlim_tan_at_right
tff(fact_10105_UN__atLeast__UNIV,axiom,
    comple7399068483239264473et_nat(image_nat_set_nat(set_ord_atLeast_nat,top_top_set_nat)) = top_top_set_nat ).

% UN_atLeast_UNIV
tff(fact_10106_tanh__real__at__bot,axiom,
    filterlim_real_real(tanh_real,topolo2815343760600316023s_real(aa_real_real(uminus_uminus_real,one_one_real)),at_bot_real) ).

% tanh_real_at_bot
tff(fact_10107_ln__at__0,axiom,
    filterlim_real_real(ln_ln_real,at_bot_real,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ).

% ln_at_0
tff(fact_10108_filterlim__inverse__at__bot__neg,axiom,
    filterlim_real_real(inverse_inverse_real,at_bot_real,topolo2177554685111907308n_real(zero_zero_real,set_or5984915006950818249n_real(zero_zero_real))) ).

% filterlim_inverse_at_bot_neg
tff(fact_10109_artanh__real__at__right__1,axiom,
    filterlim_real_real(artanh_real,at_bot_real,topolo2177554685111907308n_real(aa_real_real(uminus_uminus_real,one_one_real),set_or5849166863359141190n_real(aa_real_real(uminus_uminus_real,one_one_real)))) ).

% artanh_real_at_right_1
tff(fact_10110_DERIV__pos__imp__increasing__at__bot,axiom,
    ! [B: real,F: fun_real_real,Flim: real] :
      ( ! [X4: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),B)
         => ? [Y5: real] :
              ( has_fi5821293074295781190e_real(F,Y5,topolo2177554685111907308n_real(X4,top_top_set_real))
              & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y5) ) )
     => ( filterlim_real_real(F,topolo2815343760600316023s_real(Flim),at_bot_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Flim),aa_real_real(F,B)) ) ) ).

% DERIV_pos_imp_increasing_at_bot
tff(fact_10111_filterlim__pow__at__bot__odd,axiom,
    ! [Nb: nat,F: fun_real_real,F5: filter_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( filterlim_real_real(F,at_bot_real,F5)
       => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
         => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vb(Nb),F),at_bot_real,F5) ) ) ) ).

% filterlim_pow_at_bot_odd
tff(fact_10112_tendsto__arctan__at__bot,axiom,
    filterlim_real_real(arctan,topolo2815343760600316023s_real(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),at_bot_real) ).

% tendsto_arctan_at_bot
tff(fact_10113_filterlim__pow__at__bot__even,axiom,
    ! [Nb: nat,F: fun_real_real,F5: filter_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( filterlim_real_real(F,at_bot_real,F5)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
         => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vb(Nb),F),at_top_real,F5) ) ) ) ).

% filterlim_pow_at_bot_even
tff(fact_10114_at__bot__le__at__infinity,axiom,
    ord_le4104064031414453916r_real(at_bot_real,at_infinity_real) ).

% at_bot_le_at_infinity
tff(fact_10115_at__top__le__at__infinity,axiom,
    ord_le4104064031414453916r_real(at_top_real,at_infinity_real) ).

% at_top_le_at_infinity
tff(fact_10116_ln__at__top,axiom,
    filterlim_real_real(ln_ln_real,at_top_real,at_top_real) ).

% ln_at_top
tff(fact_10117_sqrt__at__top,axiom,
    filterlim_real_real(sqrt,at_top_real,at_top_real) ).

% sqrt_at_top
tff(fact_10118_exp__at__top,axiom,
    filterlim_real_real(exp_real,at_top_real,at_top_real) ).

% exp_at_top
tff(fact_10119_filterlim__real__sequentially,axiom,
    filterlim_nat_real(semiri5074537144036343181t_real,at_top_real,at_top_nat) ).

% filterlim_real_sequentially
tff(fact_10120_filterlim__uminus__at__top__at__bot,axiom,
    filterlim_real_real(uminus_uminus_real,at_top_real,at_bot_real) ).

% filterlim_uminus_at_top_at_bot
tff(fact_10121_filterlim__uminus__at__bot__at__top,axiom,
    filterlim_real_real(uminus_uminus_real,at_bot_real,at_top_real) ).

% filterlim_uminus_at_bot_at_top
tff(fact_10122_tanh__real__at__top,axiom,
    filterlim_real_real(tanh_real,topolo2815343760600316023s_real(one_one_real),at_top_real) ).

% tanh_real_at_top
tff(fact_10123_ln__x__over__x__tendsto__0,axiom,
    filterlim_real_real(aTP_Lamm_vc,topolo2815343760600316023s_real(zero_zero_real),at_top_real) ).

% ln_x_over_x_tendsto_0
tff(fact_10124_artanh__real__at__left__1,axiom,
    filterlim_real_real(artanh_real,at_top_real,topolo2177554685111907308n_real(one_one_real,set_or5984915006950818249n_real(one_one_real))) ).

% artanh_real_at_left_1
tff(fact_10125_filterlim__inverse__at__top__right,axiom,
    filterlim_real_real(inverse_inverse_real,at_top_real,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ).

% filterlim_inverse_at_top_right
tff(fact_10126_filterlim__inverse__at__right__top,axiom,
    filterlim_real_real(inverse_inverse_real,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)),at_top_real) ).

% filterlim_inverse_at_right_top
tff(fact_10127_tendsto__power__div__exp__0,axiom,
    ! [K2: nat] : filterlim_real_real(aa_nat_fun_real_real(aTP_Lamm_vd,K2),topolo2815343760600316023s_real(zero_zero_real),at_top_real) ).

% tendsto_power_div_exp_0
tff(fact_10128_tendsto__exp__limit__at__top,axiom,
    ! [X: real] : filterlim_real_real(aTP_Lamm_ve(X),topolo2815343760600316023s_real(aa_real_real(exp_real,X)),at_top_real) ).

% tendsto_exp_limit_at_top
tff(fact_10129_DERIV__neg__imp__decreasing__at__top,axiom,
    ! [B: real,F: fun_real_real,Flim: real] :
      ( ! [X4: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),X4)
         => ? [Y5: real] :
              ( has_fi5821293074295781190e_real(F,Y5,topolo2177554685111907308n_real(X4,top_top_set_real))
              & aa_real_o(aa_real_fun_real_o(ord_less_real,Y5),zero_zero_real) ) )
     => ( filterlim_real_real(F,topolo2815343760600316023s_real(Flim),at_top_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Flim),aa_real_real(F,B)) ) ) ).

% DERIV_neg_imp_decreasing_at_top
tff(fact_10130_tendsto__arctan__at__top,axiom,
    filterlim_real_real(arctan,topolo2815343760600316023s_real(aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),at_top_real) ).

% tendsto_arctan_at_top
tff(fact_10131_filterlim__tan__at__left,axiom,
    filterlim_real_real(tan_real,at_top_real,topolo2177554685111907308n_real(aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),set_or5984915006950818249n_real(aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))) ).

% filterlim_tan_at_left
tff(fact_10132_lhopital__left__at__top,axiom,
    ! [G: fun_real_real,X: real,G2: fun_real_real,F: fun_real_real,F4: fun_real_real,Y2: real] :
      ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
     => ( eventually_real(aTP_Lamm_vf(G2),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_vg(F),F4),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_vg(G),G2),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vh(G2),F4),topolo2815343760600316023s_real(Y2),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vh(G),F),topolo2815343760600316023s_real(Y2),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X))) ) ) ) ) ) ).

% lhopital_left_at_top
tff(fact_10133_eventually__sequentially__Suc,axiom,
    ! [P: fun_nat_o] :
      ( eventually_nat(aTP_Lamm_vi(P),at_top_nat)
    <=> eventually_nat(P,at_top_nat) ) ).

% eventually_sequentially_Suc
tff(fact_10134_eventually__sequentially__seg,axiom,
    ! [P: fun_nat_o,K2: nat] :
      ( eventually_nat(aa_nat_fun_nat_o(aTP_Lamm_vj(P),K2),at_top_nat)
    <=> eventually_nat(P,at_top_nat) ) ).

% eventually_sequentially_seg
tff(fact_10135_sequentially__offset,axiom,
    ! [P: fun_nat_o,K2: nat] :
      ( eventually_nat(P,at_top_nat)
     => eventually_nat(aa_nat_fun_nat_o(aTP_Lamm_vj(P),K2),at_top_nat) ) ).

% sequentially_offset
tff(fact_10136_eventually__False__sequentially,axiom,
    ~ eventually_nat(aTP_Lamm_rr,at_top_nat) ).

% eventually_False_sequentially
tff(fact_10137_le__sequentially,axiom,
    ! [F5: filter_nat] :
      ( ord_le2510731241096832064er_nat(F5,at_top_nat)
    <=> ! [N8: nat] : eventually_nat(aa_nat_fun_nat_o(ord_less_eq_nat,N8),F5) ) ).

% le_sequentially
tff(fact_10138_eventually__sequentially,axiom,
    ! [P: fun_nat_o] :
      ( eventually_nat(P,at_top_nat)
    <=> ? [N8: nat] :
        ! [N4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N8),N4)
         => aa_nat_o(P,N4) ) ) ).

% eventually_sequentially
tff(fact_10139_eventually__sequentiallyI,axiom,
    ! [C: nat,P: fun_nat_o] :
      ( ! [X4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),X4)
         => aa_nat_o(P,X4) )
     => eventually_nat(P,at_top_nat) ) ).

% eventually_sequentiallyI
tff(fact_10140_eventually__at__right__to__0,axiom,
    ! [P: fun_real_o,A: real] :
      ( eventually_real(P,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
    <=> eventually_real(aa_real_fun_real_o(aTP_Lamm_vk(P),A),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ) ).

% eventually_at_right_to_0
tff(fact_10141_eventually__at__left__to__right,axiom,
    ! [P: fun_real_o,A: real] :
      ( eventually_real(P,topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
    <=> eventually_real(aTP_Lamm_vl(P),topolo2177554685111907308n_real(aa_real_real(uminus_uminus_real,A),set_or5849166863359141190n_real(aa_real_real(uminus_uminus_real,A)))) ) ).

% eventually_at_left_to_right
tff(fact_10142_eventually__at__right__real,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => eventually_real(aa_real_fun_real_o(aTP_Lamm_vm(A),B),topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A))) ) ).

% eventually_at_right_real
tff(fact_10143_eventually__at__left__real,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => eventually_real(aa_real_fun_real_o(aTP_Lamm_vm(B),A),topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A))) ) ).

% eventually_at_left_real
tff(fact_10144_eventually__at__top__to__right,axiom,
    ! [P: fun_real_o] :
      ( eventually_real(P,at_top_real)
    <=> eventually_real(aTP_Lamm_vn(P),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ) ).

% eventually_at_top_to_right
tff(fact_10145_eventually__at__right__to__top,axiom,
    ! [P: fun_real_o] :
      ( eventually_real(P,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
    <=> eventually_real(aTP_Lamm_vn(P),at_top_real) ) ).

% eventually_at_right_to_top
tff(fact_10146_lhopital__at__top__at__top,axiom,
    ! [F: fun_real_real,A: real,G: fun_real_real,F4: fun_real_real,G2: fun_real_real] :
      ( filterlim_real_real(F,at_top_real,topolo2177554685111907308n_real(A,top_top_set_real))
     => ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(A,top_top_set_real))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_vg(F),F4),topolo2177554685111907308n_real(A,top_top_set_real))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_vg(G),G2),topolo2177554685111907308n_real(A,top_top_set_real))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vo(F4),G2),at_top_real,topolo2177554685111907308n_real(A,top_top_set_real))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vo(F),G),at_top_real,topolo2177554685111907308n_real(A,top_top_set_real)) ) ) ) ) ) ).

% lhopital_at_top_at_top
tff(fact_10147_lhopital,axiom,
    ! [F: fun_real_real,X: real,G: fun_real_real,G2: fun_real_real,F4: fun_real_real,F5: filter_real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(X,top_top_set_real))
     => ( filterlim_real_real(G,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(X,top_top_set_real))
       => ( eventually_real(aTP_Lamm_vf(G),topolo2177554685111907308n_real(X,top_top_set_real))
         => ( eventually_real(aTP_Lamm_vf(G2),topolo2177554685111907308n_real(X,top_top_set_real))
           => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_vg(F),F4),topolo2177554685111907308n_real(X,top_top_set_real))
             => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_vg(G),G2),topolo2177554685111907308n_real(X,top_top_set_real))
               => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vh(G2),F4),F5,topolo2177554685111907308n_real(X,top_top_set_real))
                 => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vo(F),G),F5,topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ) ) ) ) ).

% lhopital
tff(fact_10148_lhopital__right__at__top__at__top,axiom,
    ! [F: fun_real_real,A: real,G: fun_real_real,F4: fun_real_real,G2: fun_real_real] :
      ( filterlim_real_real(F,at_top_real,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
     => ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_vg(F),F4),topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_vg(G),G2),topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vo(F4),G2),at_top_real,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vo(F),G),at_top_real,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A))) ) ) ) ) ) ).

% lhopital_right_at_top_at_top
tff(fact_10149_lhopital__at__top__at__bot,axiom,
    ! [F: fun_real_real,A: real,G: fun_real_real,F4: fun_real_real,G2: fun_real_real] :
      ( filterlim_real_real(F,at_top_real,topolo2177554685111907308n_real(A,top_top_set_real))
     => ( filterlim_real_real(G,at_bot_real,topolo2177554685111907308n_real(A,top_top_set_real))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_vg(F),F4),topolo2177554685111907308n_real(A,top_top_set_real))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_vg(G),G2),topolo2177554685111907308n_real(A,top_top_set_real))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vo(F4),G2),at_bot_real,topolo2177554685111907308n_real(A,top_top_set_real))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vo(F),G),at_bot_real,topolo2177554685111907308n_real(A,top_top_set_real)) ) ) ) ) ) ).

% lhopital_at_top_at_bot
tff(fact_10150_lhopital__left__at__top__at__top,axiom,
    ! [F: fun_real_real,A: real,G: fun_real_real,F4: fun_real_real,G2: fun_real_real] :
      ( filterlim_real_real(F,at_top_real,topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
     => ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_vg(F),F4),topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_vg(G),G2),topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vo(F4),G2),at_top_real,topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vo(F),G),at_top_real,topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A))) ) ) ) ) ) ).

% lhopital_left_at_top_at_top
tff(fact_10151_lhopital__at__top,axiom,
    ! [G: fun_real_real,X: real,G2: fun_real_real,F: fun_real_real,F4: fun_real_real,Y2: real] :
      ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(X,top_top_set_real))
     => ( eventually_real(aTP_Lamm_vf(G2),topolo2177554685111907308n_real(X,top_top_set_real))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_vg(F),F4),topolo2177554685111907308n_real(X,top_top_set_real))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_vg(G),G2),topolo2177554685111907308n_real(X,top_top_set_real))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vh(G2),F4),topolo2815343760600316023s_real(Y2),topolo2177554685111907308n_real(X,top_top_set_real))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vh(G),F),topolo2815343760600316023s_real(Y2),topolo2177554685111907308n_real(X,top_top_set_real)) ) ) ) ) ) ).

% lhopital_at_top
tff(fact_10152_lhospital__at__top__at__top,axiom,
    ! [G: fun_real_real,G2: fun_real_real,F: fun_real_real,F4: fun_real_real,X: real] :
      ( filterlim_real_real(G,at_top_real,at_top_real)
     => ( eventually_real(aTP_Lamm_vf(G2),at_top_real)
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_vg(F),F4),at_top_real)
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_vg(G),G2),at_top_real)
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vh(G2),F4),topolo2815343760600316023s_real(X),at_top_real)
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vh(G),F),topolo2815343760600316023s_real(X),at_top_real) ) ) ) ) ) ).

% lhospital_at_top_at_top
tff(fact_10153_lhopital__right,axiom,
    ! [F: fun_real_real,X: real,G: fun_real_real,G2: fun_real_real,F4: fun_real_real,F5: filter_real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
     => ( filterlim_real_real(G,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
       => ( eventually_real(aTP_Lamm_vf(G),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
         => ( eventually_real(aTP_Lamm_vf(G2),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
           => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_vg(F),F4),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
             => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_vg(G),G2),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
               => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vh(G2),F4),F5,topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
                 => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vo(F),G),F5,topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X))) ) ) ) ) ) ) ) ).

% lhopital_right
tff(fact_10154_lhopital__right__0,axiom,
    ! [F0: fun_real_real,G0: fun_real_real,G2: fun_real_real,F4: fun_real_real,F5: filter_real] :
      ( filterlim_real_real(F0,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
     => ( filterlim_real_real(G0,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
       => ( eventually_real(aTP_Lamm_vf(G0),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
         => ( eventually_real(aTP_Lamm_vf(G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
           => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_vg(F0),F4),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
             => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_vg(G0),G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
               => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vh(G2),F4),F5,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
                 => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vo(F0),G0),F5,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ) ) ) ) ) ) ) ).

% lhopital_right_0
tff(fact_10155_lhopital__left,axiom,
    ! [F: fun_real_real,X: real,G: fun_real_real,G2: fun_real_real,F4: fun_real_real,F5: filter_real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
     => ( filterlim_real_real(G,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
       => ( eventually_real(aTP_Lamm_vf(G),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
         => ( eventually_real(aTP_Lamm_vf(G2),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
           => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_vg(F),F4),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
             => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_vg(G),G2),topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
               => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vh(G2),F4),F5,topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X)))
                 => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vo(F),G),F5,topolo2177554685111907308n_real(X,set_or5984915006950818249n_real(X))) ) ) ) ) ) ) ) ).

% lhopital_left
tff(fact_10156_lhopital__right__at__top__at__bot,axiom,
    ! [F: fun_real_real,A: real,G: fun_real_real,F4: fun_real_real,G2: fun_real_real] :
      ( filterlim_real_real(F,at_top_real,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
     => ( filterlim_real_real(G,at_bot_real,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_vg(F),F4),topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_vg(G),G2),topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vo(F4),G2),at_bot_real,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vo(F),G),at_bot_real,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A))) ) ) ) ) ) ).

% lhopital_right_at_top_at_bot
tff(fact_10157_lhopital__left__at__top__at__bot,axiom,
    ! [F: fun_real_real,A: real,G: fun_real_real,F4: fun_real_real,G2: fun_real_real] :
      ( filterlim_real_real(F,at_top_real,topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
     => ( filterlim_real_real(G,at_bot_real,topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_vg(F),F4),topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_vg(G),G2),topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vo(F4),G2),at_bot_real,topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vo(F),G),at_bot_real,topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A))) ) ) ) ) ) ).

% lhopital_left_at_top_at_bot
tff(fact_10158_lhopital__right__at__top,axiom,
    ! [G: fun_real_real,X: real,G2: fun_real_real,F: fun_real_real,F4: fun_real_real,Y2: real] :
      ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
     => ( eventually_real(aTP_Lamm_vf(G2),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_vg(F),F4),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_vg(G),G2),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vh(G2),F4),topolo2815343760600316023s_real(Y2),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vh(G),F),topolo2815343760600316023s_real(Y2),topolo2177554685111907308n_real(X,set_or5849166863359141190n_real(X))) ) ) ) ) ) ).

% lhopital_right_at_top
tff(fact_10159_lhopital__right__0__at__top,axiom,
    ! [G: fun_real_real,G2: fun_real_real,F: fun_real_real,F4: fun_real_real,X: real] :
      ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
     => ( eventually_real(aTP_Lamm_vf(G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_vg(F),F4),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_vg(G),G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vh(G2),F4),topolo2815343760600316023s_real(X),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vh(G),F),topolo2815343760600316023s_real(X),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ) ) ) ) ) ).

% lhopital_right_0_at_top
tff(fact_10160_GreatestI__nat,axiom,
    ! [P: fun_nat_o,K2: nat,B: nat] :
      ( aa_nat_o(P,K2)
     => ( ! [Y4: nat] :
            ( aa_nat_o(P,Y4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y4),B) )
       => aa_nat_o(P,order_Greatest_nat(P)) ) ) ).

% GreatestI_nat
tff(fact_10161_Greatest__le__nat,axiom,
    ! [P: fun_nat_o,K2: nat,B: nat] :
      ( aa_nat_o(P,K2)
     => ( ! [Y4: nat] :
            ( aa_nat_o(P,Y4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y4),B) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),order_Greatest_nat(P)) ) ) ).

% Greatest_le_nat
tff(fact_10162_GreatestI__ex__nat,axiom,
    ! [P: fun_nat_o,B: nat] :
      ( ? [X_1: nat] : aa_nat_o(P,X_1)
     => ( ! [Y4: nat] :
            ( aa_nat_o(P,Y4)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y4),B) )
       => aa_nat_o(P,order_Greatest_nat(P)) ) ) ).

% GreatestI_ex_nat
tff(fact_10163_Bseq__eq__bounded,axiom,
    ! [F: fun_nat_real,A: real,B: real] :
      ( ord_less_eq_set_real(image_nat_real(F,top_top_set_nat),set_or1222579329274155063t_real(A,B))
     => bfun_nat_real(F,at_top_nat) ) ).

% Bseq_eq_bounded
tff(fact_10164_Bseq__realpow,axiom,
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),one_one_real)
       => bfun_nat_real(aa_real_fun_nat_real(power_power_real,X),at_top_nat) ) ) ).

% Bseq_realpow
tff(fact_10165_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ~ aa_nat_o(vEBT_VEBT_valid(X),Xa)
     => ( ( ? [Uu2: $o,Uv2: $o] : X = aa_o_VEBT_VEBT(vEBT_Leaf((Uu2)),(Uv2))
         => ( Xa = one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg,TreeList),Summary) )
             => ( ( Deg = Xa )
                & $let(
                    n: nat,
                    n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                    $let(
                      m2: nat,
                      m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
                      ( ! [X3: vEBT_VEBT] :
                          ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList))
                         => aa_nat_o(vEBT_VEBT_valid(X3),n) )
                      & aa_nat_o(vEBT_VEBT_valid(Summary),m2)
                      & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m2) )
                      & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X8: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X8)
                            & ! [X3: vEBT_VEBT] :
                                ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList))
                               => ~ ? [X8: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X8) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_vp(Deg),TreeList),Summary),n),m2))),Mima) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
tff(fact_10166_decseq__bounded,axiom,
    ! [X7: fun_nat_real,B3: real] :
      ( order_9091379641038594480t_real(X7)
     => ( ! [I3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B3),aa_nat_real(X7,I3))
       => bfun_nat_real(X7,at_top_nat) ) ) ).

% decseq_bounded
tff(fact_10167_decseq__convergent,axiom,
    ! [X7: fun_nat_real,B3: real] :
      ( order_9091379641038594480t_real(X7)
     => ( ! [I3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B3),aa_nat_real(X7,I3))
       => ~ ! [L5: real] :
              ( filterlim_nat_real(X7,topolo2815343760600316023s_real(L5),at_top_nat)
             => ~ ! [I4: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L5),aa_nat_real(X7,I4)) ) ) ) ).

% decseq_convergent
tff(fact_10168_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
    ! [Mima2: option4927543243414619207at_nat,Degb: nat,TreeListb: list_VEBT_VEBT,Summaryb: vEBT_VEBT,Deg3: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima2,Degb,TreeListb),Summaryb)),Deg3)
    <=> ( ( Degb = Deg3 )
        & $let(
            n: nat,
            n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Degb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
            $let(
              m2: nat,
              m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Degb),n),
              ( ! [X3: vEBT_VEBT] :
                  ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeListb))
                 => aa_nat_o(vEBT_VEBT_valid(X3),n) )
              & aa_nat_o(vEBT_VEBT_valid(Summaryb),m2)
              & ( size_s6755466524823107622T_VEBT(TreeListb) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m2) )
              & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X8: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summaryb),X8)
                    & ! [X3: vEBT_VEBT] :
                        ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeListb))
                       => ~ ? [X8: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X8) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_vp(Degb),TreeListb),Summaryb),n),m2))),Mima2) ) ) ) ) ) ).

% VEBT_internal.valid'.simps(2)
tff(fact_10169_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y2: $o] :
      ( ( aa_nat_o(vEBT_VEBT_valid(X),Xa)
      <=> (Y2) )
     => ( ( ? [Uu2: $o,Uv2: $o] : X = aa_o_VEBT_VEBT(vEBT_Leaf((Uu2)),(Uv2))
         => ( (Y2)
          <=> ( Xa != one_one_nat ) ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg,TreeList),Summary) )
             => ( (Y2)
              <=> ~ ( ( Deg = Xa )
                    & $let(
                        n: nat,
                        n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        $let(
                          m2: nat,
                          m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
                          ( ! [X3: vEBT_VEBT] :
                              ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList))
                             => aa_nat_o(vEBT_VEBT_valid(X3),n) )
                          & aa_nat_o(vEBT_VEBT_valid(Summary),m2)
                          & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m2) )
                          & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X8: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X8)
                                & ! [X3: vEBT_VEBT] :
                                    ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList))
                                   => ~ ? [X8: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X8) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_vp(Deg),TreeList),Summary),n),m2))),Mima) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
tff(fact_10170_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(X),Xa)
     => ( ( ? [Uu2: $o,Uv2: $o] : X = aa_o_VEBT_VEBT(vEBT_Leaf((Uu2)),(Uv2))
         => ( Xa != one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg,TreeList),Summary) )
             => ~ ( ( Deg = Xa )
                  & $let(
                      n: nat,
                      n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                      $let(
                        m2: nat,
                        m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
                        ( ! [X3: vEBT_VEBT] :
                            ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList))
                           => aa_nat_o(vEBT_VEBT_valid(X3),n) )
                        & aa_nat_o(vEBT_VEBT_valid(Summary),m2)
                        & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m2) )
                        & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X8: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X8)
                              & ! [X3: vEBT_VEBT] :
                                  ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList))
                                 => ~ ? [X8: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X8) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_vp(Deg),TreeList),Summary),n),m2))),Mima) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
tff(fact_10171_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y2: $o] :
      ( ( aa_nat_o(vEBT_VEBT_valid(X),Xa)
      <=> (Y2) )
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel),produc738532404422230701BT_nat(X,Xa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X = aa_o_VEBT_VEBT(vEBT_Leaf((Uu2)),(Uv2)) )
             => ( ( (Y2)
                <=> ( Xa = one_one_nat ) )
               => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel),produc738532404422230701BT_nat(aa_o_VEBT_VEBT(vEBT_Leaf((Uu2)),(Uv2)),Xa)) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg,TreeList),Summary) )
               => ( ( (Y2)
                  <=> ( ( Deg = Xa )
                      & $let(
                          n: nat,
                          n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                          $let(
                            m2: nat,
                            m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
                            ( ! [X3: vEBT_VEBT] :
                                ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList))
                               => aa_nat_o(vEBT_VEBT_valid(X3),n) )
                            & aa_nat_o(vEBT_VEBT_valid(Summary),m2)
                            & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m2) )
                            & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X8: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X8)
                                  & ! [X3: vEBT_VEBT] :
                                      ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList))
                                     => ~ ? [X8: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X8) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_vp(Deg),TreeList),Summary),n),m2))),Mima) ) ) ) ) )
                 => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel),produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg,TreeList),Summary),Xa)) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(1)
tff(fact_10172_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(X),Xa)
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel),produc738532404422230701BT_nat(X,Xa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X = aa_o_VEBT_VEBT(vEBT_Leaf((Uu2)),(Uv2)) )
             => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel),produc738532404422230701BT_nat(aa_o_VEBT_VEBT(vEBT_Leaf((Uu2)),(Uv2)),Xa))
               => ( Xa != one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg,TreeList),Summary) )
               => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel),produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg,TreeList),Summary),Xa))
                 => ~ ( ( Deg = Xa )
                      & $let(
                          n: nat,
                          n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                          $let(
                            m2: nat,
                            m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
                            ( ! [X3: vEBT_VEBT] :
                                ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList))
                               => aa_nat_o(vEBT_VEBT_valid(X3),n) )
                            & aa_nat_o(vEBT_VEBT_valid(Summary),m2)
                            & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m2) )
                            & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X8: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X8)
                                  & ! [X3: vEBT_VEBT] :
                                      ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList))
                                     => ~ ? [X8: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X8) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_vp(Deg),TreeList),Summary),n),m2))),Mima) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
tff(fact_10173_Sup__int__def,axiom,
    ! [X7: set_int] : complete_Sup_Sup_int(X7) = the_int(aTP_Lamm_vq(X7)) ).

% Sup_int_def
tff(fact_10174_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ~ aa_nat_o(vEBT_VEBT_valid(X),Xa)
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel),produc738532404422230701BT_nat(X,Xa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X = aa_o_VEBT_VEBT(vEBT_Leaf((Uu2)),(Uv2)) )
             => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel),produc738532404422230701BT_nat(aa_o_VEBT_VEBT(vEBT_Leaf((Uu2)),(Uv2)),Xa))
               => ( Xa = one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( X = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg,TreeList),Summary) )
               => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel),produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg,TreeList),Summary),Xa))
                 => ( ( Deg = Xa )
                    & $let(
                        n: nat,
                        n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        $let(
                          m2: nat,
                          m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
                          ( ! [X3: vEBT_VEBT] :
                              ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList))
                             => aa_nat_o(vEBT_VEBT_valid(X3),n) )
                          & aa_nat_o(vEBT_VEBT_valid(Summary),m2)
                          & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m2) )
                          & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X8: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X8)
                                & ! [X3: vEBT_VEBT] :
                                    ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList))
                                   => ~ ? [X8: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X8) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_vp(Deg),TreeList),Summary),n),m2))),Mima) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
tff(fact_10175_open__bool__def,axiom,
    topolo9180104560040979295open_o = topolo4667128019001906403logy_o(sup_sup_set_set_o(image_o_set_o(set_ord_lessThan_o,top_top_set_o),image_o_set_o(set_or6416164934427428222Than_o,top_top_set_o))) ).

% open_bool_def
tff(fact_10176_open__int__def,axiom,
    topolo4325760605701065253en_int = topolo1611008123915946401gy_int(sup_sup_set_set_int(image_int_set_int(set_ord_lessThan_int,top_top_set_int),image_int_set_int(set_or1207661135979820486an_int,top_top_set_int))) ).

% open_int_def
tff(fact_10177_open__nat__def,axiom,
    topolo4328251076210115529en_nat = topolo1613498594424996677gy_nat(sup_sup_set_set_nat(image_nat_set_nat(set_ord_lessThan_nat,top_top_set_nat),image_nat_set_nat(set_or1210151606488870762an_nat,top_top_set_nat))) ).

% open_nat_def
tff(fact_10178_GMVT,axiom,
    ! [A: real,B: real,F: fun_real_real,G: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ! [X4: real] :
            ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X4)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),B) )
           => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X4,top_top_set_real),F) )
       => ( ! [X4: real] :
              ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X4)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,X4),B) )
             => differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X4,top_top_set_real)) )
         => ( ! [X4: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X4)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),B) )
               => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X4,top_top_set_real),G) )
           => ( ! [X4: real] :
                  ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X4)
                    & aa_real_o(aa_real_fun_real_o(ord_less_real,X4),B) )
                 => differ6690327859849518006l_real(G,topolo2177554685111907308n_real(X4,top_top_set_real)) )
             => ? [G_c: real,F_c: real,C3: real] :
                  ( has_fi5821293074295781190e_real(G,G_c,topolo2177554685111907308n_real(C3,top_top_set_real))
                  & has_fi5821293074295781190e_real(F,F_c,topolo2177554685111907308n_real(C3,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,A),C3)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,C3),B)
                  & ( aa_real_real(times_times_real(aa_real_real(minus_minus_real(aa_real_real(F,B)),aa_real_real(F,A))),G_c) = aa_real_real(times_times_real(aa_real_real(minus_minus_real(aa_real_real(G,B)),aa_real_real(G,A))),F_c) ) ) ) ) ) ) ) ).

% GMVT
tff(fact_10179_MVT,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
       => ( ! [X4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X4)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),B)
               => differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X4,top_top_set_real)) ) )
         => ? [L2: real,Z: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Z)
              & aa_real_o(aa_real_fun_real_o(ord_less_real,Z),B)
              & has_fi5821293074295781190e_real(F,L2,topolo2177554685111907308n_real(Z,top_top_set_real))
              & ( aa_real_real(minus_minus_real(aa_real_real(F,B)),aa_real_real(F,A)) = aa_real_real(times_times_real(aa_real_real(minus_minus_real(B),A)),L2) ) ) ) ) ) ).

% MVT
tff(fact_10180_continuous__on__arsinh_H,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( topolo5044208981011980120l_real(A2,F)
     => topolo5044208981011980120l_real(A2,aTP_Lamm_vr(F)) ) ).

% continuous_on_arsinh'
tff(fact_10181_continuous__on__arcosh_H,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( topolo5044208981011980120l_real(A2,F)
     => ( ! [X4: real] :
            ( member_real(X4,A2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(F,X4)) )
       => topolo5044208981011980120l_real(A2,aTP_Lamm_vs(F)) ) ) ).

% continuous_on_arcosh'
tff(fact_10182_continuous__image__closed__interval,axiom,
    ! [A: real,B: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
       => ? [C3: real,D2: real] :
            ( ( image_real_real(F,set_or1222579329274155063t_real(A,B)) = set_or1222579329274155063t_real(C3,D2) )
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C3),D2) ) ) ) ).

% continuous_image_closed_interval
tff(fact_10183_continuous__on__arcosh,axiom,
    ! [A2: set_real] :
      ( ord_less_eq_set_real(A2,set_ord_atLeast_real(one_one_real))
     => topolo5044208981011980120l_real(A2,arcosh_real) ) ).

% continuous_on_arcosh
tff(fact_10184_continuous__on__arccos_H,axiom,
    topolo5044208981011980120l_real(set_or1222579329274155063t_real(aa_real_real(uminus_uminus_real,one_one_real),one_one_real),arccos) ).

% continuous_on_arccos'
tff(fact_10185_continuous__on__arcsin_H,axiom,
    topolo5044208981011980120l_real(set_or1222579329274155063t_real(aa_real_real(uminus_uminus_real,one_one_real),one_one_real),arcsin) ).

% continuous_on_arcsin'
tff(fact_10186_continuous__on__artanh_H,axiom,
    ! [A2: set_real,F: fun_real_real] :
      ( topolo5044208981011980120l_real(A2,F)
     => ( ! [X4: real] :
            ( member_real(X4,A2)
           => member_real(aa_real_real(F,X4),set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,one_one_real),one_one_real)) )
       => topolo5044208981011980120l_real(A2,aTP_Lamm_vt(F)) ) ) ).

% continuous_on_artanh'
tff(fact_10187_continuous__on__artanh,axiom,
    ! [A2: set_real] :
      ( ord_less_eq_set_real(A2,set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,one_one_real),one_one_real))
     => topolo5044208981011980120l_real(A2,artanh_real) ) ).

% continuous_on_artanh
tff(fact_10188_DERIV__isconst2,axiom,
    ! [A: real,B: real,F: fun_real_real,X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B),F)
       => ( ! [X4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X4)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),B)
               => has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(X4,top_top_set_real)) ) )
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),B)
             => ( aa_real_real(F,X) = aa_real_real(F,A) ) ) ) ) ) ) ).

% DERIV_isconst2
tff(fact_10189_uniformity__complex__def,axiom,
    topolo896644834953643431omplex = comple8358262395181532106omplex(image_5971271580939081552omplex(aTP_Lamm_vv,set_or5849166863359141190n_real(zero_zero_real))) ).

% uniformity_complex_def
tff(fact_10190_uniformity__real__def,axiom,
    topolo1511823702728130853y_real = comple2936214249959783750l_real(image_2178119161166701260l_real(aTP_Lamm_vx,set_or5849166863359141190n_real(zero_zero_real))) ).

% uniformity_real_def
tff(fact_10191_open__real__def,axiom,
    ! [U3: set_real] :
      ( topolo4860482606490270245n_real(U3)
    <=> ! [X3: real] :
          ( member_real(X3,U3)
         => eventu3244425730907250241l_real(produc5414030515140494994real_o(aa_rea1652550732481586312real_o(aTP_Lamm_vy(U3),X3)),topolo1511823702728130853y_real) ) ) ).

% open_real_def
tff(fact_10192_open__complex__def,axiom,
    ! [U3: set_complex] :
      ( topolo4110288021797289639omplex(U3)
    <=> ! [X3: complex] :
          ( member_complex(X3,U3)
         => eventu5826381225784669381omplex(produc6771430404735790350plex_o(aa_com7221863818706917250plex_o(aTP_Lamm_vz(U3),X3)),topolo896644834953643431omplex) ) ) ).

% open_complex_def
tff(fact_10193_eventually__prod__sequentially,axiom,
    ! [P: fun_Pr4796943543331071514_nat_o] :
      ( eventu1038000079068216329at_nat(P,prod_filter_nat_nat(at_top_nat,at_top_nat))
    <=> ? [N8: nat] :
        ! [M3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N8),M3)
         => ! [N4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N8),N4)
             => aa_Pro6150637604445964947_nat_o(P,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,N4),M3)) ) ) ) ).

% eventually_prod_sequentially
tff(fact_10194_incseq__bounded,axiom,
    ! [X7: fun_nat_real,B3: real] :
      ( order_mono_nat_real(X7)
     => ( ! [I3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X7,I3)),B3)
       => bfun_nat_real(X7,at_top_nat) ) ) ).

% incseq_bounded
tff(fact_10195_mono__times__nat,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => order_mono_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb)) ) ).

% mono_times_nat
tff(fact_10196_mono__Suc,axiom,
    order_mono_nat_nat(suc) ).

% mono_Suc
tff(fact_10197_incseq__convergent,axiom,
    ! [X7: fun_nat_real,B3: real] :
      ( order_mono_nat_real(X7)
     => ( ! [I3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X7,I3)),B3)
       => ~ ! [L5: real] :
              ( filterlim_nat_real(X7,topolo2815343760600316023s_real(L5),at_top_nat)
             => ~ ! [I4: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X7,I4)),L5) ) ) ) ).

% incseq_convergent
tff(fact_10198_mono__ge2__power__minus__self,axiom,
    ! [K2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K2)
     => order_mono_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_wa,K2)) ) ).

% mono_ge2_power_minus_self
tff(fact_10199_tendsto__at__topI__sequentially__real,axiom,
    ! [F: fun_real_real,Y2: real] :
      ( order_mono_real_real(F)
     => ( filterlim_nat_real(aTP_Lamm_wb(F),topolo2815343760600316023s_real(Y2),at_top_nat)
       => filterlim_real_real(F,topolo2815343760600316023s_real(Y2),at_top_real) ) ) ).

% tendsto_at_topI_sequentially_real
tff(fact_10200_and__not__num_Oelims,axiom,
    ! [X: num,Xa: num,Y2: option_num] :
      ( ( bit_and_not_num(X,Xa) = Y2 )
     => ( ( ( X = one )
         => ( ( Xa = one )
           => ( Y2 != none_num ) ) )
       => ( ( ( X = one )
           => ( ? [N: num] : Xa = aa_num_num(bit0,N)
             => ( Y2 != some_num(one) ) ) )
         => ( ( ( X = one )
             => ( ? [N: num] : Xa = aa_num_num(bit1,N)
               => ( Y2 != none_num ) ) )
           => ( ! [M2: num] :
                  ( ( X = aa_num_num(bit0,M2) )
                 => ( ( Xa = one )
                   => ( Y2 != some_num(aa_num_num(bit0,M2)) ) ) )
             => ( ! [M2: num] :
                    ( ( X = aa_num_num(bit0,M2) )
                   => ! [N: num] :
                        ( ( Xa = aa_num_num(bit0,N) )
                       => ( Y2 != map_option_num_num(bit0,bit_and_not_num(M2,N)) ) ) )
               => ( ! [M2: num] :
                      ( ( X = aa_num_num(bit0,M2) )
                     => ! [N: num] :
                          ( ( Xa = aa_num_num(bit1,N) )
                         => ( Y2 != map_option_num_num(bit0,bit_and_not_num(M2,N)) ) ) )
                 => ( ! [M2: num] :
                        ( ( X = aa_num_num(bit1,M2) )
                       => ( ( Xa = one )
                         => ( Y2 != some_num(aa_num_num(bit0,M2)) ) ) )
                   => ( ! [M2: num] :
                          ( ( X = aa_num_num(bit1,M2) )
                         => ! [N: num] :
                              ( ( Xa = aa_num_num(bit0,N) )
                             => ( Y2 != case_o6005452278849405969um_num(some_num(one),aTP_Lamm_tl,bit_and_not_num(M2,N)) ) ) )
                     => ~ ! [M2: num] :
                            ( ( X = aa_num_num(bit1,M2) )
                           => ! [N: num] :
                                ( ( Xa = aa_num_num(bit1,N) )
                               => ( Y2 != map_option_num_num(bit0,bit_and_not_num(M2,N)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_not_num.elims
tff(fact_10201_nonneg__incseq__Bseq__subseq__iff,axiom,
    ! [F: fun_nat_real,G: fun_nat_nat] :
      ( ! [X4: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X4))
     => ( order_mono_nat_real(F)
       => ( order_5726023648592871131at_nat(G)
         => ( bfun_nat_real(aa_fun1538629612289296340t_real(aTP_Lamm_wc(F),G),at_top_nat)
          <=> bfun_nat_real(F,at_top_nat) ) ) ) ) ).

% nonneg_incseq_Bseq_subseq_iff
tff(fact_10202_ATP_Olambda__1,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_eg,Uu) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uu)),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat))) ).

% ATP.lambda_1
tff(fact_10203_ATP_Olambda__2,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_gk,Uu) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_nat(suc,Uu)) ).

% ATP.lambda_2
tff(fact_10204_ATP_Olambda__3,axiom,
    ! [Uu: real] :
      ( aa_real_o(aTP_Lamm_hr,Uu)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Uu)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uu),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
        & ( aa_real_real(cos_real,Uu) = zero_zero_real ) ) ) ).

% ATP.lambda_3
tff(fact_10205_ATP_Olambda__4,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_qn,Uu) = aa_real_real(times_times_real(aa_nat_real(cos_coeff,Uu)),aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Uu)) ).

% ATP.lambda_4
tff(fact_10206_ATP_Olambda__5,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_ui,Uu) = aa_real_real(divide_divide_real(aa_real_real(cos_real,Uu)),aa_real_real(sin_real,Uu)) ).

% ATP.lambda_5
tff(fact_10207_ATP_Olambda__6,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_vc,Uu) = aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,Uu)),Uu) ).

% ATP.lambda_6
tff(fact_10208_ATP_Olambda__7,axiom,
    ! [Uu: vEBT_VEBT] : aa_VEB5832953974749077948T_VEBT(aTP_Lamm_a,Uu) = vEBT_VEBT_elim_dead(Uu,aa_nat_Extended_enat(extended_enat2,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))) ).

% ATP.lambda_7
tff(fact_10209_ATP_Olambda__8,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_uk,Uu) = aa_real_real(aa_nat_fun_real_real(root,Uu),aa_nat_real(semiri5074537144036343181t_real,Uu)) ).

% ATP.lambda_8
tff(fact_10210_ATP_Olambda__9,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_rf,Uu) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),aa_nat_nat(suc,zero_zero_nat)) ).

% ATP.lambda_9
tff(fact_10211_ATP_Olambda__10,axiom,
    ! [Uu: extended_enat] : aa_Ext7969882492699152700d_enat(aTP_Lamm_gz,Uu) = plus_p3455044024723400733d_enat(Uu,one_on7984719198319812577d_enat) ).

% ATP.lambda_10
tff(fact_10212_ATP_Olambda__11,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_gv,Uu) = aa_complex_complex(plus_plus_complex(Uu),one_one_complex) ).

% ATP.lambda_11
tff(fact_10213_ATP_Olambda__12,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_gx,Uu) = aa_real_real(plus_plus_real(Uu),one_one_real) ).

% ATP.lambda_12
tff(fact_10214_ATP_Olambda__13,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_gw,Uu) = aa_rat_rat(plus_plus_rat(Uu),one_one_rat) ).

% ATP.lambda_13
tff(fact_10215_ATP_Olambda__14,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_ha,Uu) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),one_one_nat) ).

% ATP.lambda_14
tff(fact_10216_ATP_Olambda__15,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_gy,Uu) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),one_one_int) ).

% ATP.lambda_15
tff(fact_10217_ATP_Olambda__16,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_um,Uu) = aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,Uu)) ).

% ATP.lambda_16
tff(fact_10218_ATP_Olambda__17,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_ug,Uu) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_ed,Uu)) ).

% ATP.lambda_17
tff(fact_10219_ATP_Olambda__18,axiom,
    ! [Uu: real] : aa_rea5428163979925208277l_real(aTP_Lamm_vx,Uu) = princi6114159922880469582l_real(collec3799799289383736868l_real(produc5414030515140494994real_o(aTP_Lamm_vw(Uu)))) ).

% ATP.lambda_18
tff(fact_10220_ATP_Olambda__19,axiom,
    ! [Uu: real] : aa_rea5117059003799598681omplex(aTP_Lamm_vv,Uu) = princi3496590319149328850omplex(collec8663557070575231912omplex(produc6771430404735790350plex_o(aTP_Lamm_vu(Uu)))) ).

% ATP.lambda_19
tff(fact_10221_ATP_Olambda__20,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_uo,Uu) = aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uu))) ).

% ATP.lambda_20
tff(fact_10222_ATP_Olambda__21,axiom,
    ! [Uu: nat] : aa_nat_complex(aTP_Lamm_ig,Uu) = aa_complex_complex(invers8013647133539491842omplex,aa_nat_complex(semiri5044797733671781792omplex,Uu)) ).

% ATP.lambda_21
tff(fact_10223_ATP_Olambda__22,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_ih,Uu) = aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Uu)) ).

% ATP.lambda_22
tff(fact_10224_ATP_Olambda__23,axiom,
    ! [Uu: num] : aa_num_option_num(aTP_Lamm_tl,Uu) = some_num(aa_num_num(bit1,Uu)) ).

% ATP.lambda_23
tff(fact_10225_ATP_Olambda__24,axiom,
    ! [Uu: num] : aa_num_option_num(aTP_Lamm_th,Uu) = some_num(aa_num_num(bit0,Uu)) ).

% ATP.lambda_24
tff(fact_10226_ATP_Olambda__25,axiom,
    ! [Uu: int] : aa_int2565445551602696444nt_int(aTP_Lamm_qd,Uu) = aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(uminus_uminus_int,Uu)) ).

% ATP.lambda_25
tff(fact_10227_ATP_Olambda__26,axiom,
    ! [Uu: int] : aa_int2565445551602696444nt_int(aTP_Lamm_pr,Uu) = aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(abs_abs_int,Uu)) ).

% ATP.lambda_26
tff(fact_10228_ATP_Olambda__27,axiom,
    ! [Uu: nat] : aa_nat8803788312118491388at_nat(aTP_Lamm_dk,Uu) = aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_nat_nat(suc,Uu)) ).

% ATP.lambda_27
tff(fact_10229_ATP_Olambda__28,axiom,
    ! [Uu: nat] : aa_nat_option_num(aTP_Lamm_tj,Uu) = some_num(one) ).

% ATP.lambda_28
tff(fact_10230_ATP_Olambda__29,axiom,
    ! [Uu: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_to(Uu),Uua) = case_num_option_num(some_num(one),aTP_Lamm_tm(Uua),aTP_Lamm_tn(Uua),Uu) ).

% ATP.lambda_29
tff(fact_10231_ATP_Olambda__30,axiom,
    ! [Uu: complex,Uua: nat] :
      aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_jd,Uu),Uua) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Uua))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uu),Uua)),zero_zero_complex) ).

% ATP.lambda_30
tff(fact_10232_ATP_Olambda__31,axiom,
    ! [Uu: real,Uua: nat] :
      aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_jc,Uu),Uua) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),aa_real_real(real_V1485227260804924795R_real(aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Uua))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)),zero_zero_real) ).

% ATP.lambda_31
tff(fact_10233_ATP_Olambda__32,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_pw,Uu),Uua) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),zero_zero_complex) ).

% ATP.lambda_32
tff(fact_10234_ATP_Olambda__33,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_pz,Uu),Uua) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),zero_zero_real) ).

% ATP.lambda_33
tff(fact_10235_ATP_Olambda__34,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_rat(aTP_Lamm_px(Uu),Uua) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),zero_zero_rat) ).

% ATP.lambda_34
tff(fact_10236_ATP_Olambda__35,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_int(aTP_Lamm_py(Uu),Uua) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),zero_zero_int) ).

% ATP.lambda_35
tff(fact_10237_ATP_Olambda__36,axiom,
    ! [Uu: complex,Uua: nat] :
      aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_jf,Uu),Uua) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),zero_zero_complex,aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Uua))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uu),Uua))) ).

% ATP.lambda_36
tff(fact_10238_ATP_Olambda__37,axiom,
    ! [Uu: real,Uua: nat] :
      aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_je,Uu),Uua) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),zero_zero_real,aa_real_real(real_V1485227260804924795R_real(aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Uua))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua))) ).

% ATP.lambda_37
tff(fact_10239_ATP_Olambda__38,axiom,
    ! [Uu: fun_nat_real,Uua: nat] :
      aa_nat_real(aTP_Lamm_gl(Uu),Uua) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),zero_zero_real,aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% ATP.lambda_38
tff(fact_10240_ATP_Olambda__39,axiom,
    ! [Uu: int,Uua: int] :
      aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_ke,Uu),Uua) = $ite(Uu = zero_zero_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,Uu)),Uua)),aa_int_int(abs_abs_int,Uu))) ).

% ATP.lambda_39
tff(fact_10241_ATP_Olambda__40,axiom,
    ! [Uu: extended_enat,Uua: nat] :
      aa_nat_Extended_enat(aTP_Lamm_ay(Uu),Uua) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(aa_fun2186929853097002313d_enat(extend3600170679010898289d_enat,aTP_Lamm_ax(Uua)),
          $ite(Uua = zero_zero_nat,zero_z5237406670263579293d_enat,extend5688581933313929465d_enat)),Uu) ).

% ATP.lambda_40
tff(fact_10242_ATP_Olambda__41,axiom,
    ! [Uu: extended_enat,Uua: nat] : aa_nat_Extended_enat(aTP_Lamm_bq(Uu),Uua) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(aa_fun2186929853097002313d_enat(extend3600170679010898289d_enat,aTP_Lamm_bp(Uua)),zero_z5237406670263579293d_enat),Uu) ).

% ATP.lambda_41
tff(fact_10243_ATP_Olambda__42,axiom,
    ! [Uu: extended_enat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_an(Uu),Uua)
    <=> aa_Extended_enat_o(aa_o_f2257658052187763793enat_o(aa_fun2142289715968585979enat_o(extended_case_enat_o,aa_nat_fun_nat_o(aTP_Lamm_am,Uua)),$false),Uu) ) ).

% ATP.lambda_42
tff(fact_10244_ATP_Olambda__43,axiom,
    ! [Uu: extended_enat,Uua: nat] : aa_nat_Extended_enat(aTP_Lamm_al(Uu),Uua) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(aa_fun2186929853097002313d_enat(extend3600170679010898289d_enat,aTP_Lamm_ak(Uua)),extend5688581933313929465d_enat),Uu) ).

% ATP.lambda_43
tff(fact_10245_ATP_Olambda__44,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_ps,Uu),Uua) = $ite(~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),zero_zero_complex) ).

% ATP.lambda_44
tff(fact_10246_ATP_Olambda__45,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_pv,Uu),Uua) = $ite(~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),zero_zero_real) ).

% ATP.lambda_45
tff(fact_10247_ATP_Olambda__46,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_rat(aTP_Lamm_pt(Uu),Uua) = $ite(~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),zero_zero_rat) ).

% ATP.lambda_46
tff(fact_10248_ATP_Olambda__47,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_int(aTP_Lamm_pu(Uu),Uua) = $ite(~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),zero_zero_int) ).

% ATP.lambda_47
tff(fact_10249_ATP_Olambda__48,axiom,
    ! [Uu: extended_enat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_aj(Uu),Uua)
    <=> aa_Extended_enat_o(aa_o_f2257658052187763793enat_o(aa_fun2142289715968585979enat_o(extended_case_enat_o,aa_nat_fun_nat_o(ord_less_nat,Uua)),$true),Uu) ) ).

% ATP.lambda_48
tff(fact_10250_ATP_Olambda__49,axiom,
    ! [Uu: nat,Uua: num] : aa_num_option_num(aa_nat3713591252005393295on_num(aTP_Lamm_tp,Uu),Uua) = case_nat_option_num(none_num,aTP_Lamm_to(Uua),Uu) ).

% ATP.lambda_49
tff(fact_10251_ATP_Olambda__50,axiom,
    ! [Uu: nat,Uua: num] : aa_num_option_num(aTP_Lamm_tm(Uu),Uua) = case_o6005452278849405969um_num(none_num,aTP_Lamm_th,bit_take_bit_num(Uu,Uua)) ).

% ATP.lambda_50
tff(fact_10252_ATP_Olambda__51,axiom,
    ! [Uu: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_ti(Uu),Uua) = case_o6005452278849405969um_num(none_num,aTP_Lamm_th,bit_take_bit_num(Uua,Uu)) ).

% ATP.lambda_51
tff(fact_10253_ATP_Olambda__52,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ec,Uu),Uua) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(Uu),one_one_real)),aa_nat_nat(suc,Uua))) ).

% ATP.lambda_52
tff(fact_10254_ATP_Olambda__53,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_gu,Uu),Uua) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat))) ).

% ATP.lambda_53
tff(fact_10255_ATP_Olambda__54,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_hj,Uu),Uua) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))) ).

% ATP.lambda_54
tff(fact_10256_ATP_Olambda__55,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_pc(Uu),Uua) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Uua)),aa_nat_Code_integer(semiri4939895301339042750nteger,Uua))),aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_55
tff(fact_10257_ATP_Olambda__56,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_pa,Uu),Uua) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Uua)),aa_nat_complex(semiri8010041392384452111omplex,Uua))),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_56
tff(fact_10258_ATP_Olambda__57,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_pe,Uu),Uua) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri5074537144036343181t_real,Uua))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_57
tff(fact_10259_ATP_Olambda__58,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_pb(Uu),Uua) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Uua)),aa_nat_rat(semiri681578069525770553at_rat,Uua))),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_58
tff(fact_10260_ATP_Olambda__59,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_int(aTP_Lamm_pd(Uu),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),Uua)),aa_nat_int(semiri1314217659103216013at_int,Uua))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_59
tff(fact_10261_ATP_Olambda__60,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_pf,Uu),Uua) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua))),Uua)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,numera6690914467698888265omplex(aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_60
tff(fact_10262_ATP_Olambda__61,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_ph,Uu),Uua) = aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua))),Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_61
tff(fact_10263_ATP_Olambda__62,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_pg(Uu),Uua) = aa_rat_rat(divide_divide_rat(aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua))),Uua)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_62
tff(fact_10264_ATP_Olambda__63,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_hs,Uu),Uua)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Uua)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uua),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
        & ( aa_real_real(sin_real,Uua) = Uu ) ) ) ).

% ATP.lambda_63
tff(fact_10265_ATP_Olambda__64,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_hn,Uu),Uua)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Uua)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Uua),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
        & ( aa_real_real(tan_real,Uua) = Uu ) ) ) ).

% ATP.lambda_64
tff(fact_10266_ATP_Olambda__65,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa_Code_integer_int(aa_Cod1242632294114111009er_int(aTP_Lamm_rk,Uu),Uua) = $let(
        l2: int,
        l2:= aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),code_int_of_integer(Uu)),
        $ite(Uua = zero_z3403309356797280102nteger,l2,aa_int_int(aa_int_fun_int_int(plus_plus_int,l2),one_one_int)) ) ).

% ATP.lambda_65
tff(fact_10267_ATP_Olambda__66,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_Extended_enat(aa_nat2025370463774033377d_enat(aTP_Lamm_dz,Uu),Uua) = $let(
        m3: extended_enat,
        m3:= aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(bit0,one))),aa_nat_Extended_enat(semiri4216267220026989637d_enat,Uu)),
        $ite(Uua = zero_zero_nat,m3,plus_p3455044024723400733d_enat(m3,one_on7984719198319812577d_enat)) ) ).

% ATP.lambda_66
tff(fact_10268_ATP_Olambda__67,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_dv,Uu),Uua) = $let(
        m3: complex,
        m3:= aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_complex(semiri8010041392384452111omplex,Uu)),
        $ite(Uua = zero_zero_nat,m3,aa_complex_complex(plus_plus_complex(m3),one_one_complex)) ) ).

% ATP.lambda_67
tff(fact_10269_ATP_Olambda__68,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_dx,Uu),Uua) = $let(
        m3: real,
        m3:= aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Uu)),
        $ite(Uua = zero_zero_nat,m3,aa_real_real(plus_plus_real(m3),one_one_real)) ) ).

% ATP.lambda_68
tff(fact_10270_ATP_Olambda__69,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_dw,Uu),Uua) = $let(
        m3: rat,
        m3:= aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_nat_rat(semiri681578069525770553at_rat,Uu)),
        $ite(Uua = zero_zero_nat,m3,aa_rat_rat(plus_plus_rat(m3),one_one_rat)) ) ).

% ATP.lambda_69
tff(fact_10271_ATP_Olambda__70,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ea,Uu),Uua) = $let(
        m3: nat,
        m3:= aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(semiri1316708129612266289at_nat,Uu)),
        $ite(Uua = zero_zero_nat,m3,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,m3),one_one_nat)) ) ).

% ATP.lambda_70
tff(fact_10272_ATP_Olambda__71,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_dy,Uu),Uua) = $let(
        m3: int,
        m3:= aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(semiri1314217659103216013at_int,Uu)),
        $ite(Uua = zero_zero_nat,m3,aa_int_int(aa_int_fun_int_int(plus_plus_int,m3),one_one_int)) ) ).

% ATP.lambda_71
tff(fact_10273_ATP_Olambda__72,axiom,
    ! [Uu: complex,Uua: real] :
      ( aa_real_o(aTP_Lamm_im(Uu),Uua)
    <=> ( ( aa_complex_complex(sgn_sgn_complex,Uu) = aa_real_complex(cis,Uua) )
        & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Uua)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uua),pi) ) ) ).

% ATP.lambda_72
tff(fact_10274_ATP_Olambda__73,axiom,
    ! [Uu: real,Uua: int] :
      ( aa_int_o(aTP_Lamm_ic(Uu),Uua)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Uua)),Uu)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Uu),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,Uua),one_one_int))) ) ) ).

% ATP.lambda_73
tff(fact_10275_ATP_Olambda__74,axiom,
    ! [Uu: rat,Uua: int] :
      ( aa_int_o(aTP_Lamm_id(Uu),Uua)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Uua)),Uu)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Uu),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,Uua),one_one_int))) ) ) ).

% ATP.lambda_74
tff(fact_10276_ATP_Olambda__75,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ed,Uu),Uua) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat)))) ).

% ATP.lambda_75
tff(fact_10277_ATP_Olambda__76,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_uh,Uu),Uua) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% ATP.lambda_76
tff(fact_10278_ATP_Olambda__77,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_uj(Uu),Uua) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(Uu,Uua)) ).

% ATP.lambda_77
tff(fact_10279_ATP_Olambda__78,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_po(Uu),Uua) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Uua)),aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_78
tff(fact_10280_ATP_Olambda__79,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_pm,Uu),Uua) = aa_complex_complex(times_times_complex(aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Uua)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_79
tff(fact_10281_ATP_Olambda__80,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_pq,Uu),Uua) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_80
tff(fact_10282_ATP_Olambda__81,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_pn(Uu),Uua) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Uua)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_81
tff(fact_10283_ATP_Olambda__82,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_int(aTP_Lamm_pp(Uu),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),Uua)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_82
tff(fact_10284_ATP_Olambda__83,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_me,Uu),Uua) = aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,aa_complex_complex(plus_plus_complex(Uu),aa_nat_complex(semiri8010041392384452111omplex,Uua))),Uua) ).

% ATP.lambda_83
tff(fact_10285_ATP_Olambda__84,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_mg,Uu),Uua) = aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(plus_plus_real(Uu),aa_nat_real(semiri5074537144036343181t_real,Uua))),Uua) ).

% ATP.lambda_84
tff(fact_10286_ATP_Olambda__85,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_mf(Uu),Uua) = aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua))),Uua) ).

% ATP.lambda_85
tff(fact_10287_ATP_Olambda__86,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_qa,Uu),Uua) = aa_complex_complex(times_times_complex(aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,Uu),Uua)),aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(Uu),numera6690914467698888265omplex(aa_num_num(bit0,one)))),aa_nat_complex(semiri8010041392384452111omplex,Uua))) ).

% ATP.lambda_86
tff(fact_10288_ATP_Olambda__87,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_qc,Uu),Uua) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(gbinomial_real,Uu),Uua)),aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(Uu),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Uua))) ).

% ATP.lambda_87
tff(fact_10289_ATP_Olambda__88,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_qb(Uu),Uua) = aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(Uu),Uua)),aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(Uu),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).

% ATP.lambda_88
tff(fact_10290_ATP_Olambda__89,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_nk,Uu),Uua) = aa_complex_complex(times_times_complex(aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,Uu),Uua)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Uua)) ).

% ATP.lambda_89
tff(fact_10291_ATP_Olambda__90,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_nm,Uu),Uua) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(gbinomial_real,Uu),Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)) ).

% ATP.lambda_90
tff(fact_10292_ATP_Olambda__91,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_nl(Uu),Uua) = aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(Uu),Uua)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Uua)) ).

% ATP.lambda_91
tff(fact_10293_ATP_Olambda__92,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_sh,Uu),Uua)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uu),Uua)
        & ( Uu != Uua ) ) ) ).

% ATP.lambda_92
tff(fact_10294_ATP_Olambda__93,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_nv,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% ATP.lambda_93
tff(fact_10295_ATP_Olambda__94,axiom,
    ! [Uu: set_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_vq(Uu),Uua)
    <=> ( member_int(Uua,Uu)
        & ! [X3: int] :
            ( member_int(X3,Uu)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Uua) ) ) ) ).

% ATP.lambda_94
tff(fact_10296_ATP_Olambda__95,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_qs,Uu),Uua) = aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)),aa_nat_real(semiri2265585572941072030t_real,Uua)) ).

% ATP.lambda_95
tff(fact_10297_ATP_Olambda__96,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_vd,Uu),Uua) = aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu)),aa_real_real(exp_real,Uua)) ).

% ATP.lambda_96
tff(fact_10298_ATP_Olambda__97,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_wa,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uu),Uua)),Uua) ).

% ATP.lambda_97
tff(fact_10299_ATP_Olambda__98,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_md,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua)),Uu) ).

% ATP.lambda_98
tff(fact_10300_ATP_Olambda__99,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_mc,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua)),Uua) ).

% ATP.lambda_99
tff(fact_10301_ATP_Olambda__100,axiom,
    ! [Uu: nat,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_qk(Uu),Uua)
    <=> ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uua),Uu) = one_one_complex ) ) ).

% ATP.lambda_100
tff(fact_10302_ATP_Olambda__101,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ut,Uu),Uua) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(plus_plus_real(one_one_real),aa_real_real(divide_divide_real(Uu),aa_nat_real(semiri5074537144036343181t_real,Uua)))),Uua) ).

% ATP.lambda_101
tff(fact_10303_ATP_Olambda__102,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aTP_Lamm_va(Uu),Uua) = aa_real_real(powr_real(aa_real_real(plus_plus_real(one_one_real),aa_real_real(times_times_real(Uu),Uua))),aa_real_real(divide_divide_real(one_one_real),Uua)) ).

% ATP.lambda_102
tff(fact_10304_ATP_Olambda__103,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aTP_Lamm_ve(Uu),Uua) = aa_real_real(powr_real(aa_real_real(plus_plus_real(one_one_real),aa_real_real(divide_divide_real(Uu),Uua))),Uua) ).

% ATP.lambda_103
tff(fact_10305_ATP_Olambda__104,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_eb,Uu),Uua) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat))) ).

% ATP.lambda_104
tff(fact_10306_ATP_Olambda__105,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_hq,Uu),Uua)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Uua)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uua),pi)
        & ( aa_real_real(cos_real,Uua) = Uu ) ) ) ).

% ATP.lambda_105
tff(fact_10307_ATP_Olambda__106,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_nb(Uu),Uua) = aa_real_real(plus_plus_real(aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_real(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_106
tff(fact_10308_ATP_Olambda__107,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_my(Uu),Uua) = aa_rat_rat(plus_plus_rat(aa_nat_rat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_rat(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_107
tff(fact_10309_ATP_Olambda__108,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_na(Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_nat(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_108
tff(fact_10310_ATP_Olambda__109,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_mz(Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_int(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_109
tff(fact_10311_ATP_Olambda__110,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_ee(Uu),Uua) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,Uua)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,zero_zero_complex),Uua)) ).

% ATP.lambda_110
tff(fact_10312_ATP_Olambda__111,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ef(Uu),Uua) = aa_real_real(times_times_real(aa_nat_real(Uu,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Uua)) ).

% ATP.lambda_111
tff(fact_10313_ATP_Olambda__112,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_fr(Uu),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,Uua)),aa_nat_int(aa_int_fun_nat_int(power_power_int,zero_zero_int),Uua)) ).

% ATP.lambda_112
tff(fact_10314_ATP_Olambda__113,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_lt(Uu),Uua) = aa_complex_complex(minus_minus_complex(aa_nat_complex(Uu,Uua)),aa_nat_complex(Uu,aa_nat_nat(suc,Uua))) ).

% ATP.lambda_113
tff(fact_10315_ATP_Olambda__114,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_lw(Uu),Uua) = aa_real_real(minus_minus_real(aa_nat_real(Uu,Uua)),aa_nat_real(Uu,aa_nat_nat(suc,Uua))) ).

% ATP.lambda_114
tff(fact_10316_ATP_Olambda__115,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_lu(Uu),Uua) = aa_rat_rat(minus_minus_rat(aa_nat_rat(Uu,Uua)),aa_nat_rat(Uu,aa_nat_nat(suc,Uua))) ).

% ATP.lambda_115
tff(fact_10317_ATP_Olambda__116,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_lv(Uu),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Uu,Uua)),aa_nat_int(Uu,aa_nat_nat(suc,Uua))) ).

% ATP.lambda_116
tff(fact_10318_ATP_Olambda__117,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_uz(Uu),Uua) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_uj(Uu)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat))) ).

% ATP.lambda_117
tff(fact_10319_ATP_Olambda__118,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_uy(Uu),Uua) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_uj(Uu)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))) ).

% ATP.lambda_118
tff(fact_10320_ATP_Olambda__119,axiom,
    ! [Uu: code_integer,Uua: $o] : aa_o_char(aa_Cod7842134615204088097o_char(aTP_Lamm_tf,Uu),(Uua)) = produc4188289175737317920o_char(aTP_Lamm_te((Uua)),code_bit_cut_integer(Uu)) ).

% ATP.lambda_119
tff(fact_10321_ATP_Olambda__120,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_il,Uu),Uua) = aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% ATP.lambda_120
tff(fact_10322_ATP_Olambda__121,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ik,Uu),Uua) = aa_real_real(real_V1485227260804924795R_real(aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% ATP.lambda_121
tff(fact_10323_ATP_Olambda__122,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_jn,Uu),Uua) = aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(suc,Uua)))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uu),aa_nat_nat(suc,Uua))) ).

% ATP.lambda_122
tff(fact_10324_ATP_Olambda__123,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_jm,Uu),Uua) = aa_real_real(real_V1485227260804924795R_real(aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(suc,Uua)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(suc,Uua))) ).

% ATP.lambda_123
tff(fact_10325_ATP_Olambda__124,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_it,Uu),Uua) = aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Uua))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uu),Uua)) ).

% ATP.lambda_124
tff(fact_10326_ATP_Olambda__125,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_is,Uu),Uua) = aa_real_real(real_V1485227260804924795R_real(aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Uua))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_125
tff(fact_10327_ATP_Olambda__126,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_ii,Uu),Uua) = aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,aa_nat_complex(semiri5044797733671781792omplex,Uua))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uu),Uua)) ).

% ATP.lambda_126
tff(fact_10328_ATP_Olambda__127,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ij,Uu),Uua) = aa_real_real(times_times_real(aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Uua))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_127
tff(fact_10329_ATP_Olambda__128,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_fy,Uu),Uua) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,Uua))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uu),Uua)) ).

% ATP.lambda_128
tff(fact_10330_ATP_Olambda__129,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_fx,Uu),Uua) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_129
tff(fact_10331_ATP_Olambda__130,axiom,
    ! [Uu: num,Uua: num] : aa_num_int(aTP_Lamm_tg(Uu),Uua) = aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,Uu)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,Uu))),aa_num_int(numeral_numeral_int,Uua))) ).

% ATP.lambda_130
tff(fact_10332_ATP_Olambda__131,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_jl,Uu),Uua) = aa_complex_complex(real_V2046097035970521341omplex(aa_nat_real(cos_coeff,Uua)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,Uu)),Uua)) ).

% ATP.lambda_131
tff(fact_10333_ATP_Olambda__132,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_jk,Uu),Uua) = aa_real_real(real_V1485227260804924795R_real(aa_nat_real(cos_coeff,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,Uu)),Uua)) ).

% ATP.lambda_132
tff(fact_10334_ATP_Olambda__133,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_sx,Uu),Uua) = aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Uua)),Uu)) ).

% ATP.lambda_133
tff(fact_10335_ATP_Olambda__134,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_iv,Uu),Uua) = aa_complex_complex(real_V2046097035970521341omplex(aa_nat_real(sin_coeff,Uua)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uu),Uua)) ).

% ATP.lambda_134
tff(fact_10336_ATP_Olambda__135,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_iu,Uu),Uua) = aa_real_real(real_V1485227260804924795R_real(aa_nat_real(sin_coeff,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_135
tff(fact_10337_ATP_Olambda__136,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_ix,Uu),Uua) = aa_complex_complex(real_V2046097035970521341omplex(aa_nat_real(cos_coeff,Uua)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uu),Uua)) ).

% ATP.lambda_136
tff(fact_10338_ATP_Olambda__137,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_iw,Uu),Uua) = aa_real_real(real_V1485227260804924795R_real(aa_nat_real(cos_coeff,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_137
tff(fact_10339_ATP_Olambda__138,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_qm,Uu),Uua) = aa_real_real(times_times_real(aa_nat_real(sin_coeff,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_138
tff(fact_10340_ATP_Olambda__139,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ql,Uu),Uua) = aa_real_real(times_times_real(aa_nat_real(cos_coeff,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_139
tff(fact_10341_ATP_Olambda__140,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa_Code_integer_num(aa_Cod6128727250572145423er_num(aTP_Lamm_rl,Uu),Uua) = $let(
        l2: num,
        l2:= code_num_of_integer(Uu),
        $let(
          l3: num,
          l3:= aa_num_num(plus_plus_num(l2),l2),
          $ite(Uua = zero_z3403309356797280102nteger,l3,aa_num_num(plus_plus_num(l3),one)) ) ) ).

% ATP.lambda_140
tff(fact_10342_ATP_Olambda__141,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa_Code_integer_nat(aa_Cod5420483313623307717er_nat(aTP_Lamm_rn,Uu),Uua) = $let(
        l2: nat,
        l2:= code_nat_of_integer(Uu),
        $let(
          l3: nat,
          l3:= aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,l2),l2),
          $ite(Uua = zero_z3403309356797280102nteger,l3,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,l3),one_one_nat)) ) ) ).

% ATP.lambda_141
tff(fact_10343_ATP_Olambda__142,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_ho,Uu),Uua)
    <=> ( aa_real_real(exp_real,Uua) = Uu ) ) ).

% ATP.lambda_142
tff(fact_10344_ATP_Olambda__143,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_cg,Uu),Uua) = produc1086072967326762835nteger(Uu,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Uua),one_one_Code_integer)) ).

% ATP.lambda_143
tff(fact_10345_ATP_Olambda__144,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_cf,Uu),Uua) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat)) ).

% ATP.lambda_144
tff(fact_10346_ATP_Olambda__145,axiom,
    ! [Uu: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_ce,Uu),Uua) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uu),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua)),one_one_int)) ).

% ATP.lambda_145
tff(fact_10347_ATP_Olambda__146,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_cj,Uu),Uua) = produc1086072967326762835nteger(Uu,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_146
tff(fact_10348_ATP_Olambda__147,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_ci,Uu),Uua) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uu),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_147
tff(fact_10349_ATP_Olambda__148,axiom,
    ! [Uu: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_ch,Uu),Uua) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uu),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_148
tff(fact_10350_ATP_Olambda__149,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_pl,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)) ).

% ATP.lambda_149
tff(fact_10351_ATP_Olambda__150,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_uu,Uu),Uua) = aa_real_real(times_times_real(Uu),aa_real_real(plus_plus_real(one_one_real),aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua)))))) ).

% ATP.lambda_150
tff(fact_10352_ATP_Olambda__151,axiom,
    ! [Uu: nat,Uua: vEBT_VEBT] : aa_VEB5832953974749077948T_VEBT(aTP_Lamm_aa(Uu),Uua) = vEBT_VEBT_elim_dead(Uua,aa_nat_Extended_enat(extended_enat2,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% ATP.lambda_151
tff(fact_10353_ATP_Olambda__152,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_dn,Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),aa_o_int(zero_n2684676970156552555ol_int,Uua != zero_zero_int)) ).

% ATP.lambda_152
tff(fact_10354_ATP_Olambda__153,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_us,Uu),Uua) = aa_real_real(plus_plus_real(Uu),aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua))))) ).

% ATP.lambda_153
tff(fact_10355_ATP_Olambda__154,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_up,Uu),Uua) = aa_real_real(plus_plus_real(Uu),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua)))) ).

% ATP.lambda_154
tff(fact_10356_ATP_Olambda__155,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_sv(Uu),Uua) = aa_set_nat_real(groups6591440286371151544t_real(Uu),aa_nat_set_nat(set_ord_lessThan_nat,Uua)) ).

% ATP.lambda_155
tff(fact_10357_ATP_Olambda__156,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_lx(Uu),Uua)
    <=> aa_set_nat_o(ord_less_eq_set_nat(Uua),Uu) ) ).

% ATP.lambda_156
tff(fact_10358_ATP_Olambda__157,axiom,
    ! [Uu: set_int,Uua: set_int] :
      ( aa_set_int_o(aTP_Lamm_ly(Uu),Uua)
    <=> ord_less_eq_set_int(Uua,Uu) ) ).

% ATP.lambda_157
tff(fact_10359_ATP_Olambda__158,axiom,
    ! [Uu: rat,Uua: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(aTP_Lamm_lz,Uu),Uua)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Uua),Uu) ) ).

% ATP.lambda_158
tff(fact_10360_ATP_Olambda__159,axiom,
    ! [Uu: num,Uua: num] :
      ( aa_num_o(aa_num_fun_num_o(aTP_Lamm_ma,Uu),Uua)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Uua),Uu) ) ).

% ATP.lambda_159
tff(fact_10361_ATP_Olambda__160,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_am,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uua),Uu) ) ).

% ATP.lambda_160
tff(fact_10362_ATP_Olambda__161,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_mb,Uu),Uua)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uua),Uu) ) ).

% ATP.lambda_161
tff(fact_10363_ATP_Olambda__162,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_sw,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Uua),Uu) ).

% ATP.lambda_162
tff(fact_10364_ATP_Olambda__163,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_qi,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uu) ) ).

% ATP.lambda_163
tff(fact_10365_ATP_Olambda__164,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aTP_Lamm_ca(Uu),Uua) = aa_real_real(times_times_real(Uua),Uu) ).

% ATP.lambda_164
tff(fact_10366_ATP_Olambda__165,axiom,
    ! [Uu: rat,Uua: rat] : aa_rat_rat(aTP_Lamm_cb(Uu),Uua) = aa_rat_rat(times_times_rat(Uua),Uu) ).

% ATP.lambda_165
tff(fact_10367_ATP_Olambda__166,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_cc,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uu) ).

% ATP.lambda_166
tff(fact_10368_ATP_Olambda__167,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_cd,Uu),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uu) ).

% ATP.lambda_167
tff(fact_10369_ATP_Olambda__168,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_st,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uu) ).

% ATP.lambda_168
tff(fact_10370_ATP_Olambda__169,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_tr,Uu),Uua) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu) ).

% ATP.lambda_169
tff(fact_10371_ATP_Olambda__170,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_re,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uu) ).

% ATP.lambda_170
tff(fact_10372_ATP_Olambda__171,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_su,Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uua),Uu) ).

% ATP.lambda_171
tff(fact_10373_ATP_Olambda__172,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aTP_Lamm_tt(Uu),Uua) = aa_real_real(powr_real(Uua),Uu) ).

% ATP.lambda_172
tff(fact_10374_ATP_Olambda__173,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(aTP_Lamm_bh,Uu),Uua)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Uua),Uu) ) ).

% ATP.lambda_173
tff(fact_10375_ATP_Olambda__174,axiom,
    ! [Uu: complex,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_bf(Uu),Uua)
    <=> aa_complex_o(dvd_dvd_complex(Uua),Uu) ) ).

% ATP.lambda_174
tff(fact_10376_ATP_Olambda__175,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_bg,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uua),Uu) ) ).

% ATP.lambda_175
tff(fact_10377_ATP_Olambda__176,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_bi,Uu),Uua)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uua),Uu) ) ).

% ATP.lambda_176
tff(fact_10378_ATP_Olambda__177,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_sk,Uu),Uua) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uua),Uu) ).

% ATP.lambda_177
tff(fact_10379_ATP_Olambda__178,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_kx,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uua),Uu) ).

% ATP.lambda_178
tff(fact_10380_ATP_Olambda__179,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_un,Uu),Uua) = aa_real_real(aa_nat_fun_real_real(root,Uua),Uu) ).

% ATP.lambda_179
tff(fact_10381_ATP_Olambda__180,axiom,
    ! [Uu: set_complex,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_ac(Uu),Uua)
    <=> member_complex(Uua,Uu) ) ).

% ATP.lambda_180
tff(fact_10382_ATP_Olambda__181,axiom,
    ! [Uu: set_real,Uua: real] :
      ( aa_real_o(aTP_Lamm_ab(Uu),Uua)
    <=> member_real(Uua,Uu) ) ).

% ATP.lambda_181
tff(fact_10383_ATP_Olambda__182,axiom,
    ! [Uu: set_list_nat,Uua: list_nat] :
      ( aa_list_nat_o(aTP_Lamm_ad(Uu),Uua)
    <=> member_list_nat(Uua,Uu) ) ).

% ATP.lambda_182
tff(fact_10384_ATP_Olambda__183,axiom,
    ! [Uu: set_set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_ae(Uu),Uua)
    <=> member_set_nat(Uua,Uu) ) ).

% ATP.lambda_183
tff(fact_10385_ATP_Olambda__184,axiom,
    ! [Uu: set_nat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_af(Uu),Uua)
    <=> aa_set_nat_o(member_nat(Uua),Uu) ) ).

% ATP.lambda_184
tff(fact_10386_ATP_Olambda__185,axiom,
    ! [Uu: set_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_ag(Uu),Uua)
    <=> member_int(Uua,Uu) ) ).

% ATP.lambda_185
tff(fact_10387_ATP_Olambda__186,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_ht,Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_186
tff(fact_10388_ATP_Olambda__187,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_hu,Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_187
tff(fact_10389_ATP_Olambda__188,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_qr(Uu),Uua) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat)) ).

% ATP.lambda_188
tff(fact_10390_ATP_Olambda__189,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_qq(Uu),Uua) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_189
tff(fact_10391_ATP_Olambda__190,axiom,
    ! [Uu: fun_real_o,Uua: real] :
      ( aa_real_o(aTP_Lamm_vn(Uu),Uua)
    <=> aa_real_o(Uu,aa_real_real(inverse_inverse_real,Uua)) ) ).

% ATP.lambda_190
tff(fact_10392_ATP_Olambda__191,axiom,
    ! [Uu: fun_real_real,Uua: nat] : aa_nat_real(aTP_Lamm_wb(Uu),Uua) = aa_real_real(Uu,aa_nat_real(semiri5074537144036343181t_real,Uua)) ).

% ATP.lambda_191
tff(fact_10393_ATP_Olambda__192,axiom,
    ! [Uu: fun_real_real,Uua: real] : aa_real_real(aTP_Lamm_tq(Uu),Uua) = aa_real_real(Uu,aa_real_real(uminus_uminus_real,Uua)) ).

% ATP.lambda_192
tff(fact_10394_ATP_Olambda__193,axiom,
    ! [Uu: fun_real_o,Uua: real] :
      ( aa_real_o(aTP_Lamm_vl(Uu),Uua)
    <=> aa_real_o(Uu,aa_real_real(uminus_uminus_real,Uua)) ) ).

% ATP.lambda_193
tff(fact_10395_ATP_Olambda__194,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_fl(Uu),Uua) = aa_nat_complex(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_194
tff(fact_10396_ATP_Olambda__195,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_fk(Uu),Uua) = aa_nat_real(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_195
tff(fact_10397_ATP_Olambda__196,axiom,
    ! [Uu: fun_nat_o,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_vi(Uu),Uua)
    <=> aa_nat_o(Uu,aa_nat_nat(suc,Uua)) ) ).

% ATP.lambda_196
tff(fact_10398_ATP_Olambda__197,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_ls(Uu),Uua) = aa_nat_rat(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_197
tff(fact_10399_ATP_Olambda__198,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_gd(Uu),Uua) = aa_nat_nat(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_198
tff(fact_10400_ATP_Olambda__199,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_ge(Uu),Uua) = aa_nat_int(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_199
tff(fact_10401_ATP_Olambda__200,axiom,
    ! [Uu: nat,Uua: num] : aa_num_option_num(aTP_Lamm_tn(Uu),Uua) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(Uu,Uua))) ).

% ATP.lambda_200
tff(fact_10402_ATP_Olambda__201,axiom,
    ! [Uu: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_tk(Uu),Uua) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(Uua,Uu))) ).

% ATP.lambda_201
tff(fact_10403_ATP_Olambda__202,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_ss,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_sr(Uu),Uua)) ).

% ATP.lambda_202
tff(fact_10404_ATP_Olambda__203,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_sq,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_sp(Uu),Uua)) ).

% ATP.lambda_203
tff(fact_10405_ATP_Olambda__204,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aTP_Lamm_so,Uu),Uua) = produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_sn(Uu),Uua)) ).

% ATP.lambda_204
tff(fact_10406_ATP_Olambda__205,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aTP_Lamm_sm,Uu),Uua) = produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_sl(Uu),Uua)) ).

% ATP.lambda_205
tff(fact_10407_ATP_Olambda__206,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_sj,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_si(Uu),Uua)) ).

% ATP.lambda_206
tff(fact_10408_ATP_Olambda__207,axiom,
    ! [Uu: fun_nat_real,Uua: real] : aa_real_real(aTP_Lamm_ty(Uu),Uua) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_tx(Uu),Uua)) ).

% ATP.lambda_207
tff(fact_10409_ATP_Olambda__208,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_qt,Uu),Uua) = aa_real_complex(cis,aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),aa_nat_real(semiri5074537144036343181t_real,Uua))),aa_nat_real(semiri5074537144036343181t_real,Uu))) ).

% ATP.lambda_208
tff(fact_10410_ATP_Olambda__209,axiom,
    ! [Uu: fun_real_real,Uua: real] :
      ( aa_real_o(aTP_Lamm_vf(Uu),Uua)
    <=> ( aa_real_real(Uu,Uua) != zero_zero_real ) ) ).

% ATP.lambda_209
tff(fact_10411_ATP_Olambda__210,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_real(aa_com20334259897932645t_real(aTP_Lamm_jh,Uu),Uua) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Uua))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uu),Uua))) ).

% ATP.lambda_210
tff(fact_10412_ATP_Olambda__211,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_jg,Uu),Uua) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(real_V1485227260804924795R_real(aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Uua))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua))) ).

% ATP.lambda_211
tff(fact_10413_ATP_Olambda__212,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_bo,Uu),Uua)
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))) ) ).

% ATP.lambda_212
tff(fact_10414_ATP_Olambda__213,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_jj,Uu),Uua) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(real_V2046097035970521341omplex(aa_nat_real(sin_coeff,Uua)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,Uu)),Uua))) ).

% ATP.lambda_213
tff(fact_10415_ATP_Olambda__214,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ji,Uu),Uua) = aa_real_real(uminus_uminus_real,aa_real_real(real_V1485227260804924795R_real(aa_nat_real(sin_coeff,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,Uu)),Uua))) ).

% ATP.lambda_214
tff(fact_10416_ATP_Olambda__215,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_real(aa_com20334259897932645t_real(aTP_Lamm_iz,Uu),Uua) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(real_V2046097035970521341omplex(aa_nat_real(sin_coeff,Uua)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uu),Uua))) ).

% ATP.lambda_215
tff(fact_10417_ATP_Olambda__216,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_iy,Uu),Uua) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(real_V1485227260804924795R_real(aa_nat_real(sin_coeff,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua))) ).

% ATP.lambda_216
tff(fact_10418_ATP_Olambda__217,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_real(aa_com20334259897932645t_real(aTP_Lamm_jb,Uu),Uua) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(real_V2046097035970521341omplex(aa_nat_real(cos_coeff,Uua)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uu),Uua))) ).

% ATP.lambda_217
tff(fact_10419_ATP_Olambda__218,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ja,Uu),Uua) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(real_V1485227260804924795R_real(aa_nat_real(cos_coeff,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua))) ).

% ATP.lambda_218
tff(fact_10420_ATP_Olambda__219,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_hk(Uu),Uua) = times_times_complex(aa_complex_complex(minus_minus_complex(Uu),aa_nat_complex(semiri8010041392384452111omplex,Uua))) ).

% ATP.lambda_219
tff(fact_10421_ATP_Olambda__220,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_hm(Uu),Uua) = times_times_real(aa_real_real(minus_minus_real(Uu),aa_nat_real(semiri5074537144036343181t_real,Uua))) ).

% ATP.lambda_220
tff(fact_10422_ATP_Olambda__221,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_hl(Uu),Uua) = times_times_rat(aa_rat_rat(minus_minus_rat(Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).

% ATP.lambda_221
tff(fact_10423_ATP_Olambda__222,axiom,
    ! [Uu: extended_enat,Uua: nat] : aa_nat9102221060835343569d_enat(aTP_Lamm_hh(Uu),Uua) = times_7803423173614009249d_enat(plus_p3455044024723400733d_enat(Uu,aa_nat_Extended_enat(semiri4216267220026989637d_enat,Uua))) ).

% ATP.lambda_222
tff(fact_10424_ATP_Olambda__223,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_hd(Uu),Uua) = times_times_complex(aa_complex_complex(plus_plus_complex(Uu),aa_nat_complex(semiri8010041392384452111omplex,Uua))) ).

% ATP.lambda_223
tff(fact_10425_ATP_Olambda__224,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_hf(Uu),Uua) = times_times_real(aa_real_real(plus_plus_real(Uu),aa_nat_real(semiri5074537144036343181t_real,Uua))) ).

% ATP.lambda_224
tff(fact_10426_ATP_Olambda__225,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_he(Uu),Uua) = times_times_rat(aa_rat_rat(plus_plus_rat(Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).

% ATP.lambda_225
tff(fact_10427_ATP_Olambda__226,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_hi(Uu),Uua) = aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),aa_nat_nat(semiri1316708129612266289at_nat,Uua))) ).

% ATP.lambda_226
tff(fact_10428_ATP_Olambda__227,axiom,
    ! [Uu: int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_hg(Uu),Uua) = aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),aa_nat_int(semiri1314217659103216013at_int,Uua))) ).

% ATP.lambda_227
tff(fact_10429_ATP_Olambda__228,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ur,Uu),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_228
tff(fact_10430_ATP_Olambda__229,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_Extended_enat(aTP_Lamm_ax(Uu),Uua) = aa_nat_Extended_enat(extended_enat2,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu),Uua)) ).

% ATP.lambda_229
tff(fact_10431_ATP_Olambda__230,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_Extended_enat(aTP_Lamm_bp(Uu),Uua) = aa_nat_Extended_enat(extended_enat2,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uua)) ).

% ATP.lambda_230
tff(fact_10432_ATP_Olambda__231,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_Extended_enat(aTP_Lamm_ak(Uu),Uua) = aa_nat_Extended_enat(extended_enat2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua)) ).

% ATP.lambda_231
tff(fact_10433_ATP_Olambda__232,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_rs,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uu),Uua)) ).

% ATP.lambda_232
tff(fact_10434_ATP_Olambda__233,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_rt,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uua),Uu)) ).

% ATP.lambda_233
tff(fact_10435_ATP_Olambda__234,axiom,
    ! [Uu: set_complex,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_cq(Uu),Uua)
    <=> ~ member_complex(Uua,Uu) ) ).

% ATP.lambda_234
tff(fact_10436_ATP_Olambda__235,axiom,
    ! [Uu: set_real,Uua: real] :
      ( aa_real_o(aTP_Lamm_cp(Uu),Uua)
    <=> ~ member_real(Uua,Uu) ) ).

% ATP.lambda_235
tff(fact_10437_ATP_Olambda__236,axiom,
    ! [Uu: set_list_nat,Uua: list_nat] :
      ( aa_list_nat_o(aTP_Lamm_cr(Uu),Uua)
    <=> ~ member_list_nat(Uua,Uu) ) ).

% ATP.lambda_236
tff(fact_10438_ATP_Olambda__237,axiom,
    ! [Uu: set_set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_cs(Uu),Uua)
    <=> ~ member_set_nat(Uua,Uu) ) ).

% ATP.lambda_237
tff(fact_10439_ATP_Olambda__238,axiom,
    ! [Uu: set_nat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_ct(Uu),Uua)
    <=> ~ aa_set_nat_o(member_nat(Uua),Uu) ) ).

% ATP.lambda_238
tff(fact_10440_ATP_Olambda__239,axiom,
    ! [Uu: set_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_cu(Uu),Uua)
    <=> ~ member_int(Uua,Uu) ) ).

% ATP.lambda_239
tff(fact_10441_ATP_Olambda__240,axiom,
    ! [Uu: fun_complex_complex,Uua: complex] : aa_complex_real(aTP_Lamm_kw(Uu),Uua) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(Uu,Uua)) ).

% ATP.lambda_240
tff(fact_10442_ATP_Olambda__241,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_real(aTP_Lamm_ev(Uu),Uua) = aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(Uu,Uua)) ).

% ATP.lambda_241
tff(fact_10443_ATP_Olambda__242,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_eu(Uu),Uua) = aa_real_real(real_V7735802525324610683m_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_242
tff(fact_10444_ATP_Olambda__243,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ul(Uu),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_243
tff(fact_10445_ATP_Olambda__244,axiom,
    ! [Uu: fun_complex_nat,Uua: complex] : aa_complex_complex(aTP_Lamm_kf(Uu),Uua) = aa_nat_complex(semiri8010041392384452111omplex,aa_complex_nat(Uu,Uua)) ).

% ATP.lambda_244
tff(fact_10446_ATP_Olambda__245,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_Extended_enat(aTP_Lamm_ki(Uu),Uua) = aa_nat_Extended_enat(semiri4216267220026989637d_enat,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_245
tff(fact_10447_ATP_Olambda__246,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_real(aTP_Lamm_kk(Uu),Uua) = aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_246
tff(fact_10448_ATP_Olambda__247,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_kj(Uu),Uua) = aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_247
tff(fact_10449_ATP_Olambda__248,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_int(aTP_Lamm_kh(Uu),Uua) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_248
tff(fact_10450_ATP_Olambda__249,axiom,
    ! [Uu: fun_int_nat,Uua: int] : aa_int_int(aTP_Lamm_kg(Uu),Uua) = aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(Uu,Uua)) ).

% ATP.lambda_249
tff(fact_10451_ATP_Olambda__250,axiom,
    ! [Uu: fun_complex_real,Uua: complex] : aa_complex_complex(aTP_Lamm_kq(Uu),Uua) = aa_real_complex(real_V4546457046886955230omplex,aa_complex_real(Uu,Uua)) ).

% ATP.lambda_250
tff(fact_10452_ATP_Olambda__251,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_complex(aTP_Lamm_fz(Uu),Uua) = aa_real_complex(real_V4546457046886955230omplex,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_251
tff(fact_10453_ATP_Olambda__252,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ga(Uu),Uua) = aa_real_real(real_V1803761363581548252l_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_252
tff(fact_10454_ATP_Olambda__253,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_gp(Uu),Uua) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(Uu,Uua)) ).

% ATP.lambda_253
tff(fact_10455_ATP_Olambda__254,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_fj(Uu),Uua) = aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(Uu,Uua)) ).

% ATP.lambda_254
tff(fact_10456_ATP_Olambda__255,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_fi(Uu),Uua) = aa_real_real(uminus_uminus_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_255
tff(fact_10457_ATP_Olambda__256,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_go(Uu),Uua) = aa_rat_rat(uminus_uminus_rat,aa_nat_rat(Uu,Uua)) ).

% ATP.lambda_256
tff(fact_10458_ATP_Olambda__257,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_gn(Uu),Uua) = aa_int_int(uminus_uminus_int,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_257
tff(fact_10459_ATP_Olambda__258,axiom,
    ! [Uu: fun_complex_int,Uua: complex] : aa_complex_complex(aTP_Lamm_kl(Uu),Uua) = aa_int_complex(ring_17405671764205052669omplex,aa_complex_int(Uu,Uua)) ).

% ATP.lambda_258
tff(fact_10460_ATP_Olambda__259,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_real(aTP_Lamm_km(Uu),Uua) = aa_int_real(ring_1_of_int_real,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_259
tff(fact_10461_ATP_Olambda__260,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_real(aTP_Lamm_kn(Uu),Uua) = aa_int_real(ring_1_of_int_real,aa_int_int(Uu,Uua)) ).

% ATP.lambda_260
tff(fact_10462_ATP_Olambda__261,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_rat(aTP_Lamm_ko(Uu),Uua) = aa_int_rat(ring_1_of_int_rat,aa_int_int(Uu,Uua)) ).

% ATP.lambda_261
tff(fact_10463_ATP_Olambda__262,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_int(aTP_Lamm_kp(Uu),Uua) = aa_int_int(ring_1_of_int_int,aa_int_int(Uu,Uua)) ).

% ATP.lambda_262
tff(fact_10464_ATP_Olambda__263,axiom,
    ! [Uu: fun_real_real,Uua: real] : aa_real_real(aTP_Lamm_vt(Uu),Uua) = aa_real_real(artanh_real,aa_real_real(Uu,Uua)) ).

% ATP.lambda_263
tff(fact_10465_ATP_Olambda__264,axiom,
    ! [Uu: fun_real_real,Uua: real] : aa_real_real(aTP_Lamm_vr(Uu),Uua) = aa_real_real(arsinh_real,aa_real_real(Uu,Uua)) ).

% ATP.lambda_264
tff(fact_10466_ATP_Olambda__265,axiom,
    ! [Uu: fun_real_real,Uua: real] : aa_real_real(aTP_Lamm_vs(Uu),Uua) = aa_real_real(arcosh_real,aa_real_real(Uu,Uua)) ).

% ATP.lambda_265
tff(fact_10467_ATP_Olambda__266,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_fm(Uu),Uua) = aa_real_real(abs_abs_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_266
tff(fact_10468_ATP_Olambda__267,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ux(Uu),Uua) = aa_real_real(cos_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_267
tff(fact_10469_ATP_Olambda__268,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_real(aTP_Lamm_rh(Uu),Uua) = aa_complex_real(re,aa_nat_complex(Uu,Uua)) ).

% ATP.lambda_268
tff(fact_10470_ATP_Olambda__269,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_real(aTP_Lamm_ri(Uu),Uua) = aa_complex_real(im,aa_nat_complex(Uu,Uua)) ).

% ATP.lambda_269
tff(fact_10471_ATP_Olambda__270,axiom,
    ! [Uu: fun_co3258106519541737638omplex,Uua: complex] : aa_complex_complex(aTP_Lamm_mq(Uu),Uua) = suminf_complex(aa_com8181001469867462503omplex(Uu,Uua)) ).

% ATP.lambda_270
tff(fact_10472_ATP_Olambda__271,axiom,
    ! [Uu: fun_co8378349014452122532t_real,Uua: complex] : aa_complex_real(aTP_Lamm_mj(Uu),Uua) = suminf_real(aa_com20334259897932645t_real(Uu,Uua)) ).

% ATP.lambda_271
tff(fact_10473_ATP_Olambda__272,axiom,
    ! [Uu: fun_re2084383175013312036omplex,Uua: real] : aa_real_complex(aTP_Lamm_mm(Uu),Uua) = suminf_complex(aa_rea1531280491672885477omplex(Uu,Uua)) ).

% ATP.lambda_272
tff(fact_10474_ATP_Olambda__273,axiom,
    ! [Uu: fun_re4238438383274954274t_real,Uua: real] : aa_real_real(aTP_Lamm_mk(Uu),Uua) = suminf_real(aa_real_fun_nat_real(Uu,Uua)) ).

% ATP.lambda_273
tff(fact_10475_ATP_Olambda__274,axiom,
    ! [Uu: fun_na1142814060743658952omplex,Uua: nat] : aa_nat_complex(aTP_Lamm_mn(Uu),Uua) = suminf_complex(aa_nat6454746962644518025omplex(Uu,Uua)) ).

% ATP.lambda_274
tff(fact_10476_ATP_Olambda__275,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_mr(Uu),Uua) = suminf_real(aa_nat_fun_nat_real(Uu,Uua)) ).

% ATP.lambda_275
tff(fact_10477_ATP_Olambda__276,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_mp(Uu),Uua) = suminf_nat(aa_nat_fun_nat_nat(Uu,Uua)) ).

% ATP.lambda_276
tff(fact_10478_ATP_Olambda__277,axiom,
    ! [Uu: fun_in8056368165505483812omplex,Uua: int] : aa_int_complex(aTP_Lamm_mo(Uu),Uua) = suminf_complex(aa_int4144929030551567077omplex(Uu,Uua)) ).

% ATP.lambda_277
tff(fact_10479_ATP_Olambda__278,axiom,
    ! [Uu: fun_int_fun_nat_real,Uua: int] : aa_int_real(aTP_Lamm_ml(Uu),Uua) = suminf_real(aa_int_fun_nat_real(Uu,Uua)) ).

% ATP.lambda_278
tff(fact_10480_ATP_Olambda__279,axiom,
    ! [Uu: fun_int_fun_nat_int,Uua: int] : aa_int_int(aTP_Lamm_ms(Uu),Uua) = suminf_int(aa_int_fun_nat_int(Uu,Uua)) ).

% ATP.lambda_279
tff(fact_10481_ATP_Olambda__280,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_rj(Uu),Uua) = aa_complex_complex(cnj,aa_nat_complex(Uu,Uua)) ).

% ATP.lambda_280
tff(fact_10482_ATP_Olambda__281,axiom,
    ! [Uu: fun_complex_o,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_ck(Uu),Uua)
    <=> ~ aa_complex_o(Uu,Uua) ) ).

% ATP.lambda_281
tff(fact_10483_ATP_Olambda__282,axiom,
    ! [Uu: fun_real_o,Uua: real] :
      ( aa_real_o(aTP_Lamm_in(Uu),Uua)
    <=> ~ aa_real_o(Uu,Uua) ) ).

% ATP.lambda_282
tff(fact_10484_ATP_Olambda__283,axiom,
    ! [Uu: fun_list_nat_o,Uua: list_nat] :
      ( aa_list_nat_o(aTP_Lamm_cl(Uu),Uua)
    <=> ~ aa_list_nat_o(Uu,Uua) ) ).

% ATP.lambda_283
tff(fact_10485_ATP_Olambda__284,axiom,
    ! [Uu: fun_set_nat_o,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_cm(Uu),Uua)
    <=> ~ aa_set_nat_o(Uu,Uua) ) ).

% ATP.lambda_284
tff(fact_10486_ATP_Olambda__285,axiom,
    ! [Uu: fun_nat_o,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_cn(Uu),Uua)
    <=> ~ aa_nat_o(Uu,Uua) ) ).

% ATP.lambda_285
tff(fact_10487_ATP_Olambda__286,axiom,
    ! [Uu: fun_int_o,Uua: int] :
      ( aa_int_o(aTP_Lamm_co(Uu),Uua)
    <=> ~ aa_int_o(Uu,Uua) ) ).

% ATP.lambda_286
tff(fact_10488_ATP_Olambda__287,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_gm(Uu),Uua),Uub) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uub),aa_nat_real(Uua,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% ATP.lambda_287
tff(fact_10489_ATP_Olambda__288,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_qp(Uu),Uua),Uub) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uub),aa_nat_real(Uu,Uub),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_288
tff(fact_10490_ATP_Olambda__289,axiom,
    ! [Uu: num,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_bt(Uu),Uua),Uub) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(Uu)),Uub),produc1086072967326762835nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Uua),one_one_Code_integer),minus_8373710615458151222nteger(Uub,numera6620942414471956472nteger(Uu))),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Uua),Uub)) ).

% ATP.lambda_289
tff(fact_10491_ATP_Olambda__290,axiom,
    ! [Uu: num,Uua: nat,Uub: nat] :
      aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_br(Uu),Uua),Uub) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Uu)),Uub),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_num_nat(numeral_numeral_nat,Uu))),aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),Uub)) ).

% ATP.lambda_290
tff(fact_10492_ATP_Olambda__291,axiom,
    ! [Uu: num,Uua: int,Uub: int] :
      aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_bs(Uu),Uua),Uub) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Uu)),Uub),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua)),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Uub),aa_num_int(numeral_numeral_int,Uu))),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua)),Uub)) ).

% ATP.lambda_291
tff(fact_10493_ATP_Olambda__292,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: nat] :
      aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ek(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_complex(Uua,Uub),zero_zero_complex) ).

% ATP.lambda_292
tff(fact_10494_ATP_Olambda__293,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_el(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_293
tff(fact_10495_ATP_Olambda__294,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_em(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_nat(Uua,Uub),zero_zero_nat) ).

% ATP.lambda_294
tff(fact_10496_ATP_Olambda__295,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_en(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_int(Uua,Uub),zero_zero_int) ).

% ATP.lambda_295
tff(fact_10497_ATP_Olambda__296,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_rp(Uu),Uua),Uub) = $ite(Uub = zero_z3403309356797280102nteger,produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),one_one_Code_integer),minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uu),Uub))) ).

% ATP.lambda_296
tff(fact_10498_ATP_Olambda__297,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_sg(Uu),Uua),Uub) = $ite(Uub = zero_z3403309356797280102nteger,produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),one_one_Code_integer),minus_8373710615458151222nteger(abs_abs_Code_integer(Uu),Uub))) ).

% ATP.lambda_297
tff(fact_10499_ATP_Olambda__298,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_ro(Uu),Uua),Uub) = $ite(Uub = zero_z3403309356797280102nteger,produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),one_one_Code_integer),minus_8373710615458151222nteger(Uu,Uub))) ).

% ATP.lambda_298
tff(fact_10500_ATP_Olambda__299,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] :
      ( aa_real_o(aa_fun2895569901277098738real_o(aTP_Lamm_vg(Uu),Uua),Uub)
    <=> has_fi5821293074295781190e_real(Uu,aa_real_real(Uua,Uub),topolo2177554685111907308n_real(Uub,top_top_set_real)) ) ).

% ATP.lambda_299
tff(fact_10501_ATP_Olambda__300,axiom,
    ! [Uu: fun_co3258106519541737638omplex,Uua: nat,Uub: complex] : aa_complex_complex(aa_nat3829984526574172007omplex(aTP_Lamm_lm(Uu),Uua),Uub) = aa_nat_complex(aa_com8181001469867462503omplex(Uu,Uub),Uua) ).

% ATP.lambda_300
tff(fact_10502_ATP_Olambda__301,axiom,
    ! [Uu: fun_co8378349014452122532t_real,Uua: nat,Uub: complex] : aa_complex_real(aa_nat480766339200631653x_real(aTP_Lamm_ky(Uu),Uua),Uub) = aa_nat_real(aa_com20334259897932645t_real(Uu,Uub),Uua) ).

% ATP.lambda_301
tff(fact_10503_ATP_Olambda__302,axiom,
    ! [Uu: fun_re2084383175013312036omplex,Uua: nat,Uub: real] : aa_real_complex(aa_nat7791439356240649701omplex(aTP_Lamm_le(Uu),Uua),Uub) = aa_nat_complex(aa_rea1531280491672885477omplex(Uu,Uub),Uua) ).

% ATP.lambda_302
tff(fact_10504_ATP_Olambda__303,axiom,
    ! [Uu: fun_re4238438383274954274t_real,Uua: nat,Uub: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_la(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(Uu,Uub),Uua) ).

% ATP.lambda_303
tff(fact_10505_ATP_Olambda__304,axiom,
    ! [Uu: fun_na1142814060743658952omplex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_lg(Uu),Uua),Uub) = aa_nat_complex(aa_nat6454746962644518025omplex(Uu,Uub),Uua) ).

% ATP.lambda_304
tff(fact_10506_ATP_Olambda__305,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_lo(Uu),Uua),Uub) = aa_nat_real(aa_nat_fun_nat_real(Uu,Uub),Uua) ).

% ATP.lambda_305
tff(fact_10507_ATP_Olambda__306,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_lk(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(Uu,Uub),Uua) ).

% ATP.lambda_306
tff(fact_10508_ATP_Olambda__307,axiom,
    ! [Uu: fun_in8056368165505483812omplex,Uua: nat,Uub: int] : aa_int_complex(aa_nat7430282427850695397omplex(aTP_Lamm_li(Uu),Uua),Uub) = aa_nat_complex(aa_int4144929030551567077omplex(Uu,Uub),Uua) ).

% ATP.lambda_307
tff(fact_10509_ATP_Olambda__308,axiom,
    ! [Uu: fun_int_fun_nat_real,Uua: nat,Uub: int] : aa_int_real(aa_nat_fun_int_real(aTP_Lamm_lc(Uu),Uua),Uub) = aa_nat_real(aa_int_fun_nat_real(Uu,Uub),Uua) ).

% ATP.lambda_308
tff(fact_10510_ATP_Olambda__309,axiom,
    ! [Uu: fun_int_fun_nat_int,Uua: nat,Uub: int] : aa_int_int(aa_nat_fun_int_int(aTP_Lamm_lq(Uu),Uua),Uub) = aa_nat_int(aa_int_fun_nat_int(Uu,Uub),Uua) ).

% ATP.lambda_309
tff(fact_10511_ATP_Olambda__310,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_mx(Uu),Uua),Uub) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_fun3023522663955195517t_real(aTP_Lamm_mw(Uu),Uua),Uub)),aa_nat_set_nat(set_ord_atMost_nat,Uub)) ).

% ATP.lambda_310
tff(fact_10512_ATP_Olambda__311,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_mv(Uu),Uua),Uub) = aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_fun5379041630539201665omplex(aTP_Lamm_mu(Uu),Uua),Uub)),aa_nat_set_nat(set_ord_atMost_nat,Uub)) ).

% ATP.lambda_311
tff(fact_10513_ATP_Olambda__312,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_kd(Uu),Uua),Uub) = aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_kc(Uu),Uua),Uub)),aa_nat_set_nat(set_ord_atMost_nat,Uub)) ).

% ATP.lambda_312
tff(fact_10514_ATP_Olambda__313,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_kb(Uu),Uua),Uub) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ka(Uu),Uua),Uub)),aa_nat_set_nat(set_ord_atMost_nat,Uub)) ).

% ATP.lambda_313
tff(fact_10515_ATP_Olambda__314,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_jz(Uu),Uua),Uub) = aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_jy(Uu),Uua),Uub)),aa_nat_set_nat(set_ord_atMost_nat,Uub)) ).

% ATP.lambda_314
tff(fact_10516_ATP_Olambda__315,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_jx(Uu),Uua),Uub) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_jw(Uu),Uua),Uub)),aa_nat_set_nat(set_ord_atMost_nat,Uub)) ).

% ATP.lambda_315
tff(fact_10517_ATP_Olambda__316,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_jv(Uu),Uua),Uub) = aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_ju(Uu),Uua),Uub)),aa_nat_set_nat(set_ord_atMost_nat,Uub)) ).

% ATP.lambda_316
tff(fact_10518_ATP_Olambda__317,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_jt(Uu),Uua),Uub) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_js(Uu),Uua),Uub)),aa_nat_set_nat(set_ord_atMost_nat,Uub)) ).

% ATP.lambda_317
tff(fact_10519_ATP_Olambda__318,axiom,
    ! [Uu: nat,Uua: complex,Uub: nat] :
      aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_gh(Uu),Uua),Uub) = aa_complex_complex(times_times_complex($ite(Uub = Uu,one_one_complex,zero_zero_complex)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uua),Uub)) ).

% ATP.lambda_318
tff(fact_10520_ATP_Olambda__319,axiom,
    ! [Uu: nat,Uua: real,Uub: nat] :
      aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_gi(Uu),Uua),Uub) = aa_real_real(times_times_real($ite(Uub = Uu,one_one_real,zero_zero_real)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_319
tff(fact_10521_ATP_Olambda__320,axiom,
    ! [Uu: nat,Uua: int,Uub: nat] :
      aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_gj(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,
          $ite(Uub = Uu,one_one_int,zero_zero_int)),aa_nat_int(aa_int_fun_nat_int(power_power_int,Uua),Uub)) ).

% ATP.lambda_320
tff(fact_10522_ATP_Olambda__321,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod3168588076859124107eger_o(aa_Cod1096751537565085541eger_o(aTP_Lamm_rm(Uu),Uua),Uub) = aa_o_P4192039665577170892eger_o(aa_Cod3473573201699832512eger_o(produc6677183202524767010eger_o,
          $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Uu),Uua,minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),Uub))),Uub = one_one_Code_integer) ).

% ATP.lambda_321
tff(fact_10523_ATP_Olambda__322,axiom,
    ! [Uu: $o,Uua: code_integer,Uub: $o] : aa_o_char(aa_Cod7842134615204088097o_char(aTP_Lamm_te((Uu)),Uua),(Uub)) = produc4188289175737317920o_char(aa_o_f2742674633914528860o_char(aTP_Lamm_td((Uu)),(Uub)),code_bit_cut_integer(Uua)) ).

% ATP.lambda_322
tff(fact_10524_ATP_Olambda__323,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_rc(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_rb(Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_323
tff(fact_10525_ATP_Olambda__324,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_ra(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_qz(Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_324
tff(fact_10526_ATP_Olambda__325,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_qy(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_qx(Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_325
tff(fact_10527_ATP_Olambda__326,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_qw(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_qv(Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_326
tff(fact_10528_ATP_Olambda__327,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_qh(Uu),Uua),Uub)
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_qg(Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_327
tff(fact_10529_ATP_Olambda__328,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_qf(Uu),Uua),Uub)
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_qe(Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_328
tff(fact_10530_ATP_Olambda__329,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_nat_int,Uub: nat] : aa_nat_int(aa_fun8710705519803144921at_int(aTP_Lamm_lr(Uu),Uua),Uub) = groups4538972089207619220nt_int(aa_nat_fun_int_int(aTP_Lamm_lq(Uua),Uub),Uu) ).

% ATP.lambda_329
tff(fact_10531_ATP_Olambda__330,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_nat_real,Uub: nat] : aa_nat_real(aa_fun7039663905532832253t_real(aTP_Lamm_lp(Uu),Uua),Uub) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_lo(Uua),Uub)),Uu) ).

% ATP.lambda_330
tff(fact_10532_ATP_Olambda__331,axiom,
    ! [Uu: set_complex,Uua: fun_co3258106519541737638omplex,Uub: nat] : aa_nat_complex(aa_fun5003373910569599327omplex(aTP_Lamm_ln(Uu),Uua),Uub) = groups7754918857620584856omplex(aa_nat3829984526574172007omplex(aTP_Lamm_lm(Uua),Uub),Uu) ).

% ATP.lambda_331
tff(fact_10533_ATP_Olambda__332,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun8466562714962205125at_nat(aTP_Lamm_ll(Uu),Uua),Uub) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_lk(Uua),Uub)),Uu) ).

% ATP.lambda_332
tff(fact_10534_ATP_Olambda__333,axiom,
    ! [Uu: set_int,Uua: fun_in8056368165505483812omplex,Uub: nat] : aa_nat_complex(aa_fun3543561798013398749omplex(aTP_Lamm_lj(Uu),Uua),Uub) = groups3049146728041665814omplex(aa_nat7430282427850695397omplex(aTP_Lamm_li(Uua),Uub),Uu) ).

% ATP.lambda_333
tff(fact_10535_ATP_Olambda__334,axiom,
    ! [Uu: set_nat,Uua: fun_na1142814060743658952omplex,Uub: nat] : aa_nat_complex(aa_fun4886358607025424513omplex(aTP_Lamm_lh(Uu),Uua),Uub) = aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aTP_Lamm_lg(Uua),Uub)),Uu) ).

% ATP.lambda_334
tff(fact_10536_ATP_Olambda__335,axiom,
    ! [Uu: set_real,Uua: fun_re2084383175013312036omplex,Uub: nat] : aa_nat_complex(aa_fun7632768522510439005omplex(aTP_Lamm_lf(Uu),Uua),Uub) = groups5754745047067104278omplex(aa_nat7791439356240649701omplex(aTP_Lamm_le(Uua),Uub),Uu) ).

% ATP.lambda_335
tff(fact_10537_ATP_Olambda__336,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_nat_real,Uub: nat] : aa_nat_real(aa_fun2842512154077127769t_real(aTP_Lamm_ld(Uu),Uua),Uub) = groups8778361861064173332t_real(aa_nat_fun_int_real(aTP_Lamm_lc(Uua),Uub),Uu) ).

% ATP.lambda_336
tff(fact_10538_ATP_Olambda__337,axiom,
    ! [Uu: set_real,Uua: fun_re4238438383274954274t_real,Uub: nat] : aa_nat_real(aa_fun628238740774635481t_real(aTP_Lamm_lb(Uu),Uua),Uub) = groups8097168146408367636l_real(aa_nat_fun_real_real(aTP_Lamm_la(Uua),Uub),Uu) ).

% ATP.lambda_337
tff(fact_10539_ATP_Olambda__338,axiom,
    ! [Uu: set_complex,Uua: fun_co8378349014452122532t_real,Uub: nat] : aa_nat_real(aa_fun471735231769099483t_real(aTP_Lamm_kz(Uu),Uua),Uub) = groups5808333547571424918x_real(aa_nat480766339200631653x_real(aTP_Lamm_ky(Uua),Uub),Uu) ).

% ATP.lambda_338
tff(fact_10540_ATP_Olambda__339,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_tz(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Uu,Uub),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_339
tff(fact_10541_ATP_Olambda__340,axiom,
    ! [Uu: real,Uua: fun_na6724076748692969250l_real,Uub: nat] : aa_nat_real(aa_fun1113820866546176217t_real(aTP_Lamm_ua(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Uua,Uub),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_340
tff(fact_10542_ATP_Olambda__341,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_tw(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(Uu,Uub)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uub)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_341
tff(fact_10543_ATP_Olambda__342,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_qo(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_nat_real(Uua,Uub)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_342
tff(fact_10544_ATP_Olambda__343,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_sd(Uu),Uua),Uub)
    <=> ( ( size_size_list_nat(Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),one_one_nat) )
        & ( groups4561878855575611511st_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_343
tff(fact_10545_ATP_Olambda__344,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_gr(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Uub)),aa_nat_complex(Uu,Uub))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_nat_nat(suc,zero_zero_nat)))) ).

% ATP.lambda_344
tff(fact_10546_ATP_Olambda__345,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_gt(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Uub)),aa_nat_real(Uu,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_nat_nat(suc,zero_zero_nat)))) ).

% ATP.lambda_345
tff(fact_10547_ATP_Olambda__346,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_se(Uu),Uua),Uub)
    <=> ( ( size_size_list_nat(Uub) = Uu )
        & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4561878855575611511st_nat(Uub)),one_one_nat) = Uua ) ) ) ).

% ATP.lambda_346
tff(fact_10548_ATP_Olambda__347,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_sc(Uu),Uua),Uub)
    <=> ( ( size_size_list_nat(Uub) = Uu )
        & ( groups4561878855575611511st_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_347
tff(fact_10549_ATP_Olambda__348,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_sb(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(aa_nat_nat(suc,Uub)),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uua) ) ) ).

% ATP.lambda_348
tff(fact_10550_ATP_Olambda__349,axiom,
    ! [Uu: nat,Uua: nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(aTP_Lamm_sf(Uu),Uua),Uub)
    <=> ( member_set_nat(Uub,pow_nat(set_or4665077453230672383an_nat(zero_zero_nat,Uu)))
        & ( finite_card_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_349
tff(fact_10551_ATP_Olambda__350,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_sa(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),aa_nat_nat(suc,Uua)) ) ) ).

% ATP.lambda_350
tff(fact_10552_ATP_Olambda__351,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_gq(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(diffs_complex(Uu),Uub)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uua),Uub)) ).

% ATP.lambda_351
tff(fact_10553_ATP_Olambda__352,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_gs(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_nat_real(diffs_real(Uu),Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_352
tff(fact_10554_ATP_Olambda__353,axiom,
    ! [Uu: complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_hc(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(diffs_complex(Uua),Uub)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uu),Uub)) ).

% ATP.lambda_353
tff(fact_10555_ATP_Olambda__354,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_hb(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_nat_real(diffs_real(Uua),Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_354
tff(fact_10556_ATP_Olambda__355,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_mh(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uub)) ).

% ATP.lambda_355
tff(fact_10557_ATP_Olambda__356,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_dp(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uu),Uua)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Uua),Uub) ) ) ).

% ATP.lambda_356
tff(fact_10558_ATP_Olambda__357,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_rw(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uu),Uub)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uub),Uua) ) ) ).

% ATP.lambda_357
tff(fact_10559_ATP_Olambda__358,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_do(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uu),Uub)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Uua),Uub) ) ) ).

% ATP.lambda_358
tff(fact_10560_ATP_Olambda__359,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_rz(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uu),Uub)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Uub),Uua) ) ) ).

% ATP.lambda_359
tff(fact_10561_ATP_Olambda__360,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_ry(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Uu),Uub)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uub),Uua) ) ) ).

% ATP.lambda_360
tff(fact_10562_ATP_Olambda__361,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_rx(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Uu),Uub)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Uub),Uua) ) ) ).

% ATP.lambda_361
tff(fact_10563_ATP_Olambda__362,axiom,
    ! [Uu: set_complex,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_ba(Uu),Uua),Uub)
    <=> ( member_complex(Uub,Uu)
        & aa_complex_o(Uua,Uub) ) ) ).

% ATP.lambda_362
tff(fact_10564_ATP_Olambda__363,axiom,
    ! [Uu: set_real,Uua: fun_real_o,Uub: real] :
      ( aa_real_o(aa_fun5868358341814070780real_o(aTP_Lamm_az(Uu),Uua),Uub)
    <=> ( member_real(Uub,Uu)
        & aa_real_o(Uua,Uub) ) ) ).

% ATP.lambda_363
tff(fact_10565_ATP_Olambda__364,axiom,
    ! [Uu: set_list_nat,Uua: fun_list_nat_o,Uub: list_nat] :
      ( aa_list_nat_o(aa_fun6155608546934989052_nat_o(aTP_Lamm_bb(Uu),Uua),Uub)
    <=> ( member_list_nat(Uub,Uu)
        & aa_list_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_364
tff(fact_10566_ATP_Olambda__365,axiom,
    ! [Uu: set_set_nat,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_bc(Uu),Uua),Uub)
    <=> ( member_set_nat(Uub,Uu)
        & aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_365
tff(fact_10567_ATP_Olambda__366,axiom,
    ! [Uu: set_nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_bd(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_366
tff(fact_10568_ATP_Olambda__367,axiom,
    ! [Uu: set_int,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_be(Uu),Uua),Uub)
    <=> ( member_int(Uub,Uu)
        & aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_367
tff(fact_10569_ATP_Olambda__368,axiom,
    ! [Uu: set_complex,Uua: set_complex,Uub: complex] :
      ( aa_complex_o(aa_set1742385355640705825plex_o(aTP_Lamm_bv(Uu),Uua),Uub)
    <=> ( member_complex(Uub,Uu)
        & ~ member_complex(Uub,Uua) ) ) ).

% ATP.lambda_368
tff(fact_10570_ATP_Olambda__369,axiom,
    ! [Uu: set_real,Uua: set_real,Uub: real] :
      ( aa_real_o(aa_set2323979645938365349real_o(aTP_Lamm_bu(Uu),Uua),Uub)
    <=> ( member_real(Uub,Uu)
        & ~ member_real(Uub,Uua) ) ) ).

% ATP.lambda_369
tff(fact_10571_ATP_Olambda__370,axiom,
    ! [Uu: set_list_nat,Uua: set_list_nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_set6675653781276780861_nat_o(aTP_Lamm_bw(Uu),Uua),Uub)
    <=> ( member_list_nat(Uub,Uu)
        & ~ member_list_nat(Uub,Uua) ) ) ).

% ATP.lambda_370
tff(fact_10572_ATP_Olambda__371,axiom,
    ! [Uu: set_set_nat,Uua: set_set_nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_set4404905041378927473_nat_o(aTP_Lamm_bx(Uu),Uua),Uub)
    <=> ( member_set_nat(Uub,Uu)
        & ~ member_set_nat(Uub,Uua) ) ) ).

% ATP.lambda_371
tff(fact_10573_ATP_Olambda__372,axiom,
    ! [Uu: set_nat,Uua: set_nat,Uub: nat] :
      ( aa_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_by(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ~ aa_set_nat_o(member_nat(Uub),Uua) ) ) ).

% ATP.lambda_372
tff(fact_10574_ATP_Olambda__373,axiom,
    ! [Uu: set_int,Uua: set_int,Uub: int] :
      ( aa_int_o(aa_set_int_fun_int_o(aTP_Lamm_bz(Uu),Uua),Uub)
    <=> ( member_int(Uub,Uu)
        & ~ member_int(Uub,Uua) ) ) ).

% ATP.lambda_373
tff(fact_10575_ATP_Olambda__374,axiom,
    ! [Uu: real,Uua: complex,Uub: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(aTP_Lamm_vu(Uu),Uua),Uub)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V3694042436643373181omplex(Uua),Uub)),Uu) ) ).

% ATP.lambda_374
tff(fact_10576_ATP_Olambda__375,axiom,
    ! [Uu: real,Uua: real,Uub: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_vw(Uu),Uua),Uub)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V975177566351809787t_real(Uua),Uub)),Uu) ) ).

% ATP.lambda_375
tff(fact_10577_ATP_Olambda__376,axiom,
    ! [Uu: set_Pr448751882837621926eger_o,Uua: code_integer,Uub: $o] :
      ( aa_o_o(aa_Cod6379977994291289310un_o_o(aTP_Lamm_dq(Uu),Uua),(Uub))
    <=> member1379723562493234055eger_o(aa_o_P4192039665577170892eger_o(aa_Cod3473573201699832512eger_o(produc6677183202524767010eger_o,Uua),(Uub)),Uu) ) ).

% ATP.lambda_376
tff(fact_10578_ATP_Olambda__377,axiom,
    ! [Uu: set_Pr8218934625190621173um_num,Uua: num,Uub: num] :
      ( aa_num_o(aa_num_fun_num_o(aTP_Lamm_dr(Uu),Uua),Uub)
    <=> member7279096912039735102um_num(aa_num3156681643174609981um_num(aa_num8681986212680877308um_num(product_Pair_num_num,Uua),Uub),Uu) ) ).

% ATP.lambda_377
tff(fact_10579_ATP_Olambda__378,axiom,
    ! [Uu: set_Pr6200539531224447659at_num,Uua: nat,Uub: num] :
      ( aa_num_o(aa_nat_fun_num_o(aTP_Lamm_ds(Uu),Uua),Uub)
    <=> member9148766508732265716at_num(aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,Uua),Uub),Uu) ) ).

% ATP.lambda_378
tff(fact_10580_ATP_Olambda__379,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_dt(Uu),Uua),Uub)
    <=> member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uua),Uub),Uu) ) ).

% ATP.lambda_379
tff(fact_10581_ATP_Olambda__380,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_du(Uu),Uua),Uub)
    <=> member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uub),Uu) ) ).

% ATP.lambda_380
tff(fact_10582_ATP_Olambda__381,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(aTP_Lamm_qj(Uu),Uua),Uub)
    <=> ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uub),Uu) = Uua ) ) ).

% ATP.lambda_381
tff(fact_10583_ATP_Olambda__382,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex] :
      ( aa_complex_o(aa_nat_fun_complex_o(aTP_Lamm_rd(Uu),Uua),Uub)
    <=> ( aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uub),Uua) = Uu ) ) ).

% ATP.lambda_382
tff(fact_10584_ATP_Olambda__383,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_fs(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,aa_nat_nat(suc,Uub))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uua),Uub)) ).

% ATP.lambda_383
tff(fact_10585_ATP_Olambda__384,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ft(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_nat_real(Uu,aa_nat_nat(suc,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_384
tff(fact_10586_ATP_Olambda__385,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_int,Uub: nat] : aa_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_uw(Uu),Uua),Uub) = aa_real_real(minus_minus_real(aa_nat_real(Uu,Uub)),aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,aa_nat_int(Uua,Uub))),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) ).

% ATP.lambda_385
tff(fact_10587_ATP_Olambda__386,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_tx(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_nat_real(Uu,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_386
tff(fact_10588_ATP_Olambda__387,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_fp(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,Uub)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uua),Uub)) ).

% ATP.lambda_387
tff(fact_10589_ATP_Olambda__388,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_fn(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_nat_real(Uu,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_388
tff(fact_10590_ATP_Olambda__389,axiom,
    ! [Uu: fun_nat_rat,Uua: rat,Uub: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aTP_Lamm_nd(Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_nat_rat(Uu,Uub)),aa_nat_rat(power_power_rat(Uua),Uub)) ).

% ATP.lambda_389
tff(fact_10591_ATP_Olambda__390,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ns(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uub)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uua),Uub)) ).

% ATP.lambda_390
tff(fact_10592_ATP_Olambda__391,axiom,
    ! [Uu: fun_nat_int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_ng(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,Uub)),aa_nat_int(aa_int_fun_nat_int(power_power_int,Uua),Uub)) ).

% ATP.lambda_391
tff(fact_10593_ATP_Olambda__392,axiom,
    ! [Uu: fun_nat_o,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_rv(Uu),Uua),Uub)
    <=> ( aa_nat_o(Uu,Uub)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uua) ) ) ).

% ATP.lambda_392
tff(fact_10594_ATP_Olambda__393,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vo(Uu),Uua),Uub) = aa_real_real(divide_divide_real(aa_real_real(Uu,Uub)),aa_real_real(Uua,Uub)) ).

% ATP.lambda_393
tff(fact_10595_ATP_Olambda__394,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vh(Uu),Uua),Uub) = aa_real_real(divide_divide_real(aa_real_real(Uua,Uub)),aa_real_real(Uu,Uub)) ).

% ATP.lambda_394
tff(fact_10596_ATP_Olambda__395,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_rg(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uub)),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_395
tff(fact_10597_ATP_Olambda__396,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_fg(Uu),Uua),Uub) = aa_complex_complex(minus_minus_complex(aa_nat_complex(Uu,Uub)),aa_nat_complex(Uua,Uub)) ).

% ATP.lambda_396
tff(fact_10598_ATP_Olambda__397,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_fh(Uu),Uua),Uub) = aa_real_real(minus_minus_real(aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_397
tff(fact_10599_ATP_Olambda__398,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_fc(Uu),Uua),Uub) = aa_complex_complex(plus_plus_complex(aa_nat_complex(Uu,Uub)),aa_nat_complex(Uua,Uub)) ).

% ATP.lambda_398
tff(fact_10600_ATP_Olambda__399,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_fd(Uu),Uua),Uub) = aa_real_real(plus_plus_real(aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_399
tff(fact_10601_ATP_Olambda__400,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_fe(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Uu,Uub)),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_400
tff(fact_10602_ATP_Olambda__401,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_ff(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(Uu,Uub)),aa_nat_int(Uua,Uub)) ).

% ATP.lambda_401
tff(fact_10603_ATP_Olambda__402,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_tv(Uu),Uua),Uub) = aa_real_real(powr_real(aa_real_real(Uu,Uub)),aa_real_real(Uua,Uub)) ).

% ATP.lambda_402
tff(fact_10604_ATP_Olambda__403,axiom,
    ! [Uu: fun_complex_real,Uua: complex,Uub: complex] : aa_complex_complex(aa_com3342990543769683525omplex(aTP_Lamm_kr(Uu),Uua),Uub) = aa_complex_complex(real_V2046097035970521341omplex(aa_complex_real(Uu,Uub)),Uua) ).

% ATP.lambda_403
tff(fact_10605_ATP_Olambda__404,axiom,
    ! [Uu: fun_nat_real,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_ir(Uu),Uua),Uub) = aa_complex_complex(real_V2046097035970521341omplex(aa_nat_real(Uu,Uub)),Uua) ).

% ATP.lambda_404
tff(fact_10606_ATP_Olambda__405,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_iq(Uu),Uua),Uub) = aa_real_real(real_V1485227260804924795R_real(aa_nat_real(Uu,Uub)),Uua) ).

% ATP.lambda_405
tff(fact_10607_ATP_Olambda__406,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ru(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(Uu,Uub)),Uua) ) ).

% ATP.lambda_406
tff(fact_10608_ATP_Olambda__407,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ks(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(Uu,Uub)),Uua) ).

% ATP.lambda_407
tff(fact_10609_ATP_Olambda__408,axiom,
    ! [Uu: fun_int_int,Uua: int,Uub: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_kt(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(Uu,Uub)),Uua) ).

% ATP.lambda_408
tff(fact_10610_ATP_Olambda__409,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_es(Uu),Uua),Uub) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(Uu,Uub)),Uua) ).

% ATP.lambda_409
tff(fact_10611_ATP_Olambda__410,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_et(Uu),Uua),Uub) = aa_real_real(divide_divide_real(aa_nat_real(Uu,Uub)),Uua) ).

% ATP.lambda_410
tff(fact_10612_ATP_Olambda__411,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_fa(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,Uub)),Uua) ).

% ATP.lambda_411
tff(fact_10613_ATP_Olambda__412,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_fb(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_nat_real(Uu,Uub)),Uua) ).

% ATP.lambda_412
tff(fact_10614_ATP_Olambda__413,axiom,
    ! [Uu: complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_gb(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(Uua,Uub)),Uu) ).

% ATP.lambda_413
tff(fact_10615_ATP_Olambda__414,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_gc(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_nat_real(Uua,Uub)),Uu) ).

% ATP.lambda_414
tff(fact_10616_ATP_Olambda__415,axiom,
    ! [Uu: fun_real_real,Uua: nat,Uub: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_ts(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(Uu,Uub)),Uua) ).

% ATP.lambda_415
tff(fact_10617_ATP_Olambda__416,axiom,
    ! [Uu: nat,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_vb(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(Uua,Uub)),Uu) ).

% ATP.lambda_416
tff(fact_10618_ATP_Olambda__417,axiom,
    ! [Uu: fun_real_real,Uua: real,Uub: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_tu(Uu),Uua),Uub) = aa_real_real(powr_real(aa_real_real(Uu,Uub)),Uua) ).

% ATP.lambda_417
tff(fact_10619_ATP_Olambda__418,axiom,
    ! [Uu: real,Uua: fun_nat_complex,Uub: nat] : aa_nat_real(aa_fun7054779078796324018t_real(aTP_Lamm_fw(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(Uua,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_418
tff(fact_10620_ATP_Olambda__419,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_uq(Uu),Uua),Uub) = aa_real_real(divide_divide_real(Uua),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_419
tff(fact_10621_ATP_Olambda__420,axiom,
    ! [Uu: complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_nz(Uu),Uua),Uub) = aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_420
tff(fact_10622_ATP_Olambda__421,axiom,
    ! [Uu: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_oc(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_421
tff(fact_10623_ATP_Olambda__422,axiom,
    ! [Uu: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_oa(Uu),Uua),Uub) = aa_nat_rat(power_power_rat(Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_422
tff(fact_10624_ATP_Olambda__423,axiom,
    ! [Uu: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_ob(Uu),Uua),Uub) = aa_nat_int(aa_int_fun_nat_int(power_power_int,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_423
tff(fact_10625_ATP_Olambda__424,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_qu(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua)) ).

% ATP.lambda_424
tff(fact_10626_ATP_Olambda__425,axiom,
    ! [Uu: real,Uua: real,Uub: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_vm(Uu),Uua),Uub)
    <=> member_real(Uub,set_or1633881224788618240n_real(Uu,Uua)) ) ).

% ATP.lambda_425
tff(fact_10627_ATP_Olambda__426,axiom,
    ! [Uu: real,Uua: fun_complex_complex,Uub: complex] : aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_kv(Uu),Uua),Uub) = aa_complex_complex(real_V2046097035970521341omplex(Uu),aa_complex_complex(Uua,Uub)) ).

% ATP.lambda_426
tff(fact_10628_ATP_Olambda__427,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ku(Uu),Uua),Uub) = aa_real_real(real_V1485227260804924795R_real(Uu),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_427
tff(fact_10629_ATP_Olambda__428,axiom,
    ! [Uu: fun_nat_complex,Uua: real,Uub: nat] : aa_nat_complex(aa_rea1531280491672885477omplex(aTP_Lamm_ip(Uu),Uua),Uub) = aa_complex_complex(real_V2046097035970521341omplex(Uua),aa_nat_complex(Uu,Uub)) ).

% ATP.lambda_428
tff(fact_10630_ATP_Olambda__429,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_io(Uu),Uua),Uub) = aa_real_real(real_V1485227260804924795R_real(Uua),aa_nat_real(Uu,Uub)) ).

% ATP.lambda_429
tff(fact_10631_ATP_Olambda__430,axiom,
    ! [Uu: complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_eq(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(Uu),aa_nat_complex(Uua,Uub)) ).

% ATP.lambda_430
tff(fact_10632_ATP_Olambda__431,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_er(Uu),Uua),Uub) = aa_real_real(times_times_real(Uu),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_431
tff(fact_10633_ATP_Olambda__432,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_ey(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(Uua),aa_nat_complex(Uu,Uub)) ).

% ATP.lambda_432
tff(fact_10634_ATP_Olambda__433,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ez(Uu),Uua),Uub) = aa_real_real(times_times_real(Uua),aa_nat_real(Uu,Uub)) ).

% ATP.lambda_433
tff(fact_10635_ATP_Olambda__434,axiom,
    ! [Uu: fun_real_o,Uua: real,Uub: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_vk(Uu),Uua),Uub)
    <=> aa_real_o(Uu,aa_real_real(plus_plus_real(Uub),Uua)) ) ).

% ATP.lambda_434
tff(fact_10636_ATP_Olambda__435,axiom,
    ! [Uu: fun_nat_complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_ep(Uu),Uua),Uub) = aa_nat_complex(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_435
tff(fact_10637_ATP_Olambda__436,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_eo(Uu),Uua),Uub) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_436
tff(fact_10638_ATP_Olambda__437,axiom,
    ! [Uu: fun_nat_o,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_vj(Uu),Uua),Uub)
    <=> aa_nat_o(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ) ).

% ATP.lambda_437
tff(fact_10639_ATP_Olambda__438,axiom,
    ! [Uu: fun_Pr3290071492984375169er_o_o,Uua: code_integer,Uub: $o] :
      ( aa_o_o(aa_Cod6379977994291289310un_o_o(aTP_Lamm_jo(Uu),Uua),(Uub))
    <=> aa_Pro5286387174248033258er_o_o(Uu,aa_o_P4192039665577170892eger_o(aa_Cod3473573201699832512eger_o(produc6677183202524767010eger_o,Uua),(Uub))) ) ).

% ATP.lambda_438
tff(fact_10640_ATP_Olambda__439,axiom,
    ! [Uu: fun_Pr6331313853612757382_num_o,Uua: num,Uub: num] :
      ( aa_num_o(aa_num_fun_num_o(aTP_Lamm_jp(Uu),Uua),Uub)
    <=> aa_Pro7685007914727650815_num_o(Uu,aa_num3156681643174609981um_num(aa_num8681986212680877308um_num(product_Pair_num_num,Uua),Uub)) ) ).

% ATP.lambda_439
tff(fact_10641_ATP_Olambda__440,axiom,
    ! [Uu: fun_Pr5555124106300582480_num_o,Uua: nat,Uub: num] :
      ( aa_num_o(aa_nat_fun_num_o(aTP_Lamm_jq(Uu),Uua),Uub)
    <=> aa_Pro6908818167415475913_num_o(Uu,aa_num5026351239867140595at_num(aa_nat1233894041389027964at_num(product_Pair_nat_num,Uua),Uub)) ) ).

% ATP.lambda_440
tff(fact_10642_ATP_Olambda__441,axiom,
    ! [Uu: fun_Pr4796943543331071514_nat_o,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_jr(Uu),Uua),Uub)
    <=> aa_Pro6150637604445964947_nat_o(Uu,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uua),Uub)) ) ).

% ATP.lambda_441
tff(fact_10643_ATP_Olambda__442,axiom,
    ! [Uu: fun_Pr2477239776880934807at_nat,Uua: nat,Uub: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_df(Uu),Uua),Uub) = aa_Pro8023792847356852296at_nat(Uu,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uua),Uub)) ).

% ATP.lambda_442
tff(fact_10644_ATP_Olambda__443,axiom,
    ! [Uu: fun_Pr2028404973800837934_nat_o,Uua: nat,Uub: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aTP_Lamm_dg(Uu),Uua),Uub) = aa_Pro989595756709598047_nat_o(Uu,aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,Uua),Uub)) ).

% ATP.lambda_443
tff(fact_10645_ATP_Olambda__444,axiom,
    ! [Uu: fun_Pr2478283820680522243nt_int,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_dh(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uub)) ).

% ATP.lambda_444
tff(fact_10646_ATP_Olambda__445,axiom,
    ! [Uu: fun_Pr3238374608701176162_int_o,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_di(Uu),Uua),Uub)
    <=> aa_Pro4592068669816069595_int_o(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uub)) ) ).

% ATP.lambda_445
tff(fact_10647_ATP_Olambda__446,axiom,
    ! [Uu: fun_Pr982841511892225912nt_int,Uua: int,Uub: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_dj(Uu),Uua),Uub) = aa_Pro6799258434465214121nt_int(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uub)) ).

% ATP.lambda_446
tff(fact_10648_ATP_Olambda__447,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_gf(Uu),Uua),Uub) = aa_nat_complex(Uua,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uu)) ).

% ATP.lambda_447
tff(fact_10649_ATP_Olambda__448,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_gg(Uu),Uua),Uub) = aa_nat_real(Uua,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uu)) ).

% ATP.lambda_448
tff(fact_10650_ATP_Olambda__449,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_nat,Uub: nat] : aa_nat_real(aa_fun1538629612289296340t_real(aTP_Lamm_wc(Uu),Uua),Uub) = aa_nat_real(Uu,aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_449
tff(fact_10651_ATP_Olambda__450,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_real(aa_com20334259897932645t_real(aTP_Lamm_fq(Uu),Uua),Uub) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(aa_nat_complex(Uu,Uub)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uua),Uub))) ).

% ATP.lambda_450
tff(fact_10652_ATP_Olambda__451,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_fo(Uu),Uua),Uub) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(times_times_real(aa_nat_real(Uu,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub))) ).

% ATP.lambda_451
tff(fact_10653_ATP_Olambda__452,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_uv(Uu),Uua),Uub) = aa_real_real(cos_real,aa_real_real(minus_minus_real(aa_nat_real(Uu,Uub)),Uua)) ).

% ATP.lambda_452
tff(fact_10654_ATP_Olambda__453,axiom,
    ! [Uu: fun_na1668100129766719005at_nat,Uua: product_prod_nat_nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_Pro6895555690143131534at_nat(aTP_Lamm_ie(Uu),Uua),Uub),Uuc) = aa_Pro262675715343573244at_nat(aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(Uu,Uub),Uuc),Uua) ).

% ATP.lambda_453
tff(fact_10655_ATP_Olambda__454,axiom,
    ! [Uu: fun_na5914473681949881652_nat_o,Uua: product_prod_nat_nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_Pro1957341097241355813_nat_o(aTP_Lamm_if(Uu),Uua),Uub),Uuc)
    <=> aa_Pro6150637604445964947_nat_o(aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(Uu,Uub),Uuc),Uua) ) ).

% ATP.lambda_454
tff(fact_10656_ATP_Olambda__455,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] :
      aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_kc(Uu),Uua),Uub),Uuc) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uub)
        & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uuc) ),
        aa_complex_complex(times_times_complex(aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(divide_divide_real(aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))))),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uu),Uuc))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),
        zero_zero_complex ) ).

% ATP.lambda_455
tff(fact_10657_ATP_Olambda__456,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] :
      aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ka(Uu),Uua),Uub),Uuc) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uub)
        & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uuc) ),
        aa_real_real(times_times_real(aa_real_real(real_V1485227260804924795R_real(aa_real_real(divide_divide_real(aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))))),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),
        zero_zero_real ) ).

% ATP.lambda_456
tff(fact_10658_ATP_Olambda__457,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] :
      aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_ju(Uu),Uua),Uub),Uuc) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uub)
        & ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uuc) ),
        aa_complex_complex(times_times_complex(aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))))),aa_nat_real(semiri2265585572941072030t_real,Uub)))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uu),Uuc))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),
        zero_zero_complex ) ).

% ATP.lambda_457
tff(fact_10659_ATP_Olambda__458,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] :
      aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_js(Uu),Uua),Uub),Uuc) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uub)
        & ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uuc) ),
        aa_real_real(times_times_real(aa_real_real(real_V1485227260804924795R_real(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))))),aa_nat_real(semiri2265585572941072030t_real,Uub)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),
        zero_zero_real ) ).

% ATP.lambda_458
tff(fact_10660_ATP_Olambda__459,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] :
      aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_jy(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uub),aa_complex_complex(times_times_complex(aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(divide_divide_real(aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))))),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uu),Uuc))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),zero_zero_complex) ).

% ATP.lambda_459
tff(fact_10661_ATP_Olambda__460,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] :
      aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_jw(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uub),aa_real_real(times_times_real(aa_real_real(real_V1485227260804924795R_real(aa_real_real(divide_divide_real(aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(aa_int_fun_nat_int(power_power_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))))),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),zero_zero_real) ).

% ATP.lambda_460
tff(fact_10662_ATP_Olambda__461,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] :
      aa_nat_complex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_od(Uu),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),
        aa_nat_complex(Uua,Uuc),
        $ite(Uuc = Uu,zero_zero_complex,aa_nat_complex(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_461
tff(fact_10663_ATP_Olambda__462,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_ol(Uu),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),
        aa_nat_real(Uua,Uuc),
        $ite(Uuc = Uu,zero_zero_real,aa_nat_real(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_462
tff(fact_10664_ATP_Olambda__463,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_of(Uu),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),
        aa_nat_rat(Uua,Uuc),
        $ite(Uuc = Uu,zero_zero_rat,aa_nat_rat(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_463
tff(fact_10665_ATP_Olambda__464,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_oj(Uu),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),
        aa_nat_nat(Uua,Uuc),
        $ite(Uuc = Uu,zero_zero_nat,aa_nat_nat(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_464
tff(fact_10666_ATP_Olambda__465,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_oh(Uu),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),
        aa_nat_int(Uua,Uuc),
        $ite(Uuc = Uu,zero_zero_int,aa_nat_int(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_465
tff(fact_10667_ATP_Olambda__466,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] :
      aa_nat_complex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_oe(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),aa_nat_complex(Uua,Uuc),aa_nat_complex(Uub,Uuc)) ).

% ATP.lambda_466
tff(fact_10668_ATP_Olambda__467,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_om(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),aa_nat_real(Uua,Uuc),aa_nat_real(Uub,Uuc)) ).

% ATP.lambda_467
tff(fact_10669_ATP_Olambda__468,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_og(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),aa_nat_rat(Uua,Uuc),aa_nat_rat(Uub,Uuc)) ).

% ATP.lambda_468
tff(fact_10670_ATP_Olambda__469,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_ok(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),aa_nat_nat(Uua,Uuc),aa_nat_nat(Uub,Uuc)) ).

% ATP.lambda_469
tff(fact_10671_ATP_Olambda__470,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_oi(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),aa_nat_int(Uua,Uuc),aa_nat_int(Uub,Uuc)) ).

% ATP.lambda_470
tff(fact_10672_ATP_Olambda__471,axiom,
    ! [Uu: $o,Uua: $o,Uub: code_integer,Uuc: $o] : aa_o_char(aa_Cod7842134615204088097o_char(aa_o_f2742674633914528860o_char(aTP_Lamm_td((Uu)),(Uua)),Uub),(Uuc)) = produc4188289175737317920o_char(aa_o_f2742674633914528860o_char(aa_o_f9003794812476387647o_char(aTP_Lamm_tc((Uu)),(Uua)),(Uuc)),code_bit_cut_integer(Uub)) ).

% ATP.lambda_471
tff(fact_10673_ATP_Olambda__472,axiom,
    ! [Uu: nat,Uua: code_integer,Uub: code_integer,Uuc: nat] :
      aa_nat_Code_integer(aa_Cod1578151519121547077nteger(aa_Cod6397266357770349983nteger(aTP_Lamm_oy(Uu),Uua),Uub),Uuc) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,
          $ite(
            Uuc = zero_zero_nat,
            aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uub),
            $ite(Uuc = Uu,one_one_Code_integer,zero_z3403309356797280102nteger) )),aa_nat_Code_integer(power_8256067586552552935nteger(Uua),Uuc)) ).

% ATP.lambda_472
tff(fact_10674_ATP_Olambda__473,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex,Uuc: nat] :
      aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aTP_Lamm_ow(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex($ite(
            Uuc = zero_zero_nat,
            aa_complex_complex(uminus1482373934393186551omplex,Uub),
            $ite(Uuc = Uu,one_one_complex,zero_zero_complex) )),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uua),Uuc)) ).

% ATP.lambda_473
tff(fact_10675_ATP_Olambda__474,axiom,
    ! [Uu: nat,Uua: real,Uub: real,Uuc: nat] :
      aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_oz(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real($ite(
            Uuc = zero_zero_nat,
            aa_real_real(uminus_uminus_real,Uub),
            $ite(Uuc = Uu,one_one_real,zero_zero_real) )),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uuc)) ).

% ATP.lambda_474
tff(fact_10676_ATP_Olambda__475,axiom,
    ! [Uu: nat,Uua: rat,Uub: rat,Uuc: nat] :
      aa_nat_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aTP_Lamm_ox(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat($ite(
            Uuc = zero_zero_nat,
            aa_rat_rat(uminus_uminus_rat,Uub),
            $ite(Uuc = Uu,one_one_rat,zero_zero_rat) )),aa_nat_rat(power_power_rat(Uua),Uuc)) ).

% ATP.lambda_475
tff(fact_10677_ATP_Olambda__476,axiom,
    ! [Uu: nat,Uua: int,Uub: int,Uuc: nat] :
      aa_nat_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aTP_Lamm_ov(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,
          $ite(
            Uuc = zero_zero_nat,
            aa_int_int(uminus_uminus_int,Uub),
            $ite(Uuc = Uu,one_one_int,zero_zero_int) )),aa_nat_int(aa_int_fun_nat_int(power_power_int,Uua),Uuc)) ).

% ATP.lambda_476
tff(fact_10678_ATP_Olambda__477,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_nu(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_nt(Uu),Uua),Uuc)),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uub),Uuc)) ).

% ATP.lambda_477
tff(fact_10679_ATP_Olambda__478,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_int,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_fun2315716387381795033at_int(aTP_Lamm_ni(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_fun6128006242533108349at_int(aTP_Lamm_nh(Uu),Uua),Uuc),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Uub),Uuc)) ).

% ATP.lambda_478
tff(fact_10680_ATP_Olambda__479,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: rat,Uuc: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_fun5559679822142647773at_rat(aTP_Lamm_nf(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_fun2663889555218712533at_rat(aTP_Lamm_ne(Uu),Uua),Uuc),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_rat(power_power_rat(Uub),Uuc)) ).

% ATP.lambda_479
tff(fact_10681_ATP_Olambda__480,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_fun4732485501138722137t_real(aTP_Lamm_nj(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_fun3023522663955195517t_real(aTP_Lamm_mw(Uu),Uua),Uuc)),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uuc)) ).

% ATP.lambda_480
tff(fact_10682_ATP_Olambda__481,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_fun3721018923249493343omplex(aTP_Lamm_nc(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_fun5379041630539201665omplex(aTP_Lamm_mu(Uu),Uua),Uuc)),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uub),Uuc)) ).

% ATP.lambda_481
tff(fact_10683_ATP_Olambda__482,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_uf(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,Uua)),Uuc)),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uuc)) ).

% ATP.lambda_482
tff(fact_10684_ATP_Olambda__483,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_ud(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uuc)),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uuc)) ).

% ATP.lambda_483
tff(fact_10685_ATP_Olambda__484,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_ub(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Uu,Uuc),Uua)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(Uub),Uua)),Uuc)) ).

% ATP.lambda_484
tff(fact_10686_ATP_Olambda__485,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_uc(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Uu,Uuc),Uub)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(Uua),Uub)),Uuc)) ).

% ATP.lambda_485
tff(fact_10687_ATP_Olambda__486,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_si(Uu),Uua),Uub),Uuc) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu),Uub)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uub))) ).

% ATP.lambda_486
tff(fact_10688_ATP_Olambda__487,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ny(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Uu),Uuc))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_487
tff(fact_10689_ATP_Olambda__488,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_nw(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_rat(comm_s4028243227959126397er_rat(Uu),Uuc))),aa_nat_rat(comm_s4028243227959126397er_rat(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_488
tff(fact_10690_ATP_Olambda__489,axiom,
    ! [Uu: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_nx(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,Uu),Uuc))),aa_nat_int(aa_int_fun_nat_int(comm_s4660882817536571857er_int,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_489
tff(fact_10691_ATP_Olambda__490,axiom,
    ! [Uu: extended_enat,Uua: extended_enat,Uub: nat,Uuc: nat] : aa_nat_Extended_enat(aa_nat2025370463774033377d_enat(aa_Ext8984936454148280526d_enat(aTP_Lamm_nq(Uu),Uua),Uub),Uuc) = aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_Ext7969882492699152700d_enat(times_7803423173614009249d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_Extended_enat(power_8040749407984259932d_enat(Uu),Uuc))),aa_nat_Extended_enat(power_8040749407984259932d_enat(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_490
tff(fact_10692_ATP_Olambda__491,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_nn(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uu),Uuc))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_491
tff(fact_10693_ATP_Olambda__492,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_nr(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_492
tff(fact_10694_ATP_Olambda__493,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_no(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_rat(power_power_rat(Uu),Uuc))),aa_nat_rat(power_power_rat(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_493
tff(fact_10695_ATP_Olambda__494,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_mt(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uu),Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_494
tff(fact_10696_ATP_Olambda__495,axiom,
    ! [Uu: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_np(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Uu),Uuc))),aa_nat_int(aa_int_fun_nat_int(power_power_int,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_495
tff(fact_10697_ATP_Olambda__496,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_ou(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uu),Uuc))),aa_complex_complex(real_V2046097035970521341omplex(aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_496
tff(fact_10698_ATP_Olambda__497,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_ot(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_real_real(real_V1485227260804924795R_real(aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uuc))),aa_real_real(real_V1485227260804924795R_real(aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_497
tff(fact_10699_ATP_Olambda__498,axiom,
    ! [Uu: $o,Uua: real,Uub: real,Uuc: real] :
      ( aa_real_o(aa_real_fun_real_o(aa_rea1652550732481586312real_o(aTP_Lamm_ia((Uu)),Uua),Uub),Uuc)
    <=> ( ( (Uu)
         => ( Uuc = Uua ) )
        & ( ~ (Uu)
         => ( Uuc = Uub ) ) ) ) ).

% ATP.lambda_498
tff(fact_10700_ATP_Olambda__499,axiom,
    ! [Uu: $o,Uua: int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_ib((Uu)),Uua),Uub),Uuc)
    <=> ( ( (Uu)
         => ( Uuc = Uua ) )
        & ( ~ (Uu)
         => ( Uuc = Uub ) ) ) ) ).

% ATP.lambda_499
tff(fact_10701_ATP_Olambda__500,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_mi(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_500
tff(fact_10702_ATP_Olambda__501,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_qg(Uu),Uua),Uub),Uuc)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uub)) ) ).

% ATP.lambda_501
tff(fact_10703_ATP_Olambda__502,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_sn(Uu),Uua),Uub),Uuc)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ) ).

% ATP.lambda_502
tff(fact_10704_ATP_Olambda__503,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_qe(Uu),Uua),Uub),Uuc)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uub)) ) ).

% ATP.lambda_503
tff(fact_10705_ATP_Olambda__504,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_sl(Uu),Uua),Uub),Uuc)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ) ).

% ATP.lambda_504
tff(fact_10706_ATP_Olambda__505,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_sp(Uu),Uua),Uub),Uuc) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uub)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uuc)) ).

% ATP.lambda_505
tff(fact_10707_ATP_Olambda__506,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_sr(Uu),Uua),Uub),Uuc) = aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(product_Pair_nat_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_506
tff(fact_10708_ATP_Olambda__507,axiom,
    ! [Uu: set_complex,Uua: complex,Uub: complex,Uuc: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(aa_com7221863818706917250plex_o(aTP_Lamm_vz(Uu),Uua),Uub),Uuc)
    <=> ( ( Uub = Uua )
       => member_complex(Uuc,Uu) ) ) ).

% ATP.lambda_507
tff(fact_10709_ATP_Olambda__508,axiom,
    ! [Uu: set_real,Uua: real,Uub: real,Uuc: real] :
      ( aa_real_o(aa_real_fun_real_o(aa_rea1652550732481586312real_o(aTP_Lamm_vy(Uu),Uua),Uub),Uuc)
    <=> ( ( Uub = Uua )
       => member_real(Uuc,Uu) ) ) ).

% ATP.lambda_508
tff(fact_10710_ATP_Olambda__509,axiom,
    ! [Uu: code_integer,Uua: $o,Uub: code_integer,Uuc: $o] :
      ( aa_o_o(aa_Cod6379977994291289310un_o_o(aa_o_f2459543391104474489un_o_o(aTP_Lamm_hv(Uu),(Uua)),Uub),(Uuc))
    <=> ( ( Uu = Uub )
        & ( (Uua)
        <=> (Uuc) ) ) ) ).

% ATP.lambda_509
tff(fact_10711_ATP_Olambda__510,axiom,
    ! [Uu: num,Uua: num,Uub: num,Uuc: num] :
      ( aa_num_o(aa_num_fun_num_o(aa_num5226579579167915582_num_o(aTP_Lamm_hw(Uu),Uua),Uub),Uuc)
    <=> ( ( Uu = Uub )
        & ( Uua = Uuc ) ) ) ).

% ATP.lambda_510
tff(fact_10712_ATP_Olambda__511,axiom,
    ! [Uu: nat,Uua: num,Uub: nat,Uuc: num] :
      ( aa_num_o(aa_nat_fun_num_o(aa_num3446304307616776968_num_o(aTP_Lamm_hx(Uu),Uua),Uub),Uuc)
    <=> ( ( Uu = Uub )
        & ( Uua = Uuc ) ) ) ).

% ATP.lambda_511
tff(fact_10713_ATP_Olambda__512,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_hy(Uu),Uua),Uub),Uuc)
    <=> ( ( Uu = Uub )
        & ( Uua = Uuc ) ) ) ).

% ATP.lambda_512
tff(fact_10714_ATP_Olambda__513,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_hz(Uu),Uua),Uub),Uuc)
    <=> ( ( Uu = Uub )
        & ( Uua = Uuc ) ) ) ).

% ATP.lambda_513
tff(fact_10715_ATP_Olambda__514,axiom,
    ! [Uu: fun_nat_complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_fu(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uua))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uub),Uuc)) ).

% ATP.lambda_514
tff(fact_10716_ATP_Olambda__515,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_fv(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uua))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uuc)) ).

% ATP.lambda_515
tff(fact_10717_ATP_Olambda__516,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_fun5379041630539201665omplex(aTP_Lamm_mu(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,Uuc)),aa_nat_complex(Uua,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_516
tff(fact_10718_ATP_Olambda__517,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_fun3023522663955195517t_real(aTP_Lamm_mw(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_nat_real(Uu,Uuc)),aa_nat_real(Uua,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_517
tff(fact_10719_ATP_Olambda__518,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_fun2663889555218712533at_rat(aTP_Lamm_ne(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_nat_rat(Uu,Uuc)),aa_nat_rat(Uua,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_518
tff(fact_10720_ATP_Olambda__519,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_nt(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uuc)),aa_nat_nat(Uua,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_519
tff(fact_10721_ATP_Olambda__520,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_fun6128006242533108349at_int(aTP_Lamm_nh(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,Uuc)),aa_nat_int(Uua,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_520
tff(fact_10722_ATP_Olambda__521,axiom,
    ! [Uu: $o,Uua: fun_int_fun_int_o,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_dl((Uu)),Uua),Uub),Uuc)
    <=> ( (Uu)
        & aa_int_o(aa_int_fun_int_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_521
tff(fact_10723_ATP_Olambda__522,axiom,
    ! [Uu: fun_Pr2478283820680522243nt_int,Uua: fun_in1696471567231710611nt_int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_fun4152872853588681340nt_int(aTP_Lamm_dd(Uu),Uua),Uub),Uuc) = aa_Pro6970504127699319548nt_int(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(Uua,Uub),Uuc)) ).

% ATP.lambda_522
tff(fact_10724_ATP_Olambda__523,axiom,
    ! [Uu: fun_Pr3238374608701176162_int_o,Uua: fun_in1696471567231710611nt_int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_fun4545072666309863515_int_o(aTP_Lamm_cz(Uu),Uua),Uub),Uuc)
    <=> aa_Pro4592068669816069595_int_o(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(Uua,Uub),Uuc)) ) ).

% ATP.lambda_523
tff(fact_10725_ATP_Olambda__524,axiom,
    ! [Uu: fun_Pr982841511892225912nt_int,Uua: fun_in1696471567231710611nt_int,Uub: int,Uuc: int] : aa_int_int(aa_int_fun_int_int(aa_fun7461304428735181397nt_int(aTP_Lamm_da(Uu),Uua),Uub),Uuc) = aa_Pro6799258434465214121nt_int(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(Uua,Uub),Uuc)) ).

% ATP.lambda_524
tff(fact_10726_ATP_Olambda__525,axiom,
    ! [Uu: fun_o_8988054265482332260nt_int,Uua: fun_int_fun_int_o,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_fun4023131440832735197nt_int(aTP_Lamm_db(Uu),Uua),Uub),Uuc) = aa_o_P3134438092330336605nt_int(Uu,aa_int_o(aa_int_fun_int_o(Uua,Uub),Uuc)) ).

% ATP.lambda_525
tff(fact_10727_ATP_Olambda__526,axiom,
    ! [Uu: fun_o_o,Uua: fun_int_fun_int_o,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_cv(Uu),Uua),Uub),Uuc)
    <=> aa_o_o(Uu,aa_int_o(aa_int_fun_int_o(Uua,Uub),Uuc)) ) ).

% ATP.lambda_526
tff(fact_10728_ATP_Olambda__527,axiom,
    ! [Uu: fun_o_int,Uua: fun_int_fun_int_o,Uub: int,Uuc: int] : aa_int_int(aa_int_fun_int_int(aa_fun7323606162024495668nt_int(aTP_Lamm_cw(Uu),Uua),Uub),Uuc) = aa_o_int(Uu,aa_int_o(aa_int_fun_int_o(Uua,Uub),Uuc)) ).

% ATP.lambda_527
tff(fact_10729_ATP_Olambda__528,axiom,
    ! [Uu: fun_in4211310926729683942nt_int,Uua: fun_int_fun_int_int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_fun1034830265760468567nt_int(aTP_Lamm_dc(Uu),Uua),Uub),Uuc) = aa_int7462565939943104719nt_int(Uu,aa_int_int(aa_int_fun_int_int(Uua,Uub),Uuc)) ).

% ATP.lambda_528
tff(fact_10730_ATP_Olambda__529,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_fun_int_int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_fun82482788810740534_int_o(aTP_Lamm_cx(Uu),Uua),Uub),Uuc)
    <=> aa_int_o(Uu,aa_int_int(aa_int_fun_int_int(Uua,Uub),Uuc)) ) ).

% ATP.lambda_529
tff(fact_10731_ATP_Olambda__530,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_fun_int_int,Uub: int,Uuc: int] : aa_int_int(aa_int_fun_int_int(aa_fun6119123525526402106nt_int(aTP_Lamm_cy(Uu),Uua),Uub),Uuc) = aa_int_int(Uu,aa_int_int(aa_int_fun_int_int(Uua,Uub),Uuc)) ).

% ATP.lambda_530
tff(fact_10732_ATP_Olambda__531,axiom,
    ! [Uu: fun_fu3275390434952296369_nat_o,Uua: fun_na5914473681949881652_nat_o,Uub: nat,Uuc: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aa_fun824319262034445606_nat_o(aTP_Lamm_de(Uu),Uua),Uub),Uuc) = aa_fun359341089595923442_nat_o(Uu,aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(Uua,Uub),Uuc)) ).

% ATP.lambda_531
tff(fact_10733_ATP_Olambda__532,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_rb(Uu),Uua),Uub),Uuc) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uub),Uua))),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uuc))) ).

% ATP.lambda_532
tff(fact_10734_ATP_Olambda__533,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_qv(Uu),Uua),Uub),Uuc) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uub),Uua))),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uuc))) ).

% ATP.lambda_533
tff(fact_10735_ATP_Olambda__534,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_qz(Uu),Uua),Uub),Uuc) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uub)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uuc))) ).

% ATP.lambda_534
tff(fact_10736_ATP_Olambda__535,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_qx(Uu),Uua),Uub),Uuc) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uub))) ).

% ATP.lambda_535
tff(fact_10737_ATP_Olambda__536,axiom,
    ! [Uu: $o,Uua: $o,Uub: $o,Uuc: code_integer,Uud: $o] : aa_o_char(aa_Cod7842134615204088097o_char(aa_o_f2742674633914528860o_char(aa_o_f9003794812476387647o_char(aTP_Lamm_tc((Uu)),(Uua)),(Uub)),Uuc),(Uud)) = produc4188289175737317920o_char(aa_o_f2742674633914528860o_char(aa_o_f9003794812476387647o_char(aa_o_f4183413333637126194o_char(aTP_Lamm_tb((Uu)),(Uua)),(Uub)),(Uud)),code_bit_cut_integer(Uuc)) ).

% ATP.lambda_536
tff(fact_10738_ATP_Olambda__537,axiom,
    ! [Uu: nat,Uua: fun_na6724076748692969250l_real,Uub: real,Uuc: nat,Uud: real] : aa_real_real(aa_nat_fun_real_real(aa_rea1103652080130114956l_real(aa_fun2567993205388704171l_real(aTP_Lamm_ue(Uu),Uua),Uub),Uuc),Uud) = aa_real_real(minus_minus_real(aa_real_real(aa_nat_fun_real_real(Uua,Uuc),Uud)),aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_ud(Uua),Uuc),Uud)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uuc)))),aa_real_real(times_times_real(Uub),aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Uud),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uuc))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uuc)))))) ).

% ATP.lambda_537
tff(fact_10739_ATP_Olambda__538,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex,Uuc: complex,Uud: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_pi(Uu),Uua),Uub),Uuc),Uud) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Uud)),Uua)),one_one_complex)),Uud)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uub),Uud))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(plus_plus_complex(Uub),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_538
tff(fact_10740_ATP_Olambda__539,axiom,
    ! [Uu: nat,Uua: real,Uub: real,Uuc: real,Uud: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_pk(Uu),Uua),Uub),Uuc),Uud) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,Uud)),Uua)),one_one_real)),Uud)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uud))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(plus_plus_real(Uub),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_539
tff(fact_10741_ATP_Olambda__540,axiom,
    ! [Uu: nat,Uua: rat,Uub: rat,Uuc: rat,Uud: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_pj(Uu),Uua),Uub),Uuc),Uud) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,Uud)),Uua)),one_one_rat)),Uud)),aa_nat_rat(power_power_rat(Uub),Uud))),aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(Uub),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_540
tff(fact_10742_ATP_Olambda__541,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex,Uuc: complex,Uud: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_on(Uu),Uua),Uub),Uuc),Uud) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Uu)),Uua)),Uud)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uub),Uud))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,Uuc),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_541
tff(fact_10743_ATP_Olambda__542,axiom,
    ! [Uu: nat,Uua: real,Uub: real,Uuc: real,Uud: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_or(Uu),Uua),Uub),Uuc),Uud) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,Uu)),Uua)),Uud)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uud))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uuc),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_542
tff(fact_10744_ATP_Olambda__543,axiom,
    ! [Uu: nat,Uua: rat,Uub: rat,Uuc: rat,Uud: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_op(Uu),Uua),Uub),Uuc),Uud) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,Uu)),Uua)),Uud)),aa_nat_rat(power_power_rat(Uub),Uud))),aa_nat_rat(power_power_rat(Uuc),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_543
tff(fact_10745_ATP_Olambda__544,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex,Uuc: complex,Uud: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_oo(Uu),Uua),Uub),Uuc),Uud) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(aa_com8181001469867462503omplex(gbinomial_complex,aa_complex_complex(uminus1482373934393186551omplex,Uua)),Uud)),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(uminus1482373934393186551omplex,Uub)),Uud))),aa_nat_complex(aa_com8181001469867462503omplex(power_power_complex,aa_complex_complex(plus_plus_complex(Uub),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_544
tff(fact_10746_ATP_Olambda__545,axiom,
    ! [Uu: nat,Uua: real,Uub: real,Uuc: real,Uud: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_os(Uu),Uua),Uub),Uuc),Uud) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(uminus_uminus_real,Uua)),Uud)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,Uub)),Uud))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(plus_plus_real(Uub),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_545
tff(fact_10747_ATP_Olambda__546,axiom,
    ! [Uu: nat,Uua: rat,Uub: rat,Uuc: rat,Uud: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_oq(Uu),Uua),Uub),Uuc),Uud) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(aa_rat_rat(uminus_uminus_rat,Uua)),Uud)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,Uub)),Uud))),aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(Uub),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_546
tff(fact_10748_ATP_Olambda__547,axiom,
    ! [Uu: $o,Uua: $o,Uub: $o,Uuc: $o,Uud: code_integer,Uue: $o] : aa_o_char(aa_Cod7842134615204088097o_char(aa_o_f2742674633914528860o_char(aa_o_f9003794812476387647o_char(aa_o_f4183413333637126194o_char(aTP_Lamm_tb((Uu)),(Uua)),(Uub)),(Uuc)),Uud),(Uue)) = produc4188289175737317920o_char(aa_o_f2742674633914528860o_char(aa_o_f9003794812476387647o_char(aa_o_f4183413333637126194o_char(aa_o_f2932186792858379797o_char(aTP_Lamm_ta((Uu)),(Uua)),(Uub)),(Uuc)),(Uue)),code_bit_cut_integer(Uud)) ).

% ATP.lambda_547
tff(fact_10749_ATP_Olambda__548,axiom,
    ! [Uu: $o,Uua: $o,Uub: $o,Uuc: $o,Uud: $o,Uue: code_integer,Uuf: $o] : aa_o_char(aa_Cod7842134615204088097o_char(aa_o_f2742674633914528860o_char(aa_o_f9003794812476387647o_char(aa_o_f4183413333637126194o_char(aa_o_f2932186792858379797o_char(aTP_Lamm_ta((Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),Uue),(Uuf)) = produc4188289175737317920o_char(aa_o_f2742674633914528860o_char(aa_o_f9003794812476387647o_char(aa_o_f4183413333637126194o_char(aa_o_f2932186792858379797o_char(aa_o_f1143307120959153672o_char(aTP_Lamm_sz((Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),(Uuf)),code_bit_cut_integer(Uue)) ).

% ATP.lambda_548
tff(fact_10750_ATP_Olambda__549,axiom,
    ! [Uu: nat,Uua: list_VEBT_VEBT,Uub: vEBT_VEBT,Uuc: nat,Uud: nat,Uue: nat,Uuf: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_vp(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uue),Uuf)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuf),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uu))
        & ! [I: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uud))
           => ( ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Uua),I)),X_12)
            <=> aa_nat_o(vEBT_V8194947554948674370ptions(Uub),I) ) )
        & $ite(
            Uue = Uuf,
            ! [X3: vEBT_VEBT] :
              ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(Uua))
             => ~ ? [X8: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X8) ),
            ( aa_nat_o(vEBT_V5917875025757280293ildren(Uuc,Uua),Uuf)
            & ! [X3: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uu))
               => ( aa_nat_o(vEBT_V5917875025757280293ildren(Uuc,Uua),X3)
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uue),X3)
                    & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Uuf) ) ) ) ) ) ) ) ).

% ATP.lambda_549
tff(fact_10751_ATP_Olambda__550,axiom,
    ! [Uu: $o,Uua: $o,Uub: $o,Uuc: $o,Uud: $o,Uue: $o,Uuf: $o,Uug: code_integer] : aa_Cod7842134615204088097o_char(aa_o_f2742674633914528860o_char(aa_o_f9003794812476387647o_char(aa_o_f4183413333637126194o_char(aa_o_f2932186792858379797o_char(aa_o_f1143307120959153672o_char(aa_o_f3038258172971468011o_char(aTP_Lamm_sy((Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),(Uue)),(Uuf)),Uug) = char2((Uu),(Uua),(Uub),(Uuc),(Uud),(Uue),(Uuf)) ).

% ATP.lambda_550
tff(fact_10752_ATP_Olambda__551,axiom,
    ! [Uu: $o,Uua: $o,Uub: $o,Uuc: $o,Uud: $o,Uue: $o,Uuf: code_integer,Uug: $o] : aa_o_char(aa_Cod7842134615204088097o_char(aa_o_f2742674633914528860o_char(aa_o_f9003794812476387647o_char(aa_o_f4183413333637126194o_char(aa_o_f2932186792858379797o_char(aa_o_f1143307120959153672o_char(aTP_Lamm_sz((Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),(Uue)),Uuf),(Uug)) = produc4188289175737317920o_char(aa_o_f2742674633914528860o_char(aa_o_f9003794812476387647o_char(aa_o_f4183413333637126194o_char(aa_o_f2932186792858379797o_char(aa_o_f1143307120959153672o_char(aa_o_f3038258172971468011o_char(aTP_Lamm_sy((Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),(Uue)),(Uug)),code_bit_cut_integer(Uuf)) ).

% ATP.lambda_551
tff(fact_10753_ATP_Olambda__552,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBT] : aa_VEB5832953974749077948T_VEBT(aTP_Lamm_bj(Uu),Uua) = Uu ).

% ATP.lambda_552
tff(fact_10754_ATP_Olambda__553,axiom,
    ! [Uu: vEBT_VEBT,Uua: $o] : aa_o_VEBT_VEBT(aTP_Lamm_bk(Uu),(Uua)) = Uu ).

% ATP.lambda_553
tff(fact_10755_ATP_Olambda__554,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat] : aa_nat_VEBT_VEBT(aTP_Lamm_bm(Uu),Uua) = Uu ).

% ATP.lambda_554
tff(fact_10756_ATP_Olambda__555,axiom,
    ! [Uu: vEBT_VEBT,Uua: int] : aa_int_VEBT_VEBT(aTP_Lamm_bn(Uu),Uua) = Uu ).

% ATP.lambda_555
tff(fact_10757_ATP_Olambda__556,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_ew,Uu),Uua) = Uu ).

% ATP.lambda_556
tff(fact_10758_ATP_Olambda__557,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ex,Uu),Uua) = Uu ).

% ATP.lambda_557
tff(fact_10759_ATP_Olambda__558,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_bl,Uu),Uua) = Uu ).

% ATP.lambda_558
tff(fact_10760_ATP_Olambda__559,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_dm,Uu),Uua)
    <=> $true ) ).

% ATP.lambda_559
tff(fact_10761_ATP_Olambda__560,axiom,
    ! [Uu: vEBT_VEBT] : aa_VEB5832953974749077948T_VEBT(aTP_Lamm_ah,Uu) = Uu ).

% ATP.lambda_560
tff(fact_10762_ATP_Olambda__561,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_ao,Uu) = Uu ).

% ATP.lambda_561
tff(fact_10763_ATP_Olambda__562,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_ap,Uu) = Uu ).

% ATP.lambda_562
tff(fact_10764_ATP_Olambda__563,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_aq,Uu) = Uu ).

% ATP.lambda_563
tff(fact_10765_ATP_Olambda__564,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_ai,Uu) = Uu ).

% ATP.lambda_564
tff(fact_10766_ATP_Olambda__565,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_ar,Uu) = Uu ).

% ATP.lambda_565
tff(fact_10767_ATP_Olambda__566,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_as,Uu) = zero_zero_complex ).

% ATP.lambda_566
tff(fact_10768_ATP_Olambda__567,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_at,Uu) = zero_zero_real ).

% ATP.lambda_567
tff(fact_10769_ATP_Olambda__568,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_au,Uu) = zero_zero_rat ).

% ATP.lambda_568
tff(fact_10770_ATP_Olambda__569,axiom,
    ! [Uu: nat] : aa_nat_complex(aTP_Lamm_eh,Uu) = zero_zero_complex ).

% ATP.lambda_569
tff(fact_10771_ATP_Olambda__570,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_ei,Uu) = zero_zero_real ).

% ATP.lambda_570
tff(fact_10772_ATP_Olambda__571,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_av,Uu) = zero_zero_nat ).

% ATP.lambda_571
tff(fact_10773_ATP_Olambda__572,axiom,
    ! [Uu: nat] : aa_nat_int(aTP_Lamm_ej,Uu) = zero_zero_int ).

% ATP.lambda_572
tff(fact_10774_ATP_Olambda__573,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_aw,Uu) = zero_zero_int ).

% ATP.lambda_573
tff(fact_10775_ATP_Olambda__574,axiom,
    ! [Uu: real] :
      ( aa_real_o(aTP_Lamm_hp,Uu)
    <=> $false ) ).

% ATP.lambda_574
tff(fact_10776_ATP_Olambda__575,axiom,
    ! [Uu: nat] :
      ( aa_nat_o(aTP_Lamm_rr,Uu)
    <=> $false ) ).

% ATP.lambda_575
tff(fact_10777_ATP_Olambda__576,axiom,
    ! [Uu: nat] :
      ( aa_nat_o(aTP_Lamm_rq,Uu)
    <=> $true ) ).

% ATP.lambda_576

% Helper facts (14)
tff(help_fequal_2_1_fequal_001t__Int__Oint_T,axiom,
    ! [X: int,Y2: int] :
      ( ( X != Y2 )
      | aa_int_o(aa_int_fun_int_o(fequal_int,X),Y2) ) ).

tff(help_fequal_1_1_fequal_001t__Int__Oint_T,axiom,
    ! [X: int,Y2: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(fequal_int,X),Y2)
      | ( X = Y2 ) ) ).

tff(help_fChoice_1_1_fChoice_001t__Int__Oint_T,axiom,
    ! [P: fun_int_o] :
      ( aa_int_o(P,fChoice_int(P))
      = ( ? [X8: int] : aa_int_o(P,X8) ) ) ).

tff(help_fChoice_1_1_fChoice_001t__Nat__Onat_T,axiom,
    ! [P: fun_nat_o] :
      ( aa_nat_o(P,fChoice_nat(P))
      = ( ? [X8: nat] : aa_nat_o(P,X8) ) ) ).

tff(help_fequal_2_1_fequal_001t__Real__Oreal_T,axiom,
    ! [X: real,Y2: real] :
      ( ( X != Y2 )
      | aa_real_o(aa_real_fun_real_o(fequal_real,X),Y2) ) ).

tff(help_fequal_1_1_fequal_001t__Real__Oreal_T,axiom,
    ! [X: real,Y2: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(fequal_real,X),Y2)
      | ( X = Y2 ) ) ).

tff(help_fChoice_1_1_fChoice_001t__Real__Oreal_T,axiom,
    ! [P: fun_real_o] :
      ( aa_real_o(P,fChoice_real(P))
      = ( ? [X8: real] : aa_real_o(P,X8) ) ) ).

tff(help_fChoice_1_1_fChoice_001t__Complex__Ocomplex_T,axiom,
    ! [P: fun_complex_o] :
      ( aa_complex_o(P,fChoice_complex(P))
      = ( ? [X8: complex] : aa_complex_o(P,X8) ) ) ).

tff(help_fChoice_1_1_fChoice_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [P: fun_set_nat_o] :
      ( aa_set_nat_o(P,fChoice_set_nat(P))
      = ( ? [X8: set_nat] : aa_set_nat_o(P,X8) ) ) ).

tff(help_fChoice_1_1_fChoice_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [P: fun_Pr3238374608701176162_int_o] :
      ( aa_Pro4592068669816069595_int_o(P,fChoic3800441565783186701nt_int(P))
      = ( ? [X8: product_prod_int_int] : aa_Pro4592068669816069595_int_o(P,X8) ) ) ).

tff(help_fChoice_1_1_fChoice_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [P: fun_Pr4796943543331071514_nat_o] :
      ( aa_Pro6150637604445964947_nat_o(P,fChoic6978938873391328853at_nat(P))
      = ( ? [X8: product_prod_nat_nat] : aa_Pro6150637604445964947_nat_o(P,X8) ) ) ).

tff(help_fChoice_1_1_fChoice_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J_T,axiom,
    ! [P: fun_Pr5555124106300582480_num_o] :
      ( aa_Pro6908818167415475913_num_o(P,fChoic7687182810340166559at_num(P))
      = ( ? [X8: product_prod_nat_num] : aa_Pro6908818167415475913_num_o(P,X8) ) ) ).

tff(help_fChoice_1_1_fChoice_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J_T,axiom,
    ! [P: fun_Pr6331313853612757382_num_o] :
      ( aa_Pro7685007914727650815_num_o(P,fChoic5817513213647635945um_num(P))
      = ( ? [X8: product_prod_num_num] : aa_Pro7685007914727650815_num_o(P,X8) ) ) ).

tff(help_fChoice_1_1_fChoice_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
    ! [P: fun_Pr3290071492984375169er_o_o] :
      ( aa_Pro5286387174248033258er_o_o(P,fChoic166683996008689692eger_o(P))
      = ( ? [X8: produc6271795597528267376eger_o] : aa_Pro5286387174248033258er_o_o(P,X8) ) ) ).

% Conjectures (1)
tff(conj_0,conjecture,
    vEBT_VEBT_elim_dead(summarya,extend5688581933313929465d_enat) = summarya ).

%------------------------------------------------------------------------------